| 1 | # -*- coding: utf-8 -*- |
| 2 | """ |
| 3 | Finite State Machines, Automata, Transducers |
| 4 | |
| 5 | This module adds support for finite state machines, automata and |
| 6 | transducers. See class :class:`FiniteStateMachine` and the examples |
| 7 | below for details creating one. |
| 8 | |
| 9 | Examples |
| 10 | ======== |
| 11 | |
| 12 | |
| 13 | A simple finite state machine |
| 14 | ----------------------------- |
| 15 | |
| 16 | We can easily create a finite state machine by |
| 17 | |
| 18 | :: |
| 19 | |
| 20 | sage: fsm = FiniteStateMachine() |
| 21 | sage: fsm |
| 22 | finite state machine with 0 states |
| 23 | |
| 24 | By default this is the empty finite state machine, so not very |
| 25 | interesting. Let's create some states and transitions:: |
| 26 | |
| 27 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 28 | sage: day = FSMState('day') |
| 29 | sage: night = FSMState('night') |
| 30 | sage: sunrise = FSMTransition(night, day) |
| 31 | sage: sunset = FSMTransition(day, night) |
| 32 | |
| 33 | And now let's add those states and transitions to our finite state machine:: |
| 34 | |
| 35 | sage: fsm.add_transition(sunrise) |
| 36 | Transition from 'night' to 'day': -|- |
| 37 | sage: fsm.add_transition(sunset) |
| 38 | Transition from 'day' to 'night': -|- |
| 39 | |
| 40 | Note that the states are added automatically, since they are present |
| 41 | in the transitions. We could add the states manually by |
| 42 | |
| 43 | :: |
| 44 | |
| 45 | sage: fsm.add_state(day) |
| 46 | 'day' |
| 47 | sage: fsm.add_state(night) |
| 48 | 'night' |
| 49 | |
| 50 | Anyhow, we got the following finite state machine:: |
| 51 | |
| 52 | sage: fsm |
| 53 | finite state machine with 2 states |
| 54 | |
| 55 | We can also visualize it as a graph by |
| 56 | |
| 57 | :: |
| 58 | |
| 59 | sage: fsm.graph() |
| 60 | Digraph on 2 vertices |
| 61 | |
| 62 | Alternatively, we could have created the finite state machine above |
| 63 | simply by |
| 64 | |
| 65 | :: |
| 66 | |
| 67 | sage: FiniteStateMachine([('night', 'day'), ('day', 'night')]) |
| 68 | finite state machine with 2 states |
| 69 | |
| 70 | or by |
| 71 | |
| 72 | :: |
| 73 | |
| 74 | sage: fsm = FiniteStateMachine() |
| 75 | sage: day = fsm.add_state('day') |
| 76 | sage: night = fsm.add_state('night') |
| 77 | sage: sunrise = fsm.add_transition(night, day) |
| 78 | sage: sunset = fsm.add_transition(day, night) |
| 79 | sage: fsm |
| 80 | finite state machine with 2 states |
| 81 | |
| 82 | A simple Automaton (recognizing NAFs) |
| 83 | --------------------------------------- |
| 84 | |
| 85 | We want to build an automaton which recognizes non-adjacent forms |
| 86 | (NAFs), i.e., sequences which have no adjacent non-zeros. |
| 87 | We use `0`, `1`, and `-1` as digits:: |
| 88 | |
| 89 | sage: NAF = Automaton( |
| 90 | ....: {'A': [('A', 0), ('B', 1), ('B', -1)], 'B': [('A', 0)]}) |
| 91 | sage: NAF.state('A').is_initial = True |
| 92 | sage: NAF.state('A').is_final = True |
| 93 | sage: NAF.state('B').is_final = True |
| 94 | sage: NAF |
| 95 | finite state machine with 2 states |
| 96 | |
| 97 | Of course, we could have specified the initial and final states |
| 98 | directly in the definition of ``NAF`` by ``initial_states=['A']`` and |
| 99 | ``final_states=['A', 'B']``. |
| 100 | |
| 101 | So let's test the automaton with some input:: |
| 102 | |
| 103 | sage: NAF([0])[0] |
| 104 | True |
| 105 | sage: NAF([0, 1])[0] |
| 106 | True |
| 107 | sage: NAF([1, -1])[0] |
| 108 | False |
| 109 | sage: NAF([0, -1, 0, 1])[0] |
| 110 | True |
| 111 | sage: NAF([0, -1, -1, -1, 0])[0] |
| 112 | False |
| 113 | sage: NAF([-1, 0, 0, 1, 1])[0] |
| 114 | False |
| 115 | |
| 116 | Alternatively, we could call that by |
| 117 | |
| 118 | :: |
| 119 | |
| 120 | sage: NAF.process([-1, 0, 0, 1, 1])[0] |
| 121 | False |
| 122 | |
| 123 | A simple transducer (binary inverter) |
| 124 | ------------------------------------- |
| 125 | |
| 126 | Let's build a simple transducer, which rewrites a binary word by |
| 127 | iverting each bit:: |
| 128 | |
| 129 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 130 | ....: initial_states=['A'], final_states=['A']) |
| 131 | |
| 132 | We can look at the states and transitions:: |
| 133 | |
| 134 | sage: inverter.states() |
| 135 | ['A'] |
| 136 | sage: for t in inverter.transitions(): |
| 137 | ....: print t |
| 138 | Transition from 'A' to 'A': 0|1 |
| 139 | Transition from 'A' to 'A': 1|0 |
| 140 | |
| 141 | Now we apply a word to it and see what the transducer does:: |
| 142 | |
| 143 | sage: inverter([0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1]) |
| 144 | (True, 'A', [1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0]) |
| 145 | |
| 146 | ``True`` means, that we landed in a final state, that state is labeled |
| 147 | ``'A'``, and we also got an output. |
| 148 | |
| 149 | |
| 150 | A transducer which performs division by `3` in binary |
| 151 | ----------------------------------------------------- |
| 152 | |
| 153 | Now we build a transducer, which divides a binary number by 3. |
| 154 | The labels of the states are the remainder of the division. |
| 155 | The transition function is |
| 156 | |
| 157 | :: |
| 158 | |
| 159 | sage: def f(state_from, read): |
| 160 | ....: if state_from + read <= 1: |
| 161 | ....: state_to = 2*state_from + read |
| 162 | ....: write = 0 |
| 163 | ....: else: |
| 164 | ....: state_to = 2*state_from + read - 3 |
| 165 | ....: write = 1 |
| 166 | ....: return (state_to, write) |
| 167 | |
| 168 | We get the transducer with |
| 169 | |
| 170 | :: |
| 171 | |
| 172 | sage: D = Transducer(f, initial_states=[0], final_states=[0], |
| 173 | ....: input_alphabet=[0, 1]) |
| 174 | |
| 175 | Now we want to divide 13 by 3:: |
| 176 | |
| 177 | sage: D([1, 1, 0, 1]) |
| 178 | (False, 1, [0, 1, 0, 0]) |
| 179 | |
| 180 | So we have 13 : 3 = 4 and the reminder is 1. ``False`` means 13 is not |
| 181 | divisible by 3. |
| 182 | |
| 183 | |
| 184 | Using the hook-functions |
| 185 | ------------------------ |
| 186 | |
| 187 | Let's use the previous example "divison by `3`" to demonstrate the |
| 188 | optional state and transition parameters ``hook``. |
| 189 | |
| 190 | First, we define, what those functions should do. In our case, this is |
| 191 | just saying in which state we are and which transition we take |
| 192 | |
| 193 | :: |
| 194 | |
| 195 | sage: def state_hook(state, process): |
| 196 | ....: print "We are now in State %s." % (state.label(),) |
| 197 | sage: from sage.combinat.finite_state_machine import FSMWordSymbol |
| 198 | sage: def transition_hook(transition, process): |
| 199 | ....: print ("Currently we go from %s to %s, " |
| 200 | ....: "reading %s and writing %s." % ( |
| 201 | ....: transition.from_state, transition.to_state, |
| 202 | ....: FSMWordSymbol(transition.word_in), |
| 203 | ....: FSMWordSymbol(transition.word_out))) |
| 204 | |
| 205 | Now, let's add these hook-functions to the existing transducer:: |
| 206 | |
| 207 | sage: for s in D.iter_states(): |
| 208 | ....: s.hook = state_hook |
| 209 | sage: for t in D.iter_transitions(): |
| 210 | ....: t.hook = transition_hook |
| 211 | |
| 212 | Rerunning the process again now gives the following output:: |
| 213 | |
| 214 | sage: D.process([1, 1, 0, 1]) |
| 215 | We are now in State 0. |
| 216 | Currently we go from 0 to 1, reading 1 and writing 0. |
| 217 | We are now in State 1. |
| 218 | Currently we go from 1 to 0, reading 1 and writing 1. |
| 219 | We are now in State 0. |
| 220 | Currently we go from 0 to 0, reading 0 and writing 0. |
| 221 | We are now in State 0. |
| 222 | Currently we go from 0 to 1, reading 1 and writing 0. |
| 223 | We are now in State 1. |
| 224 | (False, 1, [0, 1, 0, 0]) |
| 225 | |
| 226 | The example above just explains the basic idea of using |
| 227 | hook-functions. In the following, we will use those hooks more seriously. |
| 228 | |
| 229 | |
| 230 | Detecting sequences with same number of `0` and `1` |
| 231 | --------------------------------------------------- |
| 232 | |
| 233 | Suppose we have a binary input and want to accept all sequences with |
| 234 | the same number of `0` and `1`. This cannot be done with a finite |
| 235 | automaton. Anyhow, we can make usage of the hook functions to extend |
| 236 | our finite automaton by a counter:: |
| 237 | |
| 238 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 239 | sage: C = Automaton() |
| 240 | sage: def update_counter(state, process): |
| 241 | ....: l = process.read_letter() |
| 242 | ....: process.fsm.counter += 1 if l == 1 else -1 |
| 243 | ....: if process.fsm.counter > 0: |
| 244 | ....: next_state = 'positive' |
| 245 | ....: elif process.fsm.counter < 0: |
| 246 | ....: next_state = 'negative' |
| 247 | ....: else: |
| 248 | ....: next_state = 'zero' |
| 249 | ....: return FSMTransition(state, process.fsm.state(next_state), |
| 250 | ....: l, process.fsm.counter) |
| 251 | sage: C.add_state(FSMState('zero', hook=update_counter, |
| 252 | ....: is_initial=True, is_final=True)) |
| 253 | 'zero' |
| 254 | sage: C.add_state(FSMState('positive', hook=update_counter)) |
| 255 | 'positive' |
| 256 | sage: C.add_state(FSMState('negative', hook=update_counter)) |
| 257 | 'negative' |
| 258 | |
| 259 | Now, let's input some sequence:: |
| 260 | |
| 261 | sage: C.counter = 0; C([1, 1, 1, 1, 0, 0]) |
| 262 | (False, 'positive', [1, 2, 3, 4, 3, 2]) |
| 263 | |
| 264 | The result is False, since there are four `1` but only two `0`. We |
| 265 | land in the state ``positive`` and we can also see the values of the |
| 266 | counter in each step. |
| 267 | |
| 268 | Let's try some other examples:: |
| 269 | |
| 270 | sage: C.counter = 0; C([1, 1, 0, 0]) |
| 271 | (True, 'zero', [1, 2, 1, 0]) |
| 272 | sage: C.counter = 0; C([0, 1, 0, 0]) |
| 273 | (False, 'negative', [-1, 0, -1, -2]) |
| 274 | |
| 275 | |
| 276 | AUTHORS: |
| 277 | |
| 278 | - Daniel Krenn (2012-03-27): initial version |
| 279 | - Clemens Heuberger (2012-04-05): initial version |
| 280 | - Sara Kropf (2012-04-17): initial version |
| 281 | - Clemens Heuberger (2013-08-21): release candidate for Sage patch |
| 282 | - Daniel Krenn (2013-08-21): release candidate for Sage patch |
| 283 | - Sara Kropf (2013-08-21): release candidate for Sage patch |
| 284 | - Clemens Heuberger (2013-09-02): documentation improved |
| 285 | - Daniel Krenn (2013-09-13): comments from trac worked in |
| 286 | - Clemens Heuberger (2013-11-03): output (labels) of determinisation, |
| 287 | product, composition, etc. changed (for consistency), |
| 288 | representation of state changed, documentation improved |
| 289 | - Daniel Krenn (2013-11-04): whitespaces in documentation corrected |
| 290 | - Clemens Heuberger (2013-11-04): full_group_by added |
| 291 | - Daniel Krenn (2013-11-04): next release candidate for Sage patch |
| 292 | - Sara Kropf (2013-11-08): fix for adjacency matrix |
| 293 | - Clemens Heuberger (2013-11-11): fix for prepone_output |
| 294 | - Daniel Krenn (2013-11-11): comments from trac 15078 included: |
| 295 | docstring of FiniteStateMachine rewritten, Automaton and Transducer |
| 296 | inherited from FiniteStateMachine |
| 297 | |
| 298 | |
| 299 | ACKNOWLEDGEMENT: |
| 300 | |
| 301 | - Daniel Krenn, Clemens Heuberger and Sara Kropf are supported by the |
| 302 | Austrian Science Fund (FWF): P 24644-N26. |
| 303 | |
| 304 | """ |
| 305 | |
| 306 | #***************************************************************************** |
| 307 | # Copyright (C) 2012, 2013 Daniel Krenn <math+sage@danielkrenn.at> |
| 308 | # 2012, 2013 Clemens Heuberger <clemens.heuberger@aau.at> |
| 309 | # 2012, 2013 Sara Kropf <sara.kropf@aau.at> |
| 310 | # |
| 311 | # Distributed under the terms of the GNU General Public License (GPL) |
| 312 | # as published by the Free Software Foundation; either version 2 of |
| 313 | # the License, or (at your option) any later version. |
| 314 | # http://www.gnu.org/licenses/ |
| 315 | #***************************************************************************** |
| 316 | |
| 317 | from sage.structure.sage_object import SageObject |
| 318 | from sage.graphs.digraph import DiGraph |
| 319 | from sage.matrix.constructor import matrix |
| 320 | from sage.rings.integer_ring import ZZ |
| 321 | from sage.calculus.var import var |
| 322 | from sage.misc.latex import latex |
| 323 | from sage.functions.trig import cos, sin, atan2 |
| 324 | from sage.symbolic.constants import pi |
| 325 | |
| 326 | from copy import copy |
| 327 | from copy import deepcopy |
| 328 | |
| 329 | import itertools |
| 330 | from collections import defaultdict |
| 331 | |
| 332 | |
| 333 | def full_group_by(l, key=lambda x: x): |
| 334 | """ |
| 335 | Group iterable ``l`` by values of ``key``. |
| 336 | |
| 337 | INPUT: |
| 338 | |
| 339 | - iterable ``l`` |
| 340 | - key function ``key`` |
| 341 | |
| 342 | OUTPUT: |
| 343 | |
| 344 | A list of pairs ``(k, elements)`` such that ``key(e)=k`` for all |
| 345 | ``e`` in ``elements``. |
| 346 | |
| 347 | This is similar to ``itertools.groupby`` except that lists are |
| 348 | returned instead of iterables and no prior sorting is required. |
| 349 | |
| 350 | We do not require |
| 351 | |
| 352 | - that the keys are sortable (in contrast to the |
| 353 | approach via ``sorted`` and ``itertools.groupby``) and |
| 354 | - that the keys are hashable (in contrast to the |
| 355 | implementation proposed in `<http://stackoverflow.com/a/15250161>`_). |
| 356 | |
| 357 | However, it is required |
| 358 | |
| 359 | - that distinct keys have distinct ``str``-representations. |
| 360 | |
| 361 | The implementation is inspired by |
| 362 | `<http://stackoverflow.com/a/15250161>`_, but non-hashable keys are |
| 363 | allowed. |
| 364 | |
| 365 | EXAMPLES:: |
| 366 | |
| 367 | sage: from sage.combinat.finite_state_machine import full_group_by |
| 368 | sage: t = [2/x, 1/x, 2/x] |
| 369 | sage: r = full_group_by([0,1,2], key=lambda i:t[i]) |
| 370 | sage: sorted(r, key=lambda p:p[1]) |
| 371 | [(2/x, [0, 2]), (1/x, [1])] |
| 372 | sage: from itertools import groupby |
| 373 | sage: for k, elements in groupby(sorted([0,1,2], |
| 374 | ....: key=lambda i:t[i]), |
| 375 | ....: key=lambda i:t[i]): |
| 376 | ....: print k, list(elements) |
| 377 | 2/x [0] |
| 378 | 1/x [1] |
| 379 | 2/x [2] |
| 380 | |
| 381 | Note that the behavior is different from ``itertools.groupby`` |
| 382 | because neither `1/x<2/x` nor `2/x<1/x` does hold. |
| 383 | |
| 384 | Here, the result ``r`` has been sorted in order to guarantee a |
| 385 | consistent order for the doctest suite. |
| 386 | """ |
| 387 | elements = defaultdict(list) |
| 388 | original_keys = {} |
| 389 | for item in l: |
| 390 | k = key(item) |
| 391 | s = str(k) |
| 392 | if s in original_keys: |
| 393 | if original_keys[s]!=k: |
| 394 | raise ValueError("Two distinct elements with representation " |
| 395 | "%s " % s) |
| 396 | else: |
| 397 | original_keys[s]=k |
| 398 | elements[s].append(item) |
| 399 | return [(original_keys[s], values ) for (s, values) in elements.items()] |
| 400 | |
| 401 | #***************************************************************************** |
| 402 | |
| 403 | FSMEmptyWordSymbol = '-' |
| 404 | |
| 405 | def FSMLetterSymbol(letter): |
| 406 | """ |
| 407 | Returns a string associated to the input letter. |
| 408 | |
| 409 | INPUT: |
| 410 | |
| 411 | - ``letter`` -- the input letter or ``None`` (representing the |
| 412 | empty word). |
| 413 | |
| 414 | OUTPUT: |
| 415 | |
| 416 | If ``letter`` is ``None`` the symbol for the empty word |
| 417 | ``FSMEmptyWordSymbol`` is returned, otherwise the string |
| 418 | associated to the letter. |
| 419 | |
| 420 | EXAMPLES:: |
| 421 | |
| 422 | sage: from sage.combinat.finite_state_machine import FSMLetterSymbol |
| 423 | sage: FSMLetterSymbol(0) |
| 424 | '0' |
| 425 | sage: FSMLetterSymbol(None) |
| 426 | '-' |
| 427 | """ |
| 428 | return FSMEmptyWordSymbol if letter is None else repr(letter) |
| 429 | |
| 430 | |
| 431 | def FSMWordSymbol(word): |
| 432 | """ |
| 433 | Returns a string of ``word``. It may returns the symbol of the |
| 434 | empty word ``FSMEmptyWordSymbol``. |
| 435 | |
| 436 | INPUT: |
| 437 | |
| 438 | - ``word`` -- the input word. |
| 439 | |
| 440 | OUTPUT: |
| 441 | |
| 442 | A string of ``word``. |
| 443 | |
| 444 | EXAMPLES:: |
| 445 | |
| 446 | sage: from sage.combinat.finite_state_machine import FSMWordSymbol |
| 447 | sage: FSMWordSymbol([0, 1, 1]) |
| 448 | '0,1,1' |
| 449 | """ |
| 450 | if not isinstance(word, list): |
| 451 | return FSMLetterSymbol(word) |
| 452 | if len(word) == 0: |
| 453 | return FSMEmptyWordSymbol |
| 454 | s = '' |
| 455 | for letter in word: |
| 456 | s += (',' if len(s) > 0 else '') + FSMLetterSymbol(letter) |
| 457 | return s |
| 458 | |
| 459 | |
| 460 | #***************************************************************************** |
| 461 | |
| 462 | |
| 463 | def is_FSMState(S): |
| 464 | """ |
| 465 | Tests whether or not ``S`` inherits from :class:`FSMState`. |
| 466 | |
| 467 | TESTS:: |
| 468 | |
| 469 | sage: from sage.combinat.finite_state_machine import is_FSMState, FSMState |
| 470 | sage: is_FSMState(FSMState('A')) |
| 471 | True |
| 472 | """ |
| 473 | return isinstance(S, FSMState) |
| 474 | |
| 475 | |
| 476 | class FSMState(SageObject): |
| 477 | """ |
| 478 | Class for a state of a finite state machine. |
| 479 | |
| 480 | INPUT: |
| 481 | |
| 482 | - ``label`` -- the label of the state. |
| 483 | |
| 484 | - ``word_out`` -- (default: ``None``) a word that is written when |
| 485 | the state is reached. |
| 486 | |
| 487 | - ``is_initial`` -- (default: ``False``) |
| 488 | |
| 489 | - ``is_final`` -- (default: ``False``) |
| 490 | |
| 491 | - ``hook`` -- (default: ``None``) A function which is called when |
| 492 | the state is reached during processing input. |
| 493 | |
| 494 | OUTPUT: |
| 495 | |
| 496 | Returns a state of a finite state machine. |
| 497 | |
| 498 | EXAMPLES:: |
| 499 | |
| 500 | sage: from sage.combinat.finite_state_machine import FSMState |
| 501 | sage: A = FSMState('state 1', word_out=0, is_initial=True) |
| 502 | sage: A |
| 503 | 'state 1' |
| 504 | sage: A.label() |
| 505 | 'state 1' |
| 506 | sage: B = FSMState('state 2') |
| 507 | sage: A == B |
| 508 | False |
| 509 | |
| 510 | """ |
| 511 | def __init__(self, label, word_out=None, |
| 512 | is_initial=False, is_final=False, |
| 513 | hook=None): |
| 514 | """ |
| 515 | See :class:`FSMState` for more information. |
| 516 | |
| 517 | EXAMPLES:: |
| 518 | |
| 519 | sage: from sage.combinat.finite_state_machine import FSMState |
| 520 | sage: FSMState('final', is_final=True) |
| 521 | 'final' |
| 522 | """ |
| 523 | if label is None or label == "": |
| 524 | raise ValueError, "You have to specify a label for the state." |
| 525 | self._label_ = label |
| 526 | |
| 527 | if isinstance(word_out, list): |
| 528 | self.word_out = word_out |
| 529 | elif word_out is not None: |
| 530 | self.word_out = [word_out] |
| 531 | else: |
| 532 | self.word_out = [] |
| 533 | |
| 534 | self.is_initial = is_initial |
| 535 | self.is_final = is_final |
| 536 | if hook is not None: |
| 537 | if hasattr(hook, '__call__'): |
| 538 | self.hook = hook |
| 539 | else: |
| 540 | raise TypeError, 'Wrong argument for hook.' |
| 541 | |
| 542 | |
| 543 | def label(self): |
| 544 | """ |
| 545 | Returns the label of the state. |
| 546 | |
| 547 | INPUT: |
| 548 | |
| 549 | Nothing. |
| 550 | |
| 551 | OUTPUT: |
| 552 | |
| 553 | The label of the state. |
| 554 | |
| 555 | EXAMPLES:: |
| 556 | |
| 557 | sage: from sage.combinat.finite_state_machine import FSMState |
| 558 | sage: A = FSMState('state') |
| 559 | sage: A.label() |
| 560 | 'state' |
| 561 | """ |
| 562 | return self._label_ |
| 563 | |
| 564 | |
| 565 | def __copy__(self): |
| 566 | """ |
| 567 | Returns a (shallow) copy of the state. |
| 568 | |
| 569 | INPUT: |
| 570 | |
| 571 | Nothing. |
| 572 | |
| 573 | OUTPUT: |
| 574 | |
| 575 | A new state. |
| 576 | |
| 577 | EXAMPLES:: |
| 578 | |
| 579 | sage: from sage.combinat.finite_state_machine import FSMState |
| 580 | sage: A = FSMState('A') |
| 581 | sage: copy(A) |
| 582 | 'A' |
| 583 | """ |
| 584 | new = FSMState(self.label(), self.word_out, |
| 585 | self.is_initial, self.is_final) |
| 586 | if hasattr(self, 'hook'): |
| 587 | new.hook = self.hook |
| 588 | return new |
| 589 | |
| 590 | |
| 591 | copy = __copy__ |
| 592 | |
| 593 | |
| 594 | def __deepcopy__(self, memo): |
| 595 | """ |
| 596 | Returns a deep copy of the state. |
| 597 | |
| 598 | INPUT: |
| 599 | |
| 600 | - ``memo`` -- a dictionary storing already processed elements. |
| 601 | |
| 602 | OUTPUT: |
| 603 | |
| 604 | A new state. |
| 605 | |
| 606 | EXAMPLES:: |
| 607 | |
| 608 | sage: from sage.combinat.finite_state_machine import FSMState |
| 609 | sage: A = FSMState('A') |
| 610 | sage: deepcopy(A) |
| 611 | 'A' |
| 612 | """ |
| 613 | try: |
| 614 | label = self._deepcopy_relabel_ |
| 615 | except AttributeError: |
| 616 | label = deepcopy(self.label(), memo) |
| 617 | new = FSMState(label, deepcopy(self.word_out, memo), |
| 618 | self.is_initial, self.is_final) |
| 619 | if hasattr(self, 'hook'): |
| 620 | new.hook = deepcopy(self.hook, memo) |
| 621 | return new |
| 622 | |
| 623 | |
| 624 | def deepcopy(self, memo=None): |
| 625 | """ |
| 626 | Returns a deep copy of the state. |
| 627 | |
| 628 | INPUT: |
| 629 | |
| 630 | - ``memo`` -- (default: ``None``) a dictionary storing already |
| 631 | processed elements. |
| 632 | |
| 633 | OUTPUT: |
| 634 | |
| 635 | A new state. |
| 636 | |
| 637 | EXAMPLES:: |
| 638 | |
| 639 | sage: from sage.combinat.finite_state_machine import FSMState |
| 640 | sage: A = FSMState('A') |
| 641 | sage: deepcopy(A) |
| 642 | 'A' |
| 643 | """ |
| 644 | return deepcopy(self, memo) |
| 645 | |
| 646 | |
| 647 | def relabeled(self, label, memo=None): |
| 648 | """ |
| 649 | Returns a deep copy of the state with a new label. |
| 650 | |
| 651 | INPUT: |
| 652 | |
| 653 | - ``label`` -- the label of new state. |
| 654 | |
| 655 | - ``memo`` -- (default: ``None``) a dictionary storing already |
| 656 | processed elements. |
| 657 | |
| 658 | OUTPUT: |
| 659 | |
| 660 | A new state. |
| 661 | |
| 662 | EXAMPLES:: |
| 663 | |
| 664 | sage: from sage.combinat.finite_state_machine import FSMState |
| 665 | sage: A = FSMState('A') |
| 666 | sage: A.relabeled('B') |
| 667 | 'B' |
| 668 | |
| 669 | """ |
| 670 | self._deepcopy_relabel_ = label |
| 671 | new = deepcopy(self, memo) |
| 672 | del self._deepcopy_relabel_ |
| 673 | return new |
| 674 | |
| 675 | |
| 676 | def __hash__(self): |
| 677 | """ |
| 678 | Returns a hash value for the object. |
| 679 | |
| 680 | INPUT: |
| 681 | |
| 682 | Nothing. |
| 683 | |
| 684 | OUTPUT: |
| 685 | |
| 686 | The hash of this state. |
| 687 | |
| 688 | TESTS:: |
| 689 | |
| 690 | sage: from sage.combinat.finite_state_machine import FSMState |
| 691 | sage: A = FSMState('A') |
| 692 | sage: hash(A) #random |
| 693 | -269909568 |
| 694 | """ |
| 695 | return hash(self.label()) |
| 696 | |
| 697 | |
| 698 | def _repr_(self): |
| 699 | """ |
| 700 | Returns the string "label". |
| 701 | |
| 702 | INPUT: |
| 703 | |
| 704 | Nothing. |
| 705 | |
| 706 | OUTPUT: |
| 707 | |
| 708 | A string. |
| 709 | |
| 710 | TESTS: |
| 711 | |
| 712 | sage: from sage.combinat.finite_state_machine import FSMState |
| 713 | sage: FSMState('A')._repr_() |
| 714 | "'A'" |
| 715 | """ |
| 716 | return repr(self.label()) |
| 717 | |
| 718 | |
| 719 | def __eq__(left, right): |
| 720 | """ |
| 721 | Returns True if two states are the same, i.e., if they have |
| 722 | the same labels. |
| 723 | |
| 724 | Note that the hooks and whether the states are initial or |
| 725 | final are not checked. |
| 726 | |
| 727 | INPUT: |
| 728 | |
| 729 | - ``left`` -- a state. |
| 730 | |
| 731 | - ``right`` -- a state. |
| 732 | |
| 733 | OUTPUT: |
| 734 | |
| 735 | True or False. |
| 736 | |
| 737 | EXAMPLES:: |
| 738 | |
| 739 | sage: from sage.combinat.finite_state_machine import FSMState |
| 740 | sage: A = FSMState('A') |
| 741 | sage: B = FSMState('A', is_initial=True) |
| 742 | sage: A == B |
| 743 | True |
| 744 | """ |
| 745 | if not is_FSMState(right): |
| 746 | return False |
| 747 | return left.label() == right.label() |
| 748 | |
| 749 | |
| 750 | def __ne__(left, right): |
| 751 | """ |
| 752 | Tests for inequality, complement of __eq__. |
| 753 | |
| 754 | INPUT: |
| 755 | |
| 756 | - ``left`` -- a state. |
| 757 | |
| 758 | - ``right`` -- a state. |
| 759 | |
| 760 | OUTPUT: |
| 761 | |
| 762 | True or False. |
| 763 | |
| 764 | EXAMPLES:: |
| 765 | |
| 766 | sage: from sage.combinat.finite_state_machine import FSMState |
| 767 | sage: A = FSMState('A', is_initial=True) |
| 768 | sage: B = FSMState('A', is_final=True) |
| 769 | sage: A != B |
| 770 | False |
| 771 | """ |
| 772 | return (not (left == right)) |
| 773 | |
| 774 | |
| 775 | def __nonzero__(self): |
| 776 | """ |
| 777 | Returns True. |
| 778 | |
| 779 | INPUT: |
| 780 | |
| 781 | Nothing. |
| 782 | |
| 783 | OUTPUT: |
| 784 | |
| 785 | True or False. |
| 786 | |
| 787 | TESTS:: |
| 788 | |
| 789 | sage: from sage.combinat.finite_state_machine import FSMState |
| 790 | sage: FSMState('A').__nonzero__() |
| 791 | True |
| 792 | """ |
| 793 | return True # A state cannot be zero (see __init__) |
| 794 | |
| 795 | |
| 796 | #***************************************************************************** |
| 797 | |
| 798 | |
| 799 | def is_FSMTransition(T): |
| 800 | """ |
| 801 | Tests whether or not ``T`` inherits from :class:`FSMTransition`. |
| 802 | |
| 803 | TESTS:: |
| 804 | |
| 805 | sage: from sage.combinat.finite_state_machine import is_FSMTransition, FSMTransition |
| 806 | sage: is_FSMTransition(FSMTransition('A', 'B')) |
| 807 | True |
| 808 | """ |
| 809 | return isinstance(T, FSMTransition) |
| 810 | |
| 811 | |
| 812 | class FSMTransition(SageObject): |
| 813 | """ |
| 814 | Class for a transition of a finite state machine. |
| 815 | |
| 816 | INPUT: |
| 817 | |
| 818 | - ``from_state`` -- state from which transition starts. |
| 819 | |
| 820 | - ``to_state`` -- state in which transition ends. |
| 821 | |
| 822 | - ``word_in`` -- the input word of the transitions (when the |
| 823 | finite state machine is used as automaton) |
| 824 | |
| 825 | - ``word_out`` -- the output word of the transitions (when the |
| 826 | finite state machine is used as transducer) |
| 827 | |
| 828 | OUTPUT: |
| 829 | |
| 830 | A transition of a finite state machine. |
| 831 | |
| 832 | EXAMPLES:: |
| 833 | |
| 834 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 835 | sage: A = FSMState('A') |
| 836 | sage: B = FSMState('B') |
| 837 | sage: S = FSMTransition(A, B, 0, 1) |
| 838 | sage: T = FSMTransition('A', 'B', 0, 1) |
| 839 | sage: T == S |
| 840 | True |
| 841 | sage: U = FSMTransition('A', 'B', 0) |
| 842 | sage: U == T |
| 843 | False |
| 844 | |
| 845 | """ |
| 846 | def __init__(self, from_state, to_state, |
| 847 | word_in=None, word_out=None, |
| 848 | hook=None): |
| 849 | """ |
| 850 | See :class:`FSMTransition` for more information. |
| 851 | |
| 852 | EXAMPLES:: |
| 853 | |
| 854 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 855 | sage: FSMTransition('A', 'B', 0, 1) |
| 856 | Transition from 'A' to 'B': 0|1 |
| 857 | """ |
| 858 | if is_FSMState(from_state): |
| 859 | self.from_state = from_state |
| 860 | else: |
| 861 | self.from_state = FSMState(from_state) |
| 862 | if is_FSMState(to_state): |
| 863 | self.to_state = to_state |
| 864 | else: |
| 865 | self.to_state = FSMState(to_state) |
| 866 | |
| 867 | if isinstance(word_in, list): |
| 868 | self.word_in = word_in |
| 869 | elif word_in is not None: |
| 870 | self.word_in = [word_in] |
| 871 | else: |
| 872 | self.word_in = [] |
| 873 | |
| 874 | if isinstance(word_out, list): |
| 875 | self.word_out = word_out |
| 876 | elif word_out is not None: |
| 877 | self.word_out = [word_out] |
| 878 | else: |
| 879 | self.word_out = [] |
| 880 | |
| 881 | if hook is not None: |
| 882 | if hasattr(hook, '__call__'): |
| 883 | self.hook = hook |
| 884 | else: |
| 885 | raise TypeError, 'Wrong argument for hook.' |
| 886 | |
| 887 | |
| 888 | def __copy__(self): |
| 889 | """ |
| 890 | Returns a (shallow) copy of the transition. |
| 891 | |
| 892 | INPUT: |
| 893 | |
| 894 | Nothing. |
| 895 | |
| 896 | OUTPUT: |
| 897 | |
| 898 | A new transition. |
| 899 | |
| 900 | EXAMPLES:: |
| 901 | |
| 902 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 903 | sage: t = FSMTransition('A', 'B', 0) |
| 904 | sage: copy(t) |
| 905 | Transition from 'A' to 'B': 0|- |
| 906 | """ |
| 907 | new = FSMTransition(self.from_state, self.to_state, |
| 908 | self.word_in, self.word_out) |
| 909 | if hasattr(self, 'hook'): |
| 910 | new.hook = self.hook |
| 911 | return new |
| 912 | |
| 913 | |
| 914 | copy = __copy__ |
| 915 | |
| 916 | def __deepcopy__(self, memo): |
| 917 | """ |
| 918 | Returns a deep copy of the transition. |
| 919 | |
| 920 | INPUT: |
| 921 | |
| 922 | - ``memo`` -- a dictionary storing already processed elements. |
| 923 | |
| 924 | OUTPUT: |
| 925 | |
| 926 | A new transition. |
| 927 | |
| 928 | EXAMPLES:: |
| 929 | |
| 930 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 931 | sage: t = FSMTransition('A', 'B', 0) |
| 932 | sage: deepcopy(t) |
| 933 | Transition from 'A' to 'B': 0|- |
| 934 | """ |
| 935 | new = FSMTransition(deepcopy(self.from_state, memo), |
| 936 | deepcopy(self.to_state, memo), |
| 937 | deepcopy(self.word_in, memo), |
| 938 | deepcopy(self.word_out, memo)) |
| 939 | if hasattr(self, 'hook'): |
| 940 | new.hook = deepcopy(self.hook, memo) |
| 941 | return new |
| 942 | |
| 943 | |
| 944 | def deepcopy(self, memo=None): |
| 945 | """ |
| 946 | Returns a deep copy of the transition. |
| 947 | |
| 948 | INPUT: |
| 949 | |
| 950 | - ``memo`` -- (default: ``None``) a dictionary storing already |
| 951 | processed elements. |
| 952 | |
| 953 | OUTPUT: |
| 954 | |
| 955 | A new transition. |
| 956 | |
| 957 | EXAMPLES:: |
| 958 | |
| 959 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 960 | sage: t = FSMTransition('A', 'B', 0) |
| 961 | sage: deepcopy(t) |
| 962 | Transition from 'A' to 'B': 0|- |
| 963 | """ |
| 964 | return deepcopy(self, memo) |
| 965 | |
| 966 | |
| 967 | def __hash__(self): |
| 968 | """ |
| 969 | Since transitions are mutable, they should not be hashable, so |
| 970 | we return a type error. |
| 971 | |
| 972 | INPUT: |
| 973 | |
| 974 | Nothing. |
| 975 | |
| 976 | OUTPUT: |
| 977 | |
| 978 | The hash of this transition. |
| 979 | |
| 980 | EXAMPLES:: |
| 981 | |
| 982 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 983 | sage: hash(FSMTransition('A', 'B')) |
| 984 | Traceback (most recent call last): |
| 985 | ... |
| 986 | TypeError: Transitions are mutable, and thus not hashable. |
| 987 | |
| 988 | """ |
| 989 | raise TypeError, "Transitions are mutable, and thus not hashable." |
| 990 | |
| 991 | |
| 992 | def _repr_(self): |
| 993 | """ |
| 994 | Represents a transitions as from state to state and input, output. |
| 995 | |
| 996 | INPUT: |
| 997 | |
| 998 | Nothing. |
| 999 | |
| 1000 | OUTPUT: |
| 1001 | |
| 1002 | A string. |
| 1003 | |
| 1004 | EXAMPLES:: |
| 1005 | |
| 1006 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 1007 | sage: FSMTransition('A', 'B', 0, 0)._repr_() |
| 1008 | "Transition from 'A' to 'B': 0|0" |
| 1009 | |
| 1010 | """ |
| 1011 | return "Transition from %s to %s: %s" % (repr(self.from_state), |
| 1012 | repr(self.to_state), |
| 1013 | self._in_out_label_()) |
| 1014 | |
| 1015 | |
| 1016 | def _in_out_label_(self): |
| 1017 | """ |
| 1018 | Returns the input and output of a transition as |
| 1019 | "word_in|word_out". |
| 1020 | |
| 1021 | INPUT: |
| 1022 | |
| 1023 | Nothing. |
| 1024 | |
| 1025 | OUTPUT: |
| 1026 | |
| 1027 | A string of the input and output labels. |
| 1028 | |
| 1029 | EXAMPLES:: |
| 1030 | |
| 1031 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 1032 | sage: FSMTransition('A', 'B', 0, 1)._in_out_label_() |
| 1033 | '0|1' |
| 1034 | """ |
| 1035 | return "%s|%s" % (FSMWordSymbol(self.word_in), |
| 1036 | FSMWordSymbol(self.word_out)) |
| 1037 | |
| 1038 | |
| 1039 | def __eq__(left, right): |
| 1040 | """ |
| 1041 | Returns True if the two transitions are the same, i.e., if the |
| 1042 | both go from the same states to the same states and read and |
| 1043 | write the same words. |
| 1044 | |
| 1045 | Note that the hooks are not checked. |
| 1046 | |
| 1047 | INPUT: |
| 1048 | |
| 1049 | - ``left`` -- a transition. |
| 1050 | |
| 1051 | - ``right`` -- a transition. |
| 1052 | |
| 1053 | OUTPUT: |
| 1054 | |
| 1055 | True or False. |
| 1056 | |
| 1057 | EXAMPLES:: |
| 1058 | |
| 1059 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 1060 | sage: A = FSMState('A', is_initial=True) |
| 1061 | sage: t1 = FSMTransition('A', 'B', 0, 1) |
| 1062 | sage: t2 = FSMTransition(A, 'B', 0, 1) |
| 1063 | sage: t1 == t2 |
| 1064 | True |
| 1065 | """ |
| 1066 | if not is_FSMTransition(right): |
| 1067 | raise TypeError, 'Only instances of FSMTransition ' \ |
| 1068 | 'can be compared.' |
| 1069 | return left.from_state == right.from_state \ |
| 1070 | and left.to_state == right.to_state \ |
| 1071 | and left.word_in == right.word_in \ |
| 1072 | and left.word_out == right.word_out |
| 1073 | |
| 1074 | |
| 1075 | def __ne__(left, right): |
| 1076 | """ |
| 1077 | |
| 1078 | INPUT: |
| 1079 | |
| 1080 | - ``left`` -- a transition. |
| 1081 | |
| 1082 | - ``right`` -- a transition. |
| 1083 | |
| 1084 | OUTPUT: |
| 1085 | |
| 1086 | True or False. |
| 1087 | Tests for inequality, complement of __eq__. |
| 1088 | |
| 1089 | EXAMPLES:: |
| 1090 | |
| 1091 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 1092 | sage: A = FSMState('A', is_initial=True) |
| 1093 | sage: t1 = FSMTransition('A', 'B', 0, 1) |
| 1094 | sage: t2 = FSMTransition(A, 'B', 0, 1) |
| 1095 | sage: t1 != t2 |
| 1096 | False |
| 1097 | """ |
| 1098 | return (not (left == right)) |
| 1099 | |
| 1100 | |
| 1101 | def __nonzero__(self): |
| 1102 | """ |
| 1103 | Returns True. |
| 1104 | |
| 1105 | INPUT: |
| 1106 | |
| 1107 | Nothing. |
| 1108 | |
| 1109 | OUTPUT: |
| 1110 | |
| 1111 | True or False. |
| 1112 | |
| 1113 | EXAMPLES:: |
| 1114 | |
| 1115 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 1116 | sage: FSMTransition('A', 'B', 0).__nonzero__() |
| 1117 | True |
| 1118 | """ |
| 1119 | return True # A transition cannot be zero (see __init__) |
| 1120 | |
| 1121 | |
| 1122 | #***************************************************************************** |
| 1123 | |
| 1124 | |
| 1125 | def is_FiniteStateMachine(FSM): |
| 1126 | """ |
| 1127 | Tests whether or not ``FSM`` inherits from :class:`FiniteStateMachine`. |
| 1128 | |
| 1129 | TESTS:: |
| 1130 | |
| 1131 | sage: from sage.combinat.finite_state_machine import is_FiniteStateMachine |
| 1132 | sage: is_FiniteStateMachine(FiniteStateMachine()) |
| 1133 | True |
| 1134 | sage: is_FiniteStateMachine(Automaton()) |
| 1135 | True |
| 1136 | sage: is_FiniteStateMachine(Transducer()) |
| 1137 | True |
| 1138 | """ |
| 1139 | return isinstance(FSM, FiniteStateMachine) |
| 1140 | |
| 1141 | |
| 1142 | class FiniteStateMachine(SageObject): |
| 1143 | """ |
| 1144 | Class for a finite state machine. |
| 1145 | |
| 1146 | A finite state machine is a finite set of states connected by |
| 1147 | transitions. |
| 1148 | |
| 1149 | INPUT: |
| 1150 | |
| 1151 | - ``data`` -- can be any of the following: |
| 1152 | |
| 1153 | #. a dictionary of dictionaries (of transitions), |
| 1154 | |
| 1155 | #. a dictionary of lists (of states or transitions), |
| 1156 | |
| 1157 | #. a list (of transitions), |
| 1158 | |
| 1159 | #. a function (transition function), |
| 1160 | |
| 1161 | #. an other instance of a finite state machine. |
| 1162 | |
| 1163 | - ``initial_states`` and ``final_states`` -- the initial and |
| 1164 | finial states if this machine |
| 1165 | |
| 1166 | - ``input_alphabet`` and ``output_alphabet`` -- the input and |
| 1167 | output alphabets of this machine |
| 1168 | |
| 1169 | - ``determine_alphabets`` -- If True, then the function |
| 1170 | ``determine_alphabets()`` is called after ``data`` was read and |
| 1171 | processed, if ``False``, then not. If it is ``None``, then it is |
| 1172 | decided during the construction of the finite state machine |
| 1173 | whether ``determine_alphabets()`` should be called. |
| 1174 | |
| 1175 | - ``store_states_dict`` -- If ``True``, then additionally the states |
| 1176 | are stored in an interal dictionary for speed up. |
| 1177 | |
| 1178 | OUTPUT: |
| 1179 | |
| 1180 | A finite state machine. |
| 1181 | |
| 1182 | The object creation of :class:`Automaton` and :class:`Transducer` |
| 1183 | is the same as the one described here (i.e. just replace the word |
| 1184 | ``FiniteStateMachine`` by ``Automaton`` or ``Transducer``). |
| 1185 | |
| 1186 | EXAMPLES:: |
| 1187 | |
| 1188 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 1189 | |
| 1190 | See documentation for more examples. |
| 1191 | |
| 1192 | We illustrate the different input formats: |
| 1193 | |
| 1194 | #. The input-data can be a dictionary of dictionaries, where |
| 1195 | |
| 1196 | - the keys of the outer dictionary are state-labels (from-states of |
| 1197 | transitions), |
| 1198 | - the keys of the inner dictionaries are state-labels (to-states of |
| 1199 | transitions), |
| 1200 | - the values of the inner dictionaries specify the transition |
| 1201 | more precisely. |
| 1202 | |
| 1203 | The easiest is to use a tuple consisting of an input and an |
| 1204 | output word:: |
| 1205 | |
| 1206 | sage: FiniteStateMachine({'a':{'b':(0, 1), 'c':(1, 1)}}) |
| 1207 | finite state machine with 3 states |
| 1208 | |
| 1209 | Instead of the tuple anything iterable (e.g. a list) can be |
| 1210 | used as well. |
| 1211 | |
| 1212 | If you want to use the arguments of ``FSMTransition`` |
| 1213 | directly, you can use a dictionary:: |
| 1214 | |
| 1215 | sage: FiniteStateMachine({'a':{'b':{'word_in':0, 'word_out':1}, |
| 1216 | ....: 'c':{'word_in':1, 'word_out':1}}}) |
| 1217 | finite state machine with 3 states |
| 1218 | |
| 1219 | In the case you already have instances of ``FSMTransition``, it is |
| 1220 | possible to use them directly:: |
| 1221 | |
| 1222 | sage: FiniteStateMachine({'a':{'b':FSMTransition('a', 'b', 0, 1), |
| 1223 | ....: 'c':FSMTransition('a', 'c', 1, 1)}}) |
| 1224 | finite state machine with 3 states |
| 1225 | |
| 1226 | #. The input-data can be a dictionary of lists, where the keys |
| 1227 | are states or label of states. |
| 1228 | |
| 1229 | The list-elements can be states:: |
| 1230 | |
| 1231 | sage: a = FSMState('a') |
| 1232 | sage: b = FSMState('b') |
| 1233 | sage: c = FSMState('c') |
| 1234 | sage: FiniteStateMachine({a:[b, c]}) |
| 1235 | finite state machine with 3 states |
| 1236 | |
| 1237 | Or the list-elements can simply be labels of states:: |
| 1238 | |
| 1239 | sage: FiniteStateMachine({'a':['b', 'c']}) |
| 1240 | finite state machine with 3 states |
| 1241 | |
| 1242 | The list-elements can also be transitions:: |
| 1243 | |
| 1244 | sage: FiniteStateMachine({'a':[FSMTransition('a', 'b', 0, 1), |
| 1245 | ....: FSMTransition('a', 'c', 1, 1)]}) |
| 1246 | finite state machine with 3 states |
| 1247 | |
| 1248 | Or they can be tuples of a label, an input word and an output |
| 1249 | word specifying a transition:: |
| 1250 | |
| 1251 | sage: FiniteStateMachine({'a':[('b', 0, 1), ('c', 1, 1)]}) |
| 1252 | finite state machine with 3 states |
| 1253 | |
| 1254 | #. The input-data can be a list, where its elements specify |
| 1255 | transitions:: |
| 1256 | |
| 1257 | sage: FiniteStateMachine([FSMTransition('a', 'b', 0, 1), |
| 1258 | ....: FSMTransition('a', 'c', 1, 1)]) |
| 1259 | finite state machine with 3 states |
| 1260 | |
| 1261 | It is possible to skip ``FSMTransition`` in the example above:: |
| 1262 | |
| 1263 | sage: FiniteStateMachine([('a', 'b', 0, 1), ('a', 'c', 1, 1)]) |
| 1264 | finite state machine with 3 states |
| 1265 | |
| 1266 | The parameters of the transition are given in tuples. Anyhow, |
| 1267 | anything iterable (e.g. a list) is possible. |
| 1268 | |
| 1269 | You can also name the parameters of the transition. For this |
| 1270 | purpose you take a dictionary:: |
| 1271 | |
| 1272 | sage: FiniteStateMachine([{'from_state':'a', 'to_state':'b', |
| 1273 | ....: 'word_in':0, 'word_out':1}, |
| 1274 | ....: {'from_state':'a', 'to_state':'c', |
| 1275 | ....: 'word_in':1, 'word_out':1}]) |
| 1276 | finite state machine with 3 states |
| 1277 | |
| 1278 | Other arguments, which :class:`FSMTransition` accepts, can be |
| 1279 | added, too. |
| 1280 | |
| 1281 | #. The input-data can also be function acting as transition |
| 1282 | function: |
| 1283 | |
| 1284 | This function has two input arguments: |
| 1285 | |
| 1286 | #. a label of a state (from which the transition starts), |
| 1287 | |
| 1288 | #. a letter of the (input-)alphabet (as input-label of the transition). |
| 1289 | |
| 1290 | It returns a tuple with the following entries: |
| 1291 | |
| 1292 | #. a label of a state (to which state the transition goes), |
| 1293 | |
| 1294 | #. a letter of or a word over the (output-)alphabet (as |
| 1295 | output-label of the transition). |
| 1296 | |
| 1297 | It may also output a list of such tuples if several |
| 1298 | transitions from the from-state and the input letter exist |
| 1299 | (this means that the finite state machine is |
| 1300 | non-deterministic). |
| 1301 | |
| 1302 | If the transition does not exist, the function should raise a |
| 1303 | ``LookupError`` or return an empty list. |
| 1304 | |
| 1305 | When constructing a finite state machine in this way, some |
| 1306 | inital states and an input alphabet have to be specified. |
| 1307 | |
| 1308 | :: |
| 1309 | |
| 1310 | sage: def f(state_from, read): |
| 1311 | ....: if int(state_from) + read <= 2: |
| 1312 | ....: state_to = 2*int(state_from)+read |
| 1313 | ....: write = 0 |
| 1314 | ....: else: |
| 1315 | ....: state_to = 2*int(state_from) + read - 5 |
| 1316 | ....: write = 1 |
| 1317 | ....: return (str(state_to), write) |
| 1318 | sage: F = FiniteStateMachine(f, input_alphabet=[0, 1], |
| 1319 | ....: initial_states=['0'], |
| 1320 | ....: final_states=['0']) |
| 1321 | sage: F([1, 0, 1]) |
| 1322 | (True, '0', [0, 0, 1]) |
| 1323 | |
| 1324 | #. The input-data can be an other instance of a finite state machine:: |
| 1325 | |
| 1326 | sage: FiniteStateMachine(FiniteStateMachine([])) |
| 1327 | Traceback (most recent call last): |
| 1328 | ... |
| 1329 | NotImplementedError |
| 1330 | |
| 1331 | |
| 1332 | TESTS:: |
| 1333 | |
| 1334 | sage: a = FSMState('S_a', 'a') |
| 1335 | sage: b = FSMState('S_b', 'b') |
| 1336 | sage: c = FSMState('S_c', 'c') |
| 1337 | sage: d = FSMState('S_d', 'd') |
| 1338 | sage: FiniteStateMachine({a:[b, c], b:[b, c, d], |
| 1339 | ....: c:[a, b], d:[a, c]}) |
| 1340 | finite state machine with 4 states |
| 1341 | |
| 1342 | We have several constructions which lead to the same finite |
| 1343 | state machine:: |
| 1344 | |
| 1345 | sage: A = FSMState('A') |
| 1346 | sage: B = FSMState('B') |
| 1347 | sage: C = FSMState('C') |
| 1348 | sage: FSM1 = FiniteStateMachine( |
| 1349 | ....: {A:{B:{'word_in':0, 'word_out':1}, |
| 1350 | ....: C:{'word_in':1, 'word_out':1}}}) |
| 1351 | sage: FSM2 = FiniteStateMachine({A:{B:(0, 1), C:(1, 1)}}) |
| 1352 | sage: FSM3 = FiniteStateMachine( |
| 1353 | ....: {A:{B:FSMTransition(A, B, 0, 1), |
| 1354 | ....: C:FSMTransition(A, C, 1, 1)}}) |
| 1355 | sage: FSM4 = FiniteStateMachine({A:[(B, 0, 1), (C, 1, 1)]}) |
| 1356 | sage: FSM5 = FiniteStateMachine( |
| 1357 | ....: {A:[FSMTransition(A, B, 0, 1), FSMTransition(A, C, 1, 1)]}) |
| 1358 | sage: FSM6 = FiniteStateMachine( |
| 1359 | ....: [{'from_state':A, 'to_state':B, 'word_in':0, 'word_out':1}, |
| 1360 | ....: {'from_state':A, 'to_state':C, 'word_in':1, 'word_out':1}]) |
| 1361 | sage: FSM7 = FiniteStateMachine([(A, B, 0, 1), (A, C, 1, 1)]) |
| 1362 | sage: FSM8 = FiniteStateMachine( |
| 1363 | ....: [FSMTransition(A, B, 0, 1), FSMTransition(A, C, 1, 1)]) |
| 1364 | |
| 1365 | sage: FSM1 == FSM2 == FSM3 == FSM4 == FSM5 == FSM6 == FSM7 == FSM8 |
| 1366 | True |
| 1367 | |
| 1368 | It is possible to skip ``FSMTransition`` in the example above. |
| 1369 | |
| 1370 | Some more tests for different input-data:: |
| 1371 | |
| 1372 | sage: FiniteStateMachine({'a':{'a':[0, 0], 'b':[1, 1]}, |
| 1373 | ....: 'b':{'b':[1, 0]}}) |
| 1374 | finite state machine with 2 states |
| 1375 | |
| 1376 | sage: a = FSMState('S_a', 'a') |
| 1377 | sage: b = FSMState('S_b', 'b') |
| 1378 | sage: c = FSMState('S_c', 'c') |
| 1379 | sage: d = FSMState('S_d', 'd') |
| 1380 | sage: t1 = FSMTransition(a, b) |
| 1381 | sage: t2 = FSMTransition(b, c) |
| 1382 | sage: t3 = FSMTransition(b, d) |
| 1383 | sage: t4 = FSMTransition(c, d) |
| 1384 | sage: FiniteStateMachine([t1, t2, t3, t4]) |
| 1385 | finite state machine with 4 states |
| 1386 | """ |
| 1387 | |
| 1388 | #************************************************************************* |
| 1389 | # init |
| 1390 | #************************************************************************* |
| 1391 | |
| 1392 | |
| 1393 | def __init__(self, |
| 1394 | data=None, |
| 1395 | initial_states=None, final_states=None, |
| 1396 | input_alphabet=None, output_alphabet=None, |
| 1397 | determine_alphabets=None, |
| 1398 | store_states_dict=True): |
| 1399 | """ |
| 1400 | See :class:`FiniteStateMachine` for more information. |
| 1401 | |
| 1402 | TEST:: |
| 1403 | |
| 1404 | sage: FiniteStateMachine() |
| 1405 | finite state machine with 0 states |
| 1406 | """ |
| 1407 | self._states_ = [] # List of states in the finite state |
| 1408 | # machine. Each state stores a list of |
| 1409 | # outgoing transitions. |
| 1410 | if store_states_dict: |
| 1411 | self._states_dict_ = {} |
| 1412 | |
| 1413 | if initial_states is not None: |
| 1414 | if not hasattr(initial_states, '__iter__'): |
| 1415 | raise TypeError, 'Initial states must be iterable ' \ |
| 1416 | '(e.g. a list of states).' |
| 1417 | for s in initial_states: |
| 1418 | state = self.add_state(s) |
| 1419 | state.is_initial = True |
| 1420 | |
| 1421 | if final_states is not None: |
| 1422 | if not hasattr(final_states, '__iter__'): |
| 1423 | raise TypeError, 'Final states must be iterable ' \ |
| 1424 | '(e.g. a list of states).' |
| 1425 | for s in final_states: |
| 1426 | state = self.add_state(s) |
| 1427 | state.is_final = True |
| 1428 | |
| 1429 | self.input_alphabet = input_alphabet |
| 1430 | self.output_alphabet = output_alphabet |
| 1431 | |
| 1432 | if data is None: |
| 1433 | pass |
| 1434 | elif is_FiniteStateMachine(data): |
| 1435 | raise NotImplementedError |
| 1436 | elif hasattr(data, 'iteritems'): |
| 1437 | # data is a dict (or something similar), |
| 1438 | # format: key = from_state, value = iterator of transitions |
| 1439 | for (sf, iter_transitions) in data.iteritems(): |
| 1440 | self.add_state(sf) |
| 1441 | if hasattr(iter_transitions, 'iteritems'): |
| 1442 | for (st, transition) in iter_transitions.iteritems(): |
| 1443 | self.add_state(st) |
| 1444 | if is_FSMTransition(transition): |
| 1445 | self.add_transition(transition) |
| 1446 | elif hasattr(transition, 'iteritems'): |
| 1447 | self.add_transition(sf, st, **transition) |
| 1448 | elif hasattr(transition, '__iter__'): |
| 1449 | self.add_transition(sf, st, *transition) |
| 1450 | else: |
| 1451 | self.add_transition(sf, st, transition) |
| 1452 | elif hasattr(iter_transitions, '__iter__'): |
| 1453 | for transition in iter_transitions: |
| 1454 | if hasattr(transition, '__iter__'): |
| 1455 | L = [sf] |
| 1456 | L.extend(transition) |
| 1457 | elif is_FSMTransition(transition): |
| 1458 | L = transition |
| 1459 | else: |
| 1460 | L = [sf, transition] |
| 1461 | self.add_transition(L) |
| 1462 | else: |
| 1463 | raise TypeError, 'Wrong input data for transition.' |
| 1464 | if determine_alphabets is None and input_alphabet is None \ |
| 1465 | and output_alphabet is None: |
| 1466 | determine_alphabets = True |
| 1467 | elif hasattr(data, '__iter__'): |
| 1468 | # data is a something that is iterable, |
| 1469 | # items are transitions |
| 1470 | for transition in data: |
| 1471 | if is_FSMTransition(transition): |
| 1472 | self.add_transition(transition) |
| 1473 | elif hasattr(transition, 'iteritems'): |
| 1474 | self.add_transition(transition) |
| 1475 | elif hasattr(transition, '__iter__'): |
| 1476 | self.add_transition(transition) |
| 1477 | else: |
| 1478 | raise TypeError, 'Wrong input data for transition.' |
| 1479 | if determine_alphabets is None and input_alphabet is None \ |
| 1480 | and output_alphabet is None: |
| 1481 | determine_alphabets = True |
| 1482 | elif hasattr(data, '__call__'): |
| 1483 | self.add_from_transition_function(data) |
| 1484 | else: |
| 1485 | raise TypeError, 'Cannot decide what to do with data.' |
| 1486 | |
| 1487 | if determine_alphabets: |
| 1488 | self.determine_alphabets() |
| 1489 | |
| 1490 | |
| 1491 | #************************************************************************* |
| 1492 | # copy and hash |
| 1493 | #************************************************************************* |
| 1494 | |
| 1495 | |
| 1496 | def __copy__(self): |
| 1497 | """ |
| 1498 | Returns a (shallow) copy of the finite state machine. |
| 1499 | |
| 1500 | INPUT: |
| 1501 | |
| 1502 | Nothing. |
| 1503 | |
| 1504 | OUTPUT: |
| 1505 | |
| 1506 | A new finite state machine. |
| 1507 | |
| 1508 | TESTS:: |
| 1509 | |
| 1510 | sage: copy(FiniteStateMachine()) |
| 1511 | Traceback (most recent call last): |
| 1512 | ... |
| 1513 | NotImplementedError |
| 1514 | """ |
| 1515 | raise NotImplementedError |
| 1516 | |
| 1517 | |
| 1518 | copy = __copy__ |
| 1519 | |
| 1520 | def empty_copy(self, memo=None): |
| 1521 | """ |
| 1522 | Returns an empty deep copy of the finite state machine, i.e., |
| 1523 | input_alphabet, output_alphabet are preserved, but states and |
| 1524 | transitions are not. |
| 1525 | |
| 1526 | INPUT: |
| 1527 | |
| 1528 | - ``memo`` -- a dictionary storing already processed elements. |
| 1529 | |
| 1530 | OUTPUT: |
| 1531 | |
| 1532 | A new finite state machine. |
| 1533 | |
| 1534 | EXAMPLES:: |
| 1535 | |
| 1536 | sage: F = FiniteStateMachine([('A', 'A', 0, 2), ('A', 'A', 1, 3)], |
| 1537 | ....: input_alphabet=[0,1], |
| 1538 | ....: output_alphabet=[2,3]) |
| 1539 | sage: FE = F.empty_copy(); FE |
| 1540 | finite state machine with 0 states |
| 1541 | sage: FE.input_alphabet |
| 1542 | [0, 1] |
| 1543 | sage: FE.output_alphabet |
| 1544 | [2, 3] |
| 1545 | """ |
| 1546 | new = self.__class__() |
| 1547 | new.input_alphabet = deepcopy(self.input_alphabet, memo) |
| 1548 | new.output_alphabet = deepcopy(self.output_alphabet, memo) |
| 1549 | return new |
| 1550 | |
| 1551 | def __deepcopy__(self, memo): |
| 1552 | """ |
| 1553 | Returns a deep copy of the finite state machine. |
| 1554 | |
| 1555 | INPUT: |
| 1556 | |
| 1557 | - ``memo`` -- a dictionary storing already processed elements. |
| 1558 | |
| 1559 | OUTPUT: |
| 1560 | |
| 1561 | A new finite state machine. |
| 1562 | |
| 1563 | EXAMPLES:: |
| 1564 | |
| 1565 | sage: F = FiniteStateMachine([('A', 'A', 0, 1), ('A', 'A', 1, 0)]) |
| 1566 | sage: deepcopy(F) |
| 1567 | finite state machine with 1 states |
| 1568 | """ |
| 1569 | relabel = hasattr(self, '_deepcopy_relabel_') |
| 1570 | new = self.empty_copy(memo=memo) |
| 1571 | relabel_iter = itertools.count(0) |
| 1572 | for state in self.iter_states(): |
| 1573 | if relabel: |
| 1574 | state._deepcopy_relabel_ = relabel_iter.next() |
| 1575 | s = deepcopy(state, memo) |
| 1576 | if relabel: |
| 1577 | del state._deepcopy_relabel_ |
| 1578 | new.add_state(s) |
| 1579 | for transition in self.iter_transitions(): |
| 1580 | new.add_transition(deepcopy(transition, memo)) |
| 1581 | return new |
| 1582 | |
| 1583 | |
| 1584 | def deepcopy(self, memo=None): |
| 1585 | """ |
| 1586 | Returns a deep copy of the finite state machine. |
| 1587 | |
| 1588 | INPUT: |
| 1589 | |
| 1590 | - ``memo`` -- (default: ``None``) a dictionary storing already |
| 1591 | processed elements. |
| 1592 | |
| 1593 | OUTPUT: |
| 1594 | |
| 1595 | A new finite state machine. |
| 1596 | |
| 1597 | EXAMPLES:: |
| 1598 | |
| 1599 | sage: F = FiniteStateMachine([('A', 'A', 0, 1), ('A', 'A', 1, 0)]) |
| 1600 | sage: deepcopy(F) |
| 1601 | finite state machine with 1 states |
| 1602 | """ |
| 1603 | return deepcopy(self, memo) |
| 1604 | |
| 1605 | |
| 1606 | def relabeled(self, memo=None): |
| 1607 | """ |
| 1608 | Returns a deep copy of the finite state machine, but the |
| 1609 | states are relabeled by integers starting with 0. |
| 1610 | |
| 1611 | INPUT: |
| 1612 | |
| 1613 | - ``memo`` -- (default: ``None``) a dictionary storing already |
| 1614 | processed elements. |
| 1615 | |
| 1616 | OUTPUT: |
| 1617 | |
| 1618 | A new finite state machine. |
| 1619 | |
| 1620 | EXAMPLES:: |
| 1621 | |
| 1622 | sage: FSM1 = FiniteStateMachine([('A', 'B'), ('B', 'C'), ('C', 'A')]) |
| 1623 | sage: FSM1.states() |
| 1624 | ['A', 'B', 'C'] |
| 1625 | sage: FSM2 = FSM1.relabeled() |
| 1626 | sage: FSM2.states() |
| 1627 | [0, 1, 2] |
| 1628 | """ |
| 1629 | self._deepcopy_relabel_ = True |
| 1630 | new = deepcopy(self, memo) |
| 1631 | del self._deepcopy_relabel_ |
| 1632 | return new |
| 1633 | |
| 1634 | |
| 1635 | def __hash__(self): |
| 1636 | """ |
| 1637 | Since finite state machines are mutable, they should not be |
| 1638 | hashable, so we return a type error. |
| 1639 | |
| 1640 | INPUT: |
| 1641 | |
| 1642 | Nothing. |
| 1643 | |
| 1644 | OUTPUT: |
| 1645 | |
| 1646 | The hash of this finite state machine. |
| 1647 | |
| 1648 | EXAMPLES:: |
| 1649 | |
| 1650 | sage: hash(FiniteStateMachine()) |
| 1651 | Traceback (most recent call last): |
| 1652 | ... |
| 1653 | TypeError: Finite state machines are mutable, and thus not hashable. |
| 1654 | """ |
| 1655 | if getattr(self, "_immutable", False): |
| 1656 | return hash((tuple(self.states()), tuple(self.transitions()))) |
| 1657 | raise TypeError, "Finite state machines are mutable, " \ |
| 1658 | "and thus not hashable." |
| 1659 | |
| 1660 | |
| 1661 | #************************************************************************* |
| 1662 | # operators |
| 1663 | #************************************************************************* |
| 1664 | |
| 1665 | |
| 1666 | def __add__(self, other): |
| 1667 | """ |
| 1668 | Returns the disjoint union of the finite state machines self and other. |
| 1669 | |
| 1670 | INPUT: |
| 1671 | |
| 1672 | - ``other`` -- a finite state machine. |
| 1673 | |
| 1674 | OUTPUT: |
| 1675 | |
| 1676 | A new finite state machine. |
| 1677 | |
| 1678 | TESTS:: |
| 1679 | |
| 1680 | sage: FiniteStateMachine() + FiniteStateMachine([('A', 'B')]) |
| 1681 | Traceback (most recent call last): |
| 1682 | ... |
| 1683 | NotImplementedError |
| 1684 | """ |
| 1685 | if is_FiniteStateMachine(other): |
| 1686 | return self.disjoint_union(other) |
| 1687 | |
| 1688 | |
| 1689 | def __iadd__(self, other): |
| 1690 | """ |
| 1691 | TESTS:: |
| 1692 | |
| 1693 | sage: F = FiniteStateMachine() |
| 1694 | sage: F += FiniteStateMachine() |
| 1695 | Traceback (most recent call last): |
| 1696 | ... |
| 1697 | NotImplementedError |
| 1698 | """ |
| 1699 | raise NotImplementedError |
| 1700 | |
| 1701 | |
| 1702 | def __mul__(self, other): |
| 1703 | """ |
| 1704 | TESTS:: |
| 1705 | |
| 1706 | sage: FiniteStateMachine() * FiniteStateMachine([('A', 'B')]) |
| 1707 | Traceback (most recent call last): |
| 1708 | ... |
| 1709 | NotImplementedError |
| 1710 | """ |
| 1711 | if is_FiniteStateMachine(other): |
| 1712 | return self.intersection(other) |
| 1713 | |
| 1714 | |
| 1715 | def __imul__(self, other): |
| 1716 | """ |
| 1717 | TESTS:: |
| 1718 | |
| 1719 | sage: F = FiniteStateMachine() |
| 1720 | sage: F *= FiniteStateMachine() |
| 1721 | Traceback (most recent call last): |
| 1722 | ... |
| 1723 | NotImplementedError |
| 1724 | """ |
| 1725 | raise NotImplementedError |
| 1726 | |
| 1727 | |
| 1728 | def __call__(self, *args, **kwargs): |
| 1729 | """ |
| 1730 | Calls either method :meth:`.composition` or :meth:`.process`. |
| 1731 | |
| 1732 | EXAMPLES:: |
| 1733 | |
| 1734 | sage: from sage.combinat.finite_state_machine import FSMState |
| 1735 | sage: A = FSMState('A', is_initial=True, is_final=True) |
| 1736 | sage: binary_inverter = Transducer({A:[(A, 0, 1), (A, 1, 0)]}) |
| 1737 | sage: binary_inverter([0, 1, 0, 0, 1, 1]) |
| 1738 | (True, 'A', [1, 0, 1, 1, 0, 0]) |
| 1739 | |
| 1740 | :: |
| 1741 | |
| 1742 | sage: F = Transducer([('A', 'B', 1, 0), ('B', 'B', 1, 1), |
| 1743 | ....: ('B', 'B', 0, 0)], |
| 1744 | ....: initial_states=['A'], final_states=['B']) |
| 1745 | sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0), |
| 1746 | ....: (2, 2, 0, 1), (2, 1, 1, 1)], |
| 1747 | ....: initial_states=[1], final_states=[1]) |
| 1748 | sage: H = G(F) |
| 1749 | sage: H.states() |
| 1750 | [('A', 1), ('B', 1), ('B', 2)] |
| 1751 | """ |
| 1752 | if len(args) == 0: |
| 1753 | raise TypeError, "Called with too few arguments." |
| 1754 | if is_FiniteStateMachine(args[0]): |
| 1755 | return self.composition(*args, **kwargs) |
| 1756 | if hasattr(args[0], '__iter__'): |
| 1757 | return self.process(*args, **kwargs) |
| 1758 | raise TypeError, "Do not know what to do with that arguments." |
| 1759 | |
| 1760 | |
| 1761 | #************************************************************************* |
| 1762 | # tests |
| 1763 | #************************************************************************* |
| 1764 | |
| 1765 | |
| 1766 | def __nonzero__(self): |
| 1767 | """ |
| 1768 | Returns True if the finite state machine consists of at least |
| 1769 | one state. |
| 1770 | |
| 1771 | INPUT: |
| 1772 | |
| 1773 | Nothing. |
| 1774 | |
| 1775 | OUTPUT: |
| 1776 | |
| 1777 | True or False. |
| 1778 | |
| 1779 | TESTS:: |
| 1780 | |
| 1781 | sage: FiniteStateMachine().__nonzero__() |
| 1782 | False |
| 1783 | """ |
| 1784 | return len(self._states_) > 0 |
| 1785 | |
| 1786 | |
| 1787 | def __eq__(left, right): |
| 1788 | """ |
| 1789 | Returns True if the two finite state machines are equal, i.e., |
| 1790 | if they have the same states and the same transitions. |
| 1791 | |
| 1792 | INPUT: |
| 1793 | |
| 1794 | - ``left`` -- a finite state machine. |
| 1795 | |
| 1796 | - ``right`` -- a finite state machine. |
| 1797 | |
| 1798 | OUTPUT: |
| 1799 | |
| 1800 | True or False. |
| 1801 | |
| 1802 | EXAMPLES:: |
| 1803 | |
| 1804 | sage: F = FiniteStateMachine([('A', 'B', 1)]) |
| 1805 | sage: F == FiniteStateMachine() |
| 1806 | False |
| 1807 | """ |
| 1808 | if not is_FiniteStateMachine(right): |
| 1809 | raise TypeError, 'Only instances of FiniteStateMachine ' \ |
| 1810 | 'can be compared.' |
| 1811 | if len(left._states_) != len(right._states_): |
| 1812 | return False |
| 1813 | for state in left.iter_states(): |
| 1814 | if state not in right._states_: |
| 1815 | return False |
| 1816 | left_transitions = state.transitions |
| 1817 | right_transitions = right.state(state).transitions |
| 1818 | if len(left_transitions) != len(right_transitions): |
| 1819 | return False |
| 1820 | for t in left_transitions: |
| 1821 | if t not in right_transitions: |
| 1822 | return False |
| 1823 | return True |
| 1824 | |
| 1825 | |
| 1826 | def __ne__(left, right): |
| 1827 | """ |
| 1828 | Tests for inequality, complement of :meth:`.__eq__`. |
| 1829 | |
| 1830 | INPUT: |
| 1831 | |
| 1832 | - ``left`` -- a finite state machine. |
| 1833 | |
| 1834 | - ``right`` -- a finite state machine. |
| 1835 | |
| 1836 | OUTPUT: |
| 1837 | |
| 1838 | True or False. |
| 1839 | |
| 1840 | EXAMPLES:: |
| 1841 | |
| 1842 | sage: E = FiniteStateMachine([('A', 'B', 0)]) |
| 1843 | sage: F = Automaton([('A', 'B', 0)]) |
| 1844 | sage: G = Transducer([('A', 'B', 0, 1)]) |
| 1845 | sage: E == F |
| 1846 | True |
| 1847 | sage: E == G |
| 1848 | False |
| 1849 | """ |
| 1850 | return (not (left == right)) |
| 1851 | |
| 1852 | |
| 1853 | def __contains__(self, item): |
| 1854 | """ |
| 1855 | Returns true, if the finite state machine contains the |
| 1856 | state or transition item. Note that only the labels of the |
| 1857 | states and the input and output words are tested. |
| 1858 | |
| 1859 | INPUT: |
| 1860 | |
| 1861 | - ``item`` -- a state or a transition. |
| 1862 | |
| 1863 | OUTPUT: |
| 1864 | |
| 1865 | True or False. |
| 1866 | |
| 1867 | EXAMPLES:: |
| 1868 | |
| 1869 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 1870 | sage: F = FiniteStateMachine([('A', 'B', 0), ('B', 'A', 1)]) |
| 1871 | sage: FSMState('A', is_initial=True) in F |
| 1872 | True |
| 1873 | sage: 'A' in F |
| 1874 | False |
| 1875 | sage: FSMTransition('A', 'B', 0) in F |
| 1876 | True |
| 1877 | """ |
| 1878 | if is_FSMState(item): |
| 1879 | return self.has_state(item) |
| 1880 | if is_FSMTransition(item): |
| 1881 | return self.has_transition(item) |
| 1882 | return False |
| 1883 | |
| 1884 | |
| 1885 | #************************************************************************* |
| 1886 | # representations / LaTeX |
| 1887 | #************************************************************************* |
| 1888 | |
| 1889 | |
| 1890 | def _repr_(self): |
| 1891 | """ |
| 1892 | Represents the finite state machine as "finite state machine |
| 1893 | with n states" where n is the number of states. |
| 1894 | |
| 1895 | INPUT: |
| 1896 | |
| 1897 | Nothing. |
| 1898 | |
| 1899 | OUTPUT: |
| 1900 | |
| 1901 | A string. |
| 1902 | |
| 1903 | EXAMPLES:: |
| 1904 | |
| 1905 | sage: FiniteStateMachine()._repr_() |
| 1906 | 'finite state machine with 0 states' |
| 1907 | """ |
| 1908 | return "finite state machine with %s states" % len(self._states_) |
| 1909 | |
| 1910 | |
| 1911 | def _latex_(self): |
| 1912 | r""" |
| 1913 | Returns a LaTeX code for the graph of the finite state machine. |
| 1914 | |
| 1915 | INPUT: |
| 1916 | |
| 1917 | Nothing. |
| 1918 | |
| 1919 | OUTPUT: |
| 1920 | |
| 1921 | A string. |
| 1922 | |
| 1923 | EXAMPLES:: |
| 1924 | |
| 1925 | sage: F = FiniteStateMachine([('A', 'B', 1, 2)]) |
| 1926 | sage: F._latex_() |
| 1927 | '\\begin{tikzpicture}[auto]\n\\node[state] (v0) at (3.000000,0.000000) {\\text{\\texttt{A}}}\n;\\node[state] (v1) at (-3.000000,0.000000) {\\text{\\texttt{B}}}\n;\\path[->] (v0) edge node {$ $} (v1);\n\\end{tikzpicture}' |
| 1928 | """ |
| 1929 | result = "\\begin{tikzpicture}[auto]\n" |
| 1930 | j = 0; |
| 1931 | for vertex in self.states(): |
| 1932 | if not hasattr(vertex, "coordinates"): |
| 1933 | vertex.coordinates = (3*cos(2*pi*j/len(self.states())), |
| 1934 | 3*sin(2*pi*j/len(self.states()))) |
| 1935 | options = "" |
| 1936 | if vertex in self.final_states(): |
| 1937 | options += ",accepting" |
| 1938 | if hasattr(vertex, "format_label"): |
| 1939 | label = vertex.format_label() |
| 1940 | elif hasattr(self, "format_state_label"): |
| 1941 | label = self.format_state_label(vertex) |
| 1942 | else: |
| 1943 | label = latex(vertex.label()) |
| 1944 | result += "\\node[state%s] (v%d) at (%f,%f) {%s}\n;" % ( |
| 1945 | options, j, vertex.coordinates[0], |
| 1946 | vertex.coordinates[1], label) |
| 1947 | vertex._number_ = j |
| 1948 | j += 1 |
| 1949 | adjacent = {} |
| 1950 | for source in self.states(): |
| 1951 | for target in self.states(): |
| 1952 | transitions = filter(lambda transition: \ |
| 1953 | transition.to_state == target, |
| 1954 | source.transitions) |
| 1955 | adjacent[source, target] = transitions |
| 1956 | |
| 1957 | for ((source, target), transitions) in adjacent.iteritems(): |
| 1958 | if len(transitions) > 0: |
| 1959 | labels = [] |
| 1960 | for transition in transitions: |
| 1961 | if hasattr(transition, "format_label"): |
| 1962 | labels.append(transition.format_label()) |
| 1963 | continue |
| 1964 | elif hasattr(self, "format_transition_label"): |
| 1965 | format_transition_label = self.format_transition_label |
| 1966 | else: |
| 1967 | format_transition_label = latex |
| 1968 | labels.append(self._latex_transition_label_( |
| 1969 | transition, format_transition_label)) |
| 1970 | label = ", ".join(labels) |
| 1971 | if source != target: |
| 1972 | if len(adjacent[target, source]) > 0: |
| 1973 | angle = atan2( |
| 1974 | target.coordinates[1] - source.coordinates[1], |
| 1975 | target.coordinates[0]-source.coordinates[0])*180/pi |
| 1976 | angle_source = ".%.2f" % ((angle+5).n(),) |
| 1977 | angle_target = ".%.2f" % ((angle+175).n(),) |
| 1978 | else: |
| 1979 | angle_source = "" |
| 1980 | angle_target = "" |
| 1981 | result += "\\path[->] (v%d%s) edge node {$%s$} (v%d%s);\n" % ( |
| 1982 | source._number_, angle_source, label, |
| 1983 | target._number_, angle_target) |
| 1984 | else: |
| 1985 | result += "\\path[->] (v%d) edge[loop above] node {$%s$} ();\n" % ( |
| 1986 | source._number_, label) |
| 1987 | |
| 1988 | result += "\\end{tikzpicture}" |
| 1989 | return result |
| 1990 | |
| 1991 | |
| 1992 | def _latex_transition_label_(self, transition, format_function=latex): |
| 1993 | r""" |
| 1994 | Returns the proper transition label. |
| 1995 | |
| 1996 | INPUT: |
| 1997 | |
| 1998 | - ``transition`` - a transition |
| 1999 | |
| 2000 | - ``format_function'' - a function formatting the labels |
| 2001 | |
| 2002 | OUTPUT: |
| 2003 | |
| 2004 | A string. |
| 2005 | |
| 2006 | TESTS:: |
| 2007 | |
| 2008 | sage: F = FiniteStateMachine([('A', 'B', 0, 1)]) |
| 2009 | sage: t = F.transitions()[0] |
| 2010 | sage: F._latex_transition_label_(t) |
| 2011 | ' ' |
| 2012 | """ |
| 2013 | return ' ' |
| 2014 | |
| 2015 | |
| 2016 | #************************************************************************* |
| 2017 | # other |
| 2018 | #************************************************************************* |
| 2019 | |
| 2020 | |
| 2021 | def _matrix_(self, R=None): |
| 2022 | """ |
| 2023 | Returns the adjacency matrix of the finite state machine. |
| 2024 | See :meth:`.adjacency_matrix` for more information. |
| 2025 | |
| 2026 | EXAMPLES:: |
| 2027 | |
| 2028 | sage: B = FiniteStateMachine({0: {0: (0, 0), 'a': (1, 0)}, |
| 2029 | ....: 'a': {2: (0, 0), 3: (1, 0)}, |
| 2030 | ....: 2:{0:(1, 1), 4:(0, 0)}, |
| 2031 | ....: 3:{'a':(0, 1), 2:(1, 1)}, |
| 2032 | ....: 4:{4:(1, 1), 3:(0, 1)}}, |
| 2033 | ....: initial_states=[0]) |
| 2034 | sage: B._matrix_() |
| 2035 | [1 1 0 0 0] |
| 2036 | [0 0 1 1 0] |
| 2037 | [x 0 0 0 1] |
| 2038 | [0 x x 0 0] |
| 2039 | [0 0 0 x x] |
| 2040 | """ |
| 2041 | return self.adjacency_matrix() |
| 2042 | |
| 2043 | |
| 2044 | def adjacency_matrix(self, input=None, |
| 2045 | entry=(lambda transition:var('x')**transition.word_out[0])): |
| 2046 | """ |
| 2047 | Returns the adjacency matrix of the underlying graph. |
| 2048 | |
| 2049 | INPUT: |
| 2050 | |
| 2051 | - ``input`` -- Only transitions with input label ``input`` are |
| 2052 | respected. |
| 2053 | |
| 2054 | - ``entry`` -- The function ``entry`` takes a transition and |
| 2055 | the return value is written in the matrix as the entry |
| 2056 | ``(transition.from_state, transition.to_state)``. |
| 2057 | |
| 2058 | OUTPUT: |
| 2059 | |
| 2060 | A matrix. |
| 2061 | |
| 2062 | If any label of a state is not an integer, the finite state |
| 2063 | machine is relabeled at the beginning. If there are more than |
| 2064 | one transitions between two states, then the different return |
| 2065 | values of ``entry`` are added up. |
| 2066 | |
| 2067 | The default value of entry takes the variable ``x`` to the |
| 2068 | power of the output word of the transition. |
| 2069 | |
| 2070 | EXAMPLES:: |
| 2071 | |
| 2072 | sage: B = FiniteStateMachine({0:{0:(0, 0), 'a':(1, 0)}, |
| 2073 | ....: 'a':{2:(0, 0), 3:(1, 0)}, |
| 2074 | ....: 2:{0:(1, 1), 4:(0, 0)}, |
| 2075 | ....: 3:{'a':(0, 1), 2:(1, 1)}, |
| 2076 | ....: 4:{4:(1, 1), 3:(0, 1)}}, |
| 2077 | ....: initial_states=[0]) |
| 2078 | sage: B.adjacency_matrix() |
| 2079 | [1 1 0 0 0] |
| 2080 | [0 0 1 1 0] |
| 2081 | [x 0 0 0 1] |
| 2082 | [0 x x 0 0] |
| 2083 | [0 0 0 x x] |
| 2084 | sage: B.adjacency_matrix(entry=(lambda transition: 1)) |
| 2085 | [1 1 0 0 0] |
| 2086 | [0 0 1 1 0] |
| 2087 | [1 0 0 0 1] |
| 2088 | [0 1 1 0 0] |
| 2089 | [0 0 0 1 1] |
| 2090 | sage: B.adjacency_matrix(1, entry=(lambda transition: |
| 2091 | ....: exp(I*transition.word_out[0]*var('t')))) |
| 2092 | [ 0 1 0 0 0] |
| 2093 | [ 0 0 0 1 0] |
| 2094 | [e^(I*t) 0 0 0 0] |
| 2095 | [ 0 0 e^(I*t) 0 0] |
| 2096 | [ 0 0 0 0 e^(I*t)] |
| 2097 | |
| 2098 | """ |
| 2099 | relabeledFSM = self |
| 2100 | l = len(relabeledFSM.states()) |
| 2101 | for state in self.states(): |
| 2102 | if state.label() not in ZZ or state.label() >= l or state.label() < 0: |
| 2103 | relabeledFSM = self.relabeled() |
| 2104 | break |
| 2105 | dictionary = {} |
| 2106 | for transition in relabeledFSM.iter_transitions(): |
| 2107 | if input is None or transition.word_in == [input]: |
| 2108 | if (transition.from_state.label(), transition.to_state.label()) in dictionary: |
| 2109 | dictionary[(transition.from_state.label(), transition.to_state.label())] += entry(transition) |
| 2110 | else: |
| 2111 | dictionary[(transition.from_state.label(), transition.to_state.label())] = entry(transition) |
| 2112 | return matrix(len(relabeledFSM.states()), dictionary) |
| 2113 | |
| 2114 | |
| 2115 | def determine_alphabets(self, reset=True): |
| 2116 | """ |
| 2117 | Determines the input and output alphabet according to the |
| 2118 | transitions in self. |
| 2119 | |
| 2120 | INPUT: |
| 2121 | |
| 2122 | - ``reset`` -- If reset is True, then the existing input |
| 2123 | alphabet is erased, otherwise new letters are appended to |
| 2124 | the existing alphabet. |
| 2125 | |
| 2126 | OUTPUT: |
| 2127 | |
| 2128 | Nothing. |
| 2129 | |
| 2130 | After this operation the input alphabet and the output |
| 2131 | alphabet of self are a list of letters. |
| 2132 | |
| 2133 | EXAMPLES:: |
| 2134 | |
| 2135 | sage: T = Transducer([(1, 1, 1, 0), (1, 2, 2, 1), |
| 2136 | ....: (2, 2, 1, 1), (2, 2, 0, 0)], |
| 2137 | ....: determine_alphabets=False) |
| 2138 | sage: (T.input_alphabet, T.output_alphabet) |
| 2139 | (None, None) |
| 2140 | sage: T.determine_alphabets() |
| 2141 | sage: (T.input_alphabet, T.output_alphabet) |
| 2142 | ([0, 1, 2], [0, 1]) |
| 2143 | """ |
| 2144 | if reset: |
| 2145 | ain = set() |
| 2146 | aout = set() |
| 2147 | else: |
| 2148 | ain = set(self.input_alphabet) |
| 2149 | aout = set(self.output_alphabet) |
| 2150 | |
| 2151 | for t in self.iter_transitions(): |
| 2152 | for letter in t.word_in: |
| 2153 | ain.add(letter) |
| 2154 | for letter in t.word_out: |
| 2155 | aout.add(letter) |
| 2156 | self.input_alphabet = list(ain) |
| 2157 | self.output_alphabet = list(aout) |
| 2158 | |
| 2159 | |
| 2160 | #************************************************************************* |
| 2161 | # get states and transitions |
| 2162 | #************************************************************************* |
| 2163 | |
| 2164 | |
| 2165 | def states(self): |
| 2166 | """ |
| 2167 | Returns the states of the finite state machine. |
| 2168 | |
| 2169 | INPUT: |
| 2170 | |
| 2171 | Nothing. |
| 2172 | |
| 2173 | OUTPUT: |
| 2174 | |
| 2175 | The states of the finite state machine as list. |
| 2176 | |
| 2177 | EXAMPLES:: |
| 2178 | |
| 2179 | sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)]) |
| 2180 | sage: FSM.states() |
| 2181 | ['1', '2'] |
| 2182 | |
| 2183 | """ |
| 2184 | return copy(self._states_) |
| 2185 | |
| 2186 | |
| 2187 | def iter_states(self): |
| 2188 | """ |
| 2189 | Returns an iterator of the states. |
| 2190 | |
| 2191 | INPUT: |
| 2192 | |
| 2193 | Nothing. |
| 2194 | |
| 2195 | OUTPUT: |
| 2196 | |
| 2197 | An iterator of the states of the finite state machine. |
| 2198 | |
| 2199 | EXAMPLES:: |
| 2200 | |
| 2201 | sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)]) |
| 2202 | sage: [s.label() for s in FSM.iter_states()] |
| 2203 | ['1', '2'] |
| 2204 | """ |
| 2205 | return iter(self._states_) |
| 2206 | |
| 2207 | |
| 2208 | def transitions(self, from_state=None): |
| 2209 | """ |
| 2210 | Returns a list of all transitions. |
| 2211 | |
| 2212 | INPUT: |
| 2213 | |
| 2214 | - ``from_state`` -- (default: ``None``) If ``from_state`` is |
| 2215 | given, then a list of transitions starting there is given. |
| 2216 | |
| 2217 | OUTPUT: |
| 2218 | |
| 2219 | A list of all transitions. |
| 2220 | |
| 2221 | EXAMPLES:: |
| 2222 | |
| 2223 | sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)]) |
| 2224 | sage: FSM.transitions() |
| 2225 | [Transition from '1' to '2': 1|-, |
| 2226 | Transition from '2' to '2': 0|-] |
| 2227 | """ |
| 2228 | return list(self.iter_transitions(from_state)) |
| 2229 | |
| 2230 | |
| 2231 | def iter_transitions(self, from_state=None): |
| 2232 | """ |
| 2233 | Returns an iterator of all transitions. |
| 2234 | |
| 2235 | INPUT: |
| 2236 | |
| 2237 | - ``from_state`` -- (default: ``None``) If ``from_state`` is |
| 2238 | given, then a list of transitions starting there is given. |
| 2239 | |
| 2240 | OUTPUT: |
| 2241 | |
| 2242 | An iterator of all transitions. |
| 2243 | |
| 2244 | EXAMPLES:: |
| 2245 | |
| 2246 | sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)]) |
| 2247 | sage: [(t.from_state.label(), t.to_state.label()) |
| 2248 | ....: for t in FSM.iter_transitions('1')] |
| 2249 | [('1', '2')] |
| 2250 | sage: [(t.from_state.label(), t.to_state.label()) |
| 2251 | ....: for t in FSM.iter_transitions('2')] |
| 2252 | [('2', '2')] |
| 2253 | sage: [(t.from_state.label(), t.to_state.label()) |
| 2254 | ....: for t in FSM.iter_transitions()] |
| 2255 | [('1', '2'), ('2', '2')] |
| 2256 | """ |
| 2257 | if from_state is None: |
| 2258 | return self._iter_transitions_all_() |
| 2259 | else: |
| 2260 | return iter(self.state(from_state).transitions) |
| 2261 | |
| 2262 | |
| 2263 | def _iter_transitions_all_(self): |
| 2264 | """ |
| 2265 | Returns an iterator over all transitions. |
| 2266 | |
| 2267 | INPUT: |
| 2268 | |
| 2269 | Nothing. |
| 2270 | |
| 2271 | OUTPUT: |
| 2272 | |
| 2273 | An iterator over all transitions. |
| 2274 | |
| 2275 | EXAMPLES:: |
| 2276 | |
| 2277 | sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)]) |
| 2278 | sage: [(t.from_state.label(), t.to_state.label()) |
| 2279 | ....: for t in FSM._iter_transitions_all_()] |
| 2280 | [('1', '2'), ('2', '2')] |
| 2281 | """ |
| 2282 | for state in self.iter_states(): |
| 2283 | for t in state.transitions: |
| 2284 | yield t |
| 2285 | |
| 2286 | |
| 2287 | def initial_states(self): |
| 2288 | """ |
| 2289 | Returns a list of all initial states. |
| 2290 | |
| 2291 | INPUT: |
| 2292 | |
| 2293 | Nothing. |
| 2294 | |
| 2295 | OUTPUT: |
| 2296 | |
| 2297 | A list of all initial states. |
| 2298 | |
| 2299 | EXAMPLES:: |
| 2300 | |
| 2301 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2302 | sage: A = FSMState('A', is_initial=True) |
| 2303 | sage: B = FSMState('B') |
| 2304 | sage: F = FiniteStateMachine([(A, B, 1, 0)]) |
| 2305 | sage: F.initial_states() |
| 2306 | ['A'] |
| 2307 | """ |
| 2308 | return list(self.iter_initial_states()) |
| 2309 | |
| 2310 | |
| 2311 | def iter_initial_states(self): |
| 2312 | """ |
| 2313 | Returns an iterator of the initial states. |
| 2314 | |
| 2315 | INPUT: |
| 2316 | |
| 2317 | Nothing. |
| 2318 | |
| 2319 | OUTPUT: |
| 2320 | |
| 2321 | An iterator over all initial states. |
| 2322 | |
| 2323 | EXAMPLES:: |
| 2324 | |
| 2325 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2326 | sage: A = FSMState('A', is_initial=True) |
| 2327 | sage: B = FSMState('B') |
| 2328 | sage: F = FiniteStateMachine([(A, B, 1, 0)]) |
| 2329 | sage: [s.label() for s in F.iter_initial_states()] |
| 2330 | ['A'] |
| 2331 | """ |
| 2332 | return itertools.ifilter(lambda s:s.is_initial, self.iter_states()) |
| 2333 | |
| 2334 | |
| 2335 | def final_states(self): |
| 2336 | """ |
| 2337 | Returns a list of all final states. |
| 2338 | |
| 2339 | INPUT: |
| 2340 | |
| 2341 | Nothing. |
| 2342 | |
| 2343 | OUTPUT: |
| 2344 | |
| 2345 | A list of all final states. |
| 2346 | |
| 2347 | EXAMPLES:: |
| 2348 | |
| 2349 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2350 | sage: A = FSMState('A', is_final=True) |
| 2351 | sage: B = FSMState('B', is_initial=True) |
| 2352 | sage: C = FSMState('C', is_final=True) |
| 2353 | sage: F = FiniteStateMachine([(A, B), (A, C)]) |
| 2354 | sage: F.final_states() |
| 2355 | ['A', 'C'] |
| 2356 | """ |
| 2357 | return list(self.iter_final_states()) |
| 2358 | |
| 2359 | |
| 2360 | def iter_final_states(self): |
| 2361 | """ |
| 2362 | Returns an iterator of the final states. |
| 2363 | |
| 2364 | INPUT: |
| 2365 | |
| 2366 | Nothing. |
| 2367 | |
| 2368 | OUTPUT: |
| 2369 | |
| 2370 | An iterator over all initial states. |
| 2371 | |
| 2372 | EXAMPLES:: |
| 2373 | |
| 2374 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2375 | sage: A = FSMState('A', is_final=True) |
| 2376 | sage: B = FSMState('B', is_initial=True) |
| 2377 | sage: C = FSMState('C', is_final=True) |
| 2378 | sage: F = FiniteStateMachine([(A, B), (A, C)]) |
| 2379 | sage: [s.label() for s in F.iter_final_states()] |
| 2380 | ['A', 'C'] |
| 2381 | """ |
| 2382 | return itertools.ifilter(lambda s:s.is_final, self.iter_states()) |
| 2383 | |
| 2384 | |
| 2385 | def state(self, state): |
| 2386 | """ |
| 2387 | Returns the state of the finite state machine. |
| 2388 | |
| 2389 | INPUT: |
| 2390 | |
| 2391 | - ``state`` -- If ``state`` is not an instance of |
| 2392 | :class:`FSMState`, then it is assumed that it is the label |
| 2393 | of a state. |
| 2394 | |
| 2395 | OUTPUT: |
| 2396 | |
| 2397 | Returns the state of the finite state machine corresponding to |
| 2398 | ``state``. |
| 2399 | |
| 2400 | If no state is found, then a ``LookupError`` is thrown. |
| 2401 | |
| 2402 | EXAMPLES:: |
| 2403 | |
| 2404 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2405 | sage: A = FSMState('A') |
| 2406 | sage: FSM = FiniteStateMachine([(A, 'B'), ('C', A)]) |
| 2407 | sage: FSM.state('A') == A |
| 2408 | True |
| 2409 | sage: FSM.state('xyz') |
| 2410 | Traceback (most recent call last): |
| 2411 | ... |
| 2412 | LookupError: No state with label xyz found. |
| 2413 | """ |
| 2414 | def what(s, switch): |
| 2415 | if switch: |
| 2416 | return s.label() |
| 2417 | else: |
| 2418 | return s |
| 2419 | switch = is_FSMState(state) |
| 2420 | |
| 2421 | try: |
| 2422 | return self._states_dict_[what(state, switch)] |
| 2423 | except AttributeError: |
| 2424 | for s in self.iter_states(): |
| 2425 | if what(s, not switch) == state: |
| 2426 | return s |
| 2427 | except KeyError: |
| 2428 | pass |
| 2429 | raise LookupError, \ |
| 2430 | "No state with label %s found." % (what(state, switch),) |
| 2431 | |
| 2432 | |
| 2433 | def transition(self, transition): |
| 2434 | """ |
| 2435 | Returns the transition of the finite state machine. |
| 2436 | |
| 2437 | INPUT: |
| 2438 | |
| 2439 | - ``transition`` -- If ``transition`` is not an instance of |
| 2440 | :class:`FSMTransition`, then it is assumed that it is a |
| 2441 | tuple ``(from_state, to_state, word_in, word_out)``. |
| 2442 | |
| 2443 | OUTPUT: |
| 2444 | |
| 2445 | Returns the transition of the finite state machine |
| 2446 | corresponding to ``transition``. |
| 2447 | |
| 2448 | If no transition is found, then a ``LookupError`` is thrown. |
| 2449 | |
| 2450 | EXAMPLES:: |
| 2451 | |
| 2452 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 2453 | sage: t = FSMTransition('A', 'B', 0) |
| 2454 | sage: F = FiniteStateMachine([t]) |
| 2455 | sage: F.transition(('A', 'B', 0)) |
| 2456 | Transition from 'A' to 'B': 0|- |
| 2457 | sage: id(t) == id(F.transition(('A', 'B', 0))) |
| 2458 | True |
| 2459 | """ |
| 2460 | if not is_FSMTransition(transition): |
| 2461 | transition = FSMTransition(*transition) |
| 2462 | for s in self.iter_transitions(transition.from_state): |
| 2463 | if s == transition: |
| 2464 | return s |
| 2465 | raise LookupError, "No transition found." |
| 2466 | |
| 2467 | |
| 2468 | #************************************************************************* |
| 2469 | # properties (state and transitions) |
| 2470 | #************************************************************************* |
| 2471 | |
| 2472 | |
| 2473 | def has_state(self, state): |
| 2474 | """ |
| 2475 | Returns whether ``state`` is one of the states of the finite |
| 2476 | state machine. |
| 2477 | |
| 2478 | INPUT: |
| 2479 | |
| 2480 | - ``state`` can be a :class:`FSMState` or a label of a state. |
| 2481 | |
| 2482 | OUTPUT: |
| 2483 | |
| 2484 | True or False. |
| 2485 | |
| 2486 | EXAMPLES:: |
| 2487 | |
| 2488 | sage: FiniteStateMachine().has_state('A') |
| 2489 | False |
| 2490 | """ |
| 2491 | try: |
| 2492 | self.state(state) |
| 2493 | return True |
| 2494 | except LookupError: |
| 2495 | return False |
| 2496 | |
| 2497 | |
| 2498 | def has_transition(self, transition): |
| 2499 | """ |
| 2500 | Returns whether ``transition`` is one of the transitions of |
| 2501 | the finite state machine. |
| 2502 | |
| 2503 | INPUT: |
| 2504 | |
| 2505 | - ``transition`` has to be a :class:`FSMTransition`. |
| 2506 | |
| 2507 | OUTPUT: |
| 2508 | |
| 2509 | True or False. |
| 2510 | |
| 2511 | EXAMPLES:: |
| 2512 | |
| 2513 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 2514 | sage: t = FSMTransition('A', 'A', 0, 1) |
| 2515 | sage: FiniteStateMachine().has_transition(t) |
| 2516 | False |
| 2517 | sage: FiniteStateMachine().has_transition(('A', 'A', 0, 1)) |
| 2518 | Traceback (most recent call last): |
| 2519 | ... |
| 2520 | TypeError: Transition is not an instance of FSMTransition. |
| 2521 | """ |
| 2522 | if is_FSMTransition(transition): |
| 2523 | return transition in self.iter_transitions() |
| 2524 | raise TypeError, "Transition is not an instance of FSMTransition." |
| 2525 | |
| 2526 | |
| 2527 | def has_initial_state(self, state): |
| 2528 | """ |
| 2529 | Returns whether ``state`` is one of the initial states of the |
| 2530 | finite state machine. |
| 2531 | |
| 2532 | INPUT: |
| 2533 | |
| 2534 | - ``state`` can be a :class:`FSMState` or a label. |
| 2535 | |
| 2536 | OUTPUT: |
| 2537 | |
| 2538 | True or False. |
| 2539 | |
| 2540 | EXAMPLES:: |
| 2541 | |
| 2542 | sage: F = FiniteStateMachine([('A', 'A')], initial_states=['A']) |
| 2543 | sage: F.has_initial_state('A') |
| 2544 | True |
| 2545 | """ |
| 2546 | try: |
| 2547 | return self.state(state).is_initial |
| 2548 | except LookupError: |
| 2549 | return False |
| 2550 | |
| 2551 | |
| 2552 | def has_initial_states(self): |
| 2553 | """ |
| 2554 | Returns whether the finite state machine has an initial state. |
| 2555 | |
| 2556 | INPUT: |
| 2557 | |
| 2558 | Nothing. |
| 2559 | |
| 2560 | OUTPUT: |
| 2561 | |
| 2562 | True or False. |
| 2563 | |
| 2564 | EXAMPLES:: |
| 2565 | |
| 2566 | sage: FiniteStateMachine().has_initial_states() |
| 2567 | False |
| 2568 | """ |
| 2569 | return len(self.initial_states()) > 0 |
| 2570 | |
| 2571 | |
| 2572 | def has_final_state(self, state): |
| 2573 | """ |
| 2574 | Returns whether ``state`` is one of the final states of the |
| 2575 | finite state machine. |
| 2576 | |
| 2577 | INPUT: |
| 2578 | |
| 2579 | - ``state`` can be a :class:`FSMState` or a label. |
| 2580 | |
| 2581 | OUTPUT: |
| 2582 | |
| 2583 | True or False. |
| 2584 | |
| 2585 | EXAMPLES:: |
| 2586 | |
| 2587 | sage: FiniteStateMachine(final_states=['A']).has_final_state('A') |
| 2588 | True |
| 2589 | """ |
| 2590 | try: |
| 2591 | return self.state(state).is_final |
| 2592 | except LookupError: |
| 2593 | return False |
| 2594 | |
| 2595 | |
| 2596 | def has_final_states(self): |
| 2597 | """ |
| 2598 | Returns whether the finite state machine has a final state. |
| 2599 | |
| 2600 | INPUT: |
| 2601 | |
| 2602 | Nothing. |
| 2603 | |
| 2604 | OUTPUT: |
| 2605 | |
| 2606 | True or False. |
| 2607 | |
| 2608 | EXAMPLES:: |
| 2609 | |
| 2610 | sage: FiniteStateMachine().has_final_states() |
| 2611 | False |
| 2612 | """ |
| 2613 | return len(self.final_states()) > 0 |
| 2614 | |
| 2615 | |
| 2616 | #************************************************************************* |
| 2617 | # properties |
| 2618 | #************************************************************************* |
| 2619 | |
| 2620 | |
| 2621 | def is_deterministic(self): |
| 2622 | """ |
| 2623 | Returns whether the finite finite state machine is deterministic. |
| 2624 | |
| 2625 | INPUT: |
| 2626 | |
| 2627 | Nothing. |
| 2628 | |
| 2629 | OUTPUT: |
| 2630 | |
| 2631 | True or False. |
| 2632 | |
| 2633 | A finite state machine is considered to be deterministic if |
| 2634 | each transition has input label of length one and for each |
| 2635 | pair `(q,a)` where `q` is a state and `a` is an element of the |
| 2636 | input alphabet, there is at most one transition from `q` with |
| 2637 | input label `a`. |
| 2638 | |
| 2639 | TESTS:: |
| 2640 | |
| 2641 | sage: fsm = FiniteStateMachine() |
| 2642 | sage: fsm.add_transition(('A', 'B', 0, [])) |
| 2643 | Transition from 'A' to 'B': 0|- |
| 2644 | sage: fsm.is_deterministic() |
| 2645 | True |
| 2646 | sage: fsm.add_transition(('A', 'C', 0, [])) |
| 2647 | Transition from 'A' to 'C': 0|- |
| 2648 | sage: fsm.is_deterministic() |
| 2649 | False |
| 2650 | sage: fsm.add_transition(('A', 'B', [0,1], [])) |
| 2651 | Transition from 'A' to 'B': 0,1|- |
| 2652 | sage: fsm.is_deterministic() |
| 2653 | False |
| 2654 | """ |
| 2655 | for state in self.states(): |
| 2656 | for transition in state.transitions: |
| 2657 | if len(transition.word_in) != 1: |
| 2658 | return False |
| 2659 | |
| 2660 | transition_classes_by_word_in = full_group_by( |
| 2661 | state.transitions, |
| 2662 | key=lambda t:t.word_in) |
| 2663 | |
| 2664 | for key,transition_class in transition_classes_by_word_in: |
| 2665 | if len(transition_class) > 1: |
| 2666 | return False |
| 2667 | return True |
| 2668 | |
| 2669 | |
| 2670 | def is_connected(self): |
| 2671 | """ |
| 2672 | TESTS:: |
| 2673 | |
| 2674 | sage: FiniteStateMachine().is_connected() |
| 2675 | Traceback (most recent call last): |
| 2676 | ... |
| 2677 | NotImplementedError |
| 2678 | """ |
| 2679 | raise NotImplementedError |
| 2680 | |
| 2681 | |
| 2682 | #************************************************************************* |
| 2683 | # let the finite state machine work |
| 2684 | #************************************************************************* |
| 2685 | |
| 2686 | |
| 2687 | def process(self, *args, **kwargs): |
| 2688 | """ |
| 2689 | Returns whether the finite state machine accepts the input, the state |
| 2690 | where the computation stops and which output is generated. |
| 2691 | |
| 2692 | INPUT: |
| 2693 | |
| 2694 | - ``input_tape`` -- The input tape can be a list with entries from |
| 2695 | the input alphabet. |
| 2696 | |
| 2697 | - ``initial_state`` -- (default: ``None``) The state in which |
| 2698 | to start. If this parameter is ``None`` and there is only |
| 2699 | one initial state in the machine, then this state is taken. |
| 2700 | |
| 2701 | OUTPUT: |
| 2702 | |
| 2703 | A triple, where |
| 2704 | |
| 2705 | - the first entry is true if the input string is accepted, |
| 2706 | |
| 2707 | - the second gives the reached state after processing the |
| 2708 | input tape, and |
| 2709 | |
| 2710 | - the third gives a list of the output labels used during |
| 2711 | processing (in the case the finite state machine runs as |
| 2712 | transducer). |
| 2713 | |
| 2714 | Note that in the case the finite state machine is not |
| 2715 | deterministic, one possible path is gone. This means, that in |
| 2716 | this case the output can be wrong. Use |
| 2717 | :meth:`.determinisation` to get a deterministic finite state |
| 2718 | machine and try again. |
| 2719 | |
| 2720 | EXAMPLES:: |
| 2721 | |
| 2722 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2723 | sage: A = FSMState('A', is_initial = True, is_final = True) |
| 2724 | sage: binary_inverter = Transducer({A:[(A, 0, 1), (A, 1, 0)]}) |
| 2725 | sage: binary_inverter.process([0, 1, 0, 0, 1, 1]) |
| 2726 | (True, 'A', [1, 0, 1, 1, 0, 0]) |
| 2727 | |
| 2728 | Alternatively, we can invoke this function by:: |
| 2729 | |
| 2730 | sage: binary_inverter([0, 1, 0, 0, 1, 1]) |
| 2731 | (True, 'A', [1, 0, 1, 1, 0, 0]) |
| 2732 | |
| 2733 | :: |
| 2734 | |
| 2735 | sage: NAF_ = FSMState('_', is_initial = True, is_final = True) |
| 2736 | sage: NAF1 = FSMState('1', is_final = True) |
| 2737 | sage: NAF = Automaton( |
| 2738 | ....: {NAF_: [(NAF_, 0), (NAF1, 1)], NAF1: [(NAF_, 0)]}) |
| 2739 | sage: [NAF.process(w)[0] for w in [[0], [0, 1], [1, 1], [0, 1, 0, 1], |
| 2740 | ....: [0, 1, 1, 1, 0], [1, 0, 0, 1, 1]]] |
| 2741 | [True, True, False, True, False, False] |
| 2742 | |
| 2743 | """ |
| 2744 | it = self.iter_process(*args, **kwargs) |
| 2745 | for _ in it: |
| 2746 | pass |
| 2747 | return (it.accept_input, it.current_state, it.output_tape) |
| 2748 | |
| 2749 | |
| 2750 | def iter_process(self, input_tape=None, initial_state=None): |
| 2751 | """ |
| 2752 | See `process` for more informations. |
| 2753 | |
| 2754 | EXAMPLES:: |
| 2755 | |
| 2756 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 2757 | ....: initial_states=['A'], final_states=['A']) |
| 2758 | sage: it = inverter.iter_process(input_tape=[0, 1, 1]) |
| 2759 | sage: for _ in it: |
| 2760 | ....: pass |
| 2761 | sage: it.output_tape |
| 2762 | [1, 0, 0] |
| 2763 | """ |
| 2764 | return FSMProcessIterator(self, input_tape, initial_state) |
| 2765 | |
| 2766 | |
| 2767 | #************************************************************************* |
| 2768 | # change finite state machine (add/remove state/transitions) |
| 2769 | #************************************************************************* |
| 2770 | |
| 2771 | |
| 2772 | def add_state(self, state): |
| 2773 | """ |
| 2774 | Adds a state to the finite state machine and returns the new |
| 2775 | state. If the state already exists, that existing state is |
| 2776 | returned. |
| 2777 | |
| 2778 | INPUT: |
| 2779 | |
| 2780 | - ``state`` is either an instance of FSMState or, otherwise, a |
| 2781 | label of a state. |
| 2782 | |
| 2783 | OUTPUT: |
| 2784 | |
| 2785 | The new or existing state. |
| 2786 | |
| 2787 | EXAMPLES:: |
| 2788 | |
| 2789 | sage: from sage.combinat.finite_state_machine import FSMState |
| 2790 | sage: F = FiniteStateMachine() |
| 2791 | sage: A = FSMState('A', is_initial=True) |
| 2792 | sage: F.add_state(A) |
| 2793 | 'A' |
| 2794 | """ |
| 2795 | try: |
| 2796 | return self.state(state) |
| 2797 | except LookupError: |
| 2798 | pass |
| 2799 | # at this point we know that we have a new state |
| 2800 | if is_FSMState(state): |
| 2801 | s = state |
| 2802 | else: |
| 2803 | s = FSMState(state) |
| 2804 | s.transitions = list() |
| 2805 | self._states_.append(s) |
| 2806 | try: |
| 2807 | self._states_dict_[s.label()] = s |
| 2808 | except AttributeError: |
| 2809 | pass |
| 2810 | return s |
| 2811 | |
| 2812 | |
| 2813 | def add_states(self, states): |
| 2814 | """ |
| 2815 | Adds several states. See add_state for more information. |
| 2816 | |
| 2817 | INPUT: |
| 2818 | |
| 2819 | - ``states`` -- a list of states or iterator over states. |
| 2820 | |
| 2821 | OUTPUT: |
| 2822 | |
| 2823 | Nothing. |
| 2824 | |
| 2825 | EXAMPLES:: |
| 2826 | |
| 2827 | sage: F = FiniteStateMachine() |
| 2828 | sage: F.add_states(['A', 'B']) |
| 2829 | sage: F.states() |
| 2830 | ['A', 'B'] |
| 2831 | """ |
| 2832 | for state in states: |
| 2833 | self.add_state(state) |
| 2834 | |
| 2835 | |
| 2836 | def add_transition(self, *args, **kwargs): |
| 2837 | """ |
| 2838 | Adds a transition to the finite state machine and returns the |
| 2839 | new transition. If the transition already exists, that |
| 2840 | existing transition is returned. |
| 2841 | |
| 2842 | INPUT: |
| 2843 | |
| 2844 | The following forms are all accepted: |
| 2845 | |
| 2846 | :: |
| 2847 | |
| 2848 | sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition |
| 2849 | sage: A = FSMState('A') |
| 2850 | sage: B = FSMState('B') |
| 2851 | |
| 2852 | sage: FSM = FiniteStateMachine() |
| 2853 | sage: FSM.add_transition(FSMTransition(A, B, 0, 1)) |
| 2854 | Transition from 'A' to 'B': 0|1 |
| 2855 | |
| 2856 | sage: FSM = FiniteStateMachine() |
| 2857 | sage: FSM.add_transition(A, B, 0, 1) |
| 2858 | Transition from 'A' to 'B': 0|1 |
| 2859 | |
| 2860 | sage: FSM = FiniteStateMachine() |
| 2861 | sage: FSM.add_transition(A, B, word_in=0, word_out=1) |
| 2862 | Transition from 'A' to 'B': 0|1 |
| 2863 | |
| 2864 | sage: FSM = FiniteStateMachine() |
| 2865 | sage: FSM.add_transition('A', 'B', {'word_in': 0, 'word_out': 1}) |
| 2866 | Transition from 'A' to 'B': {'word_in': 0, 'word_out': 1}|- |
| 2867 | |
| 2868 | sage: FSM = FiniteStateMachine() |
| 2869 | sage: FSM.add_transition(from_state=A, to_state=B, |
| 2870 | ....: word_in=0, word_out=1) |
| 2871 | Transition from 'A' to 'B': 0|1 |
| 2872 | |
| 2873 | sage: FSM = FiniteStateMachine() |
| 2874 | sage: FSM.add_transition({'from_state': A, 'to_state': B, |
| 2875 | ....: 'word_in': 0, 'word_out': 1}) |
| 2876 | Transition from 'A' to 'B': 0|1 |
| 2877 | |
| 2878 | sage: FSM = FiniteStateMachine() |
| 2879 | sage: FSM.add_transition((A, B, 0, 1)) |
| 2880 | Transition from 'A' to 'B': 0|1 |
| 2881 | |
| 2882 | sage: FSM = FiniteStateMachine() |
| 2883 | sage: FSM.add_transition([A, B, 0, 1]) |
| 2884 | Transition from 'A' to 'B': 0|1 |
| 2885 | |
| 2886 | If the states ``A`` and ``B`` are not instances of FSMState, then |
| 2887 | it is assumed that they are labels of states. |
| 2888 | |
| 2889 | OUTPUT: |
| 2890 | |
| 2891 | The new or existing transition. |
| 2892 | """ |
| 2893 | if len(args) + len(kwargs) == 0: |
| 2894 | return |
| 2895 | if len(args) + len(kwargs) == 1: |
| 2896 | if len(args) == 1: |
| 2897 | d = args[0] |
| 2898 | if is_FSMTransition(d): |
| 2899 | return self._add_fsm_transition_(d) |
| 2900 | else: |
| 2901 | d = kwargs.itervalues().next() |
| 2902 | if hasattr(d, 'iteritems'): |
| 2903 | args = [] |
| 2904 | kwargs = d |
| 2905 | elif hasattr(d, '__iter__'): |
| 2906 | args = d |
| 2907 | kwargs = {} |
| 2908 | else: |
| 2909 | raise TypeError, "Cannot decide what to do with input." |
| 2910 | |
| 2911 | data = dict(zip( |
| 2912 | ('from_state', 'to_state', 'word_in', 'word_out', 'hook'), |
| 2913 | args)) |
| 2914 | data.update(kwargs) |
| 2915 | |
| 2916 | data['from_state'] = self.add_state(data['from_state']) |
| 2917 | data['to_state'] = self.add_state(data['to_state']) |
| 2918 | |
| 2919 | return self._add_fsm_transition_(FSMTransition(**data)) |
| 2920 | |
| 2921 | |
| 2922 | def _add_fsm_transition_(self, t): |
| 2923 | """ |
| 2924 | Adds a transition. |
| 2925 | |
| 2926 | INPUT: |
| 2927 | |
| 2928 | - ``t`` -- an instance of ``FSMTransition``. |
| 2929 | |
| 2930 | OUTPUT: |
| 2931 | |
| 2932 | The new transition. |
| 2933 | |
| 2934 | TESTS:: |
| 2935 | |
| 2936 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 2937 | sage: F = FiniteStateMachine() |
| 2938 | sage: F._add_fsm_transition_(FSMTransition('A', 'B')) |
| 2939 | Transition from 'A' to 'B': -|- |
| 2940 | """ |
| 2941 | try: |
| 2942 | return self.transition(t) |
| 2943 | except LookupError: |
| 2944 | pass |
| 2945 | from_state = self.add_state(t.from_state) |
| 2946 | self.add_state(t.to_state) |
| 2947 | from_state.transitions.append(t) |
| 2948 | return t |
| 2949 | |
| 2950 | |
| 2951 | def add_from_transition_function(self, function, initial_states=None, |
| 2952 | explore_existing_states=True): |
| 2953 | """ |
| 2954 | Constructs a finite state machine from a transition function. |
| 2955 | |
| 2956 | INPUT: |
| 2957 | |
| 2958 | - ``function`` may return a tuple (new_state, output_word) or a |
| 2959 | list of such tuples. |
| 2960 | |
| 2961 | - ``initial_states`` -- If no initial states are given, the |
| 2962 | already existing initial states of self are taken. |
| 2963 | |
| 2964 | - If ``explore_existing_states`` is True (default), then |
| 2965 | already existing states in self (e.g. already given final |
| 2966 | states) will also be processed if they are reachable from |
| 2967 | the initial states. |
| 2968 | |
| 2969 | OUTPUT: |
| 2970 | |
| 2971 | Nothing. |
| 2972 | |
| 2973 | EXAMPLES:: |
| 2974 | |
| 2975 | sage: F = FiniteStateMachine(initial_states=['A'], |
| 2976 | ....: input_alphabet=[0, 1]) |
| 2977 | sage: def f(state, input): |
| 2978 | ....: return [('A', input), ('B', 1-input)] |
| 2979 | sage: F.add_from_transition_function(f) |
| 2980 | sage: F.transitions() |
| 2981 | [Transition from 'A' to 'A': 0|0, |
| 2982 | Transition from 'A' to 'B': 0|1, |
| 2983 | Transition from 'A' to 'A': 1|1, |
| 2984 | Transition from 'A' to 'B': 1|0, |
| 2985 | Transition from 'B' to 'A': 0|0, |
| 2986 | Transition from 'B' to 'B': 0|1, |
| 2987 | Transition from 'B' to 'A': 1|1, |
| 2988 | Transition from 'B' to 'B': 1|0] |
| 2989 | |
| 2990 | Initial states can also be given as a parameter:: |
| 2991 | |
| 2992 | sage: F = FiniteStateMachine(input_alphabet=[0,1]) |
| 2993 | sage: def f(state, input): |
| 2994 | ....: return [('A', input), ('B', 1-input)] |
| 2995 | sage: F.add_from_transition_function(f,initial_states=['A']) |
| 2996 | sage: F.initial_states() |
| 2997 | ['A'] |
| 2998 | |
| 2999 | Already existing states in the finite state machine (the final |
| 3000 | states in the example below) are also explored:: |
| 3001 | |
| 3002 | sage: F = FiniteStateMachine(initial_states=[0], |
| 3003 | ....: final_states=[1], |
| 3004 | ....: input_alphabet=[0]) |
| 3005 | sage: def transition_function(state, letter): |
| 3006 | ....: return(1-state, []) |
| 3007 | sage: F.add_from_transition_function(transition_function) |
| 3008 | sage: F.transitions() |
| 3009 | [Transition from 0 to 1: 0|-, |
| 3010 | Transition from 1 to 0: 0|-] |
| 3011 | |
| 3012 | If ``explore_existing_states=False``, however, this behavior |
| 3013 | is turned off, i.e., already existing states are not |
| 3014 | explored:: |
| 3015 | |
| 3016 | sage: F = FiniteStateMachine(initial_states=[0], |
| 3017 | ....: final_states=[1], |
| 3018 | ....: input_alphabet=[0]) |
| 3019 | sage: def transition_function(state, letter): |
| 3020 | ....: return(1-state, []) |
| 3021 | sage: F.add_from_transition_function(transition_function, |
| 3022 | ....: explore_existing_states=False) |
| 3023 | sage: F.transitions() |
| 3024 | [Transition from 0 to 1: 0|-] |
| 3025 | |
| 3026 | TEST:: |
| 3027 | |
| 3028 | sage: F = FiniteStateMachine(initial_states=['A']) |
| 3029 | sage: def f(state, input): |
| 3030 | ....: return [('A', input), ('B', 1-input)] |
| 3031 | sage: F.add_from_transition_function(f) |
| 3032 | Traceback (most recent call last): |
| 3033 | ... |
| 3034 | ValueError: No input alphabet is given. |
| 3035 | Try calling determine_alphabets(). |
| 3036 | """ |
| 3037 | if self.input_alphabet is None: |
| 3038 | raise ValueError, ("No input alphabet is given. " |
| 3039 | "Try calling determine_alphabets().") |
| 3040 | |
| 3041 | if initial_states is None: |
| 3042 | not_done = self.initial_states() |
| 3043 | elif hasattr(initial_states, '__iter__'): |
| 3044 | not_done = [] |
| 3045 | for s in initial_states: |
| 3046 | state = self.add_state(s) |
| 3047 | state.is_initial = True |
| 3048 | not_done.append(state) |
| 3049 | else: |
| 3050 | raise TypeError, 'Initial states must be iterable ' \ |
| 3051 | '(e.g. a list of states).' |
| 3052 | if len(not_done) == 0: |
| 3053 | raise ValueError, "No state is initial." |
| 3054 | if explore_existing_states: |
| 3055 | ignore_done = self.states() |
| 3056 | for s in not_done: |
| 3057 | try: |
| 3058 | ignore_done.remove(s) |
| 3059 | except ValueError: |
| 3060 | pass |
| 3061 | else: |
| 3062 | ignore_done = [] |
| 3063 | while len(not_done) > 0: |
| 3064 | s = not_done.pop(0) |
| 3065 | for letter in self.input_alphabet: |
| 3066 | try: |
| 3067 | return_value = function(s.label(), letter) |
| 3068 | except LookupError: |
| 3069 | continue |
| 3070 | if not hasattr(return_value, "pop"): |
| 3071 | return_value = [return_value] |
| 3072 | try: |
| 3073 | for (st_label, word) in return_value: |
| 3074 | if not self.has_state(st_label): |
| 3075 | not_done.append(self.add_state(st_label)) |
| 3076 | elif len(ignore_done) > 0: |
| 3077 | u = self.state(st_label) |
| 3078 | if u in ignore_done: |
| 3079 | not_done.append(u) |
| 3080 | ignore_done.remove(u) |
| 3081 | self.add_transition(s, st_label, |
| 3082 | word_in=letter, word_out=word) |
| 3083 | except TypeError: |
| 3084 | raise ValueError("The callback function for add_from_transition is expected to return a pair (new_state, output_label) or a list of such pairs. For the state %s and the input letter %s, it however returned %s, which is not acceptable." % (s.label(), letter, return_value)) |
| 3085 | |
| 3086 | |
| 3087 | def add_transitions_from_function(self, function, labels_as_input=True): |
| 3088 | """ |
| 3089 | Adds a transition if ``function(state, state)`` says that there is one. |
| 3090 | |
| 3091 | INPUT: |
| 3092 | |
| 3093 | - ``function`` -- a transition function. Given two states |
| 3094 | ``from_state`` and ``to_state`` (or their labels, if |
| 3095 | ``label_as_input`` is true), this function shall return a |
| 3096 | tuple ``(word_in, word_out)`` to add a transition from |
| 3097 | ``from_state`` to ``to_state`` with input and output labels |
| 3098 | ``word_in`` and ``word_out``, respectively. If no such |
| 3099 | addition is to be added, the transition function shall |
| 3100 | return ``None``. |
| 3101 | |
| 3102 | - ``label_as_input`` -- (default: ``True``) |
| 3103 | |
| 3104 | OUTPUT: |
| 3105 | |
| 3106 | Nothing. |
| 3107 | |
| 3108 | EXAMPLES:: |
| 3109 | |
| 3110 | sage: F = FiniteStateMachine() |
| 3111 | sage: F.add_states(['A', 'B', 'C']) |
| 3112 | sage: def f(state1, state2): |
| 3113 | ....: if state1 == 'C': |
| 3114 | ....: return None |
| 3115 | ....: return (0, 1) |
| 3116 | sage: F.add_transitions_from_function(f) |
| 3117 | sage: len(F.transitions()) |
| 3118 | 6 |
| 3119 | """ |
| 3120 | for s_from in self.iter_states(): |
| 3121 | for s_to in self.iter_states(): |
| 3122 | if labels_as_input: |
| 3123 | t = function(s_from.label(), s_to.label()) |
| 3124 | else: |
| 3125 | t = function(s_from, s_to) |
| 3126 | if hasattr(t, '__getitem__'): |
| 3127 | label_in = t[0] |
| 3128 | try: |
| 3129 | label_out = t[1] |
| 3130 | except LookupError: |
| 3131 | label_out = None |
| 3132 | self.add_transition(s_from, s_to, label_in, label_out) |
| 3133 | |
| 3134 | |
| 3135 | def delete_transition(self, t): |
| 3136 | """ |
| 3137 | Deletes a transition by removing it from the list of transitions of |
| 3138 | the state, where the transition starts. |
| 3139 | |
| 3140 | INPUT: |
| 3141 | |
| 3142 | - ``t`` -- a transition. |
| 3143 | |
| 3144 | OUTPUT: |
| 3145 | |
| 3146 | Nothing. |
| 3147 | |
| 3148 | EXAMPLES:: |
| 3149 | |
| 3150 | sage: F = FiniteStateMachine([('A', 'B', 0), ('B', 'A', 1)]) |
| 3151 | sage: F.delete_transition(('A', 'B', 0)) |
| 3152 | sage: F.transitions() |
| 3153 | [Transition from 'B' to 'A': 1|-] |
| 3154 | """ |
| 3155 | transition = self.transition(t) |
| 3156 | transition.from_state.transitions.remove(transition) |
| 3157 | |
| 3158 | |
| 3159 | def delete_state(self, s): |
| 3160 | """ |
| 3161 | Deletes a state and all transitions coming or going to this state. |
| 3162 | |
| 3163 | INPUT: |
| 3164 | |
| 3165 | - ``s`` -- s has to be a label of a state or :class:`FSMState`. |
| 3166 | |
| 3167 | OUTPUT: |
| 3168 | |
| 3169 | Nothing. |
| 3170 | |
| 3171 | EXAMPLES:: |
| 3172 | |
| 3173 | sage: from sage.combinat.finite_state_machine import FSMTransition |
| 3174 | sage: t1 = FSMTransition('A', 'B', 0) |
| 3175 | sage: t2 = FSMTransition('B', 'B', 1) |
| 3176 | sage: F = FiniteStateMachine([t1, t2]) |
| 3177 | sage: F.delete_state('A') |
| 3178 | sage: F. transitions() |
| 3179 | [Transition from 'B' to 'B': 1|-] |
| 3180 | """ |
| 3181 | state = self.state(s) |
| 3182 | for transition in self.transitions(): |
| 3183 | if transition.to_state == state: |
| 3184 | self.delete_transition(transition) |
| 3185 | self._states_.remove(state) |
| 3186 | |
| 3187 | |
| 3188 | def remove_epsilon_transitions(self): |
| 3189 | """ |
| 3190 | TESTS:: |
| 3191 | |
| 3192 | sage: FiniteStateMachine().remove_epsilon_transitions() |
| 3193 | Traceback (most recent call last): |
| 3194 | ... |
| 3195 | NotImplementedError |
| 3196 | """ |
| 3197 | raise NotImplementedError |
| 3198 | |
| 3199 | |
| 3200 | def accessible_components(self): |
| 3201 | """ |
| 3202 | Returns a new finite state machine with the accessible states |
| 3203 | of self and all transitions between those states. |
| 3204 | |
| 3205 | INPUT: |
| 3206 | |
| 3207 | Nothing. |
| 3208 | |
| 3209 | OUTPUT: |
| 3210 | |
| 3211 | A finite state machine with the accessible states of self and |
| 3212 | all transitions between those states. |
| 3213 | |
| 3214 | A state is accessible if there is a directed path from an |
| 3215 | initial state to the state. If self has no initial states then |
| 3216 | a copy of the finite state machine self is returned. |
| 3217 | |
| 3218 | EXAMPLES:: |
| 3219 | |
| 3220 | sage: F = Automaton([(0, 0, 0), (0, 1, 1), (1, 1, 0), (1, 0, 1)], |
| 3221 | ....: initial_states=[0]) |
| 3222 | sage: F.accessible_components() |
| 3223 | finite state machine with 2 states |
| 3224 | |
| 3225 | :: |
| 3226 | |
| 3227 | sage: F = Automaton([(0, 0, 1), (0, 0, 1), (1, 1, 0), (1, 0, 1)], |
| 3228 | ....: initial_states=[0]) |
| 3229 | sage: F.accessible_components() |
| 3230 | finite state machine with 1 states |
| 3231 | """ |
| 3232 | if len(self.initial_states()) == 0: |
| 3233 | return deepcopy(self) |
| 3234 | |
| 3235 | memo = {} |
| 3236 | def accessible(sf, read): |
| 3237 | trans = filter(lambda x: x.word_in[0] == read, |
| 3238 | self.transitions(sf)) |
| 3239 | return map(lambda x: (deepcopy(x.to_state, memo), x.word_out), |
| 3240 | trans) |
| 3241 | |
| 3242 | new_initial_states=map(lambda x: deepcopy(x, memo), |
| 3243 | self.initial_states()) |
| 3244 | result = self.empty_copy() |
| 3245 | result.add_from_transition_function(accessible, |
| 3246 | initial_states=new_initial_states) |
| 3247 | for final_state in self.iter_final_states(): |
| 3248 | try: |
| 3249 | new_final_state=result.state(final_state.label) |
| 3250 | new_final_state.is_final=True |
| 3251 | except LookupError: |
| 3252 | pass |
| 3253 | return result |
| 3254 | |
| 3255 | |
| 3256 | # ************************************************************************* |
| 3257 | # creating new finite state machines |
| 3258 | # ************************************************************************* |
| 3259 | |
| 3260 | |
| 3261 | def disjoint_union(self, other): |
| 3262 | """ |
| 3263 | TESTS:: |
| 3264 | |
| 3265 | sage: F = FiniteStateMachine([('A', 'A')]) |
| 3266 | sage: FiniteStateMachine().disjoint_union(F) |
| 3267 | Traceback (most recent call last): |
| 3268 | ... |
| 3269 | NotImplementedError |
| 3270 | """ |
| 3271 | raise NotImplementedError |
| 3272 | |
| 3273 | |
| 3274 | def concatenation(self, other): |
| 3275 | """ |
| 3276 | TESTS:: |
| 3277 | |
| 3278 | sage: F = FiniteStateMachine([('A', 'A')]) |
| 3279 | sage: FiniteStateMachine().concatenation(F) |
| 3280 | Traceback (most recent call last): |
| 3281 | ... |
| 3282 | NotImplementedError |
| 3283 | """ |
| 3284 | raise NotImplementedError |
| 3285 | |
| 3286 | |
| 3287 | def Kleene_closure(self): |
| 3288 | """ |
| 3289 | TESTS:: |
| 3290 | |
| 3291 | sage: FiniteStateMachine().Kleene_closure() |
| 3292 | Traceback (most recent call last): |
| 3293 | ... |
| 3294 | NotImplementedError |
| 3295 | """ |
| 3296 | raise NotImplementedError |
| 3297 | |
| 3298 | |
| 3299 | def intersection(self, other): |
| 3300 | """ |
| 3301 | TESTS:: |
| 3302 | |
| 3303 | sage: F = FiniteStateMachine([('A', 'A')]) |
| 3304 | sage: FiniteStateMachine().intersection(F) |
| 3305 | Traceback (most recent call last): |
| 3306 | ... |
| 3307 | NotImplementedError |
| 3308 | """ |
| 3309 | raise NotImplementedError |
| 3310 | |
| 3311 | |
| 3312 | def product_FiniteStateMachine(self, other, function, |
| 3313 | new_input_alphabet=None, |
| 3314 | only_accessible_components=True): |
| 3315 | """ |
| 3316 | Returns a new finite state machine whose states are |
| 3317 | pairs of states of the original finite state machines. |
| 3318 | |
| 3319 | INPUT: |
| 3320 | |
| 3321 | - ``other`` -- a finite state machine. |
| 3322 | |
| 3323 | - ``function`` has to accept two transitions from `A` to `B` |
| 3324 | and `C` to `D` and returns a pair ``(word_in, word_out)`` |
| 3325 | which is the label of the transition `(A, C)` to `(B, |
| 3326 | D)`. If there is no transition from `(A, C)` to `(B, D)`, |
| 3327 | then ``function`` should raise a ``LookupError``. |
| 3328 | |
| 3329 | - ``new_input_alphabet`` (optional)-- the new input alphabet |
| 3330 | as a list. |
| 3331 | |
| 3332 | - ``only_accessible_components`` -- If true (default), then |
| 3333 | the result is piped through ``accessible_components``. If no |
| 3334 | ``new_input_alphabet`` is given, it is determined by |
| 3335 | ``determine_alphabets``. |
| 3336 | |
| 3337 | OUTPUT: |
| 3338 | |
| 3339 | A finite state machine whose states are pairs of states of the |
| 3340 | original finite state machines. |
| 3341 | |
| 3342 | The labels of the transitions are defined by ``function``. |
| 3343 | |
| 3344 | EXAMPLES:: |
| 3345 | |
| 3346 | sage: F = Automaton([('A', 'B', 1), ('A', 'A', 0), ('B', 'A', 2)], |
| 3347 | ....: initial_states=['A'], final_states=['B'], |
| 3348 | ....: determine_alphabets=True) |
| 3349 | sage: G = Automaton([(1, 1, 1)], initial_states=[1], final_states=[1]) |
| 3350 | sage: def addition(transition1, transition2): |
| 3351 | ....: return (transition1.word_in[0] + transition2.word_in[0], |
| 3352 | ....: None) |
| 3353 | sage: H = F.product_FiniteStateMachine(G, addition, [0, 1, 2, 3], only_accessible_components=False) |
| 3354 | sage: H.transitions() |
| 3355 | [Transition from ('A', 1) to ('B', 1): 2|-, |
| 3356 | Transition from ('A', 1) to ('A', 1): 1|-, |
| 3357 | Transition from ('B', 1) to ('A', 1): 3|-] |
| 3358 | sage: H1 = F.product_FiniteStateMachine(G, addition, [0, 1, 2, 3], only_accessible_components=False) |
| 3359 | sage: H1.states()[0].label()[0] is F.states()[0] |
| 3360 | True |
| 3361 | sage: H1.states()[0].label()[1] is G.states()[0] |
| 3362 | True |
| 3363 | |
| 3364 | :: |
| 3365 | |
| 3366 | sage: F = Automaton([(0,1,1/4), (0,0,3/4), (1,1,3/4), (1,0,1/4)], |
| 3367 | ....: initial_states=[0] ) |
| 3368 | sage: G = Automaton([(0,0,1), (1,1,3/4), (1,0,1/4)], |
| 3369 | ....: initial_states=[0] ) |
| 3370 | sage: H = F.product_FiniteStateMachine( |
| 3371 | ....: G, lambda t1,t2: (t1.word_in[0]*t2.word_in[0], None)) |
| 3372 | sage: H.states() |
| 3373 | [(0, 0), (1, 0)] |
| 3374 | |
| 3375 | :: |
| 3376 | |
| 3377 | sage: F = Automaton([(0,1,1/4), (0,0,3/4), (1,1,3/4), (1,0,1/4)], |
| 3378 | ....: initial_states=[0] ) |
| 3379 | sage: G = Automaton([(0,0,1), (1,1,3/4), (1,0,1/4)], |
| 3380 | ....: initial_states=[0] ) |
| 3381 | sage: H = F.product_FiniteStateMachine(G, |
| 3382 | ....: lambda t1,t2: (t1.word_in[0]*t2.word_in[0], None), |
| 3383 | ....: only_accessible_components=False) |
| 3384 | sage: H.states() |
| 3385 | [(0, 0), (1, 0), (0, 1), (1, 1)] |
| 3386 | """ |
| 3387 | result = self.empty_copy() |
| 3388 | if new_input_alphabet is not None: |
| 3389 | result.input_alphabet = new_input_alphabet |
| 3390 | else: |
| 3391 | result.input_alphabet = None |
| 3392 | |
| 3393 | for transition1 in self.transitions(): |
| 3394 | for transition2 in other.transitions(): |
| 3395 | try: |
| 3396 | word = function(transition1, transition2) |
| 3397 | except LookupError: |
| 3398 | continue |
| 3399 | result.add_transition((transition1.from_state, |
| 3400 | transition2.from_state), |
| 3401 | (transition1.to_state, |
| 3402 | transition2.to_state), |
| 3403 | word[0], word[1]) |
| 3404 | for state in result.states(): |
| 3405 | if all(map(lambda s: s.is_initial, state.label())): |
| 3406 | state.is_initial = True |
| 3407 | if all(map(lambda s: s.is_final, state.label())): |
| 3408 | state.is_final = True |
| 3409 | |
| 3410 | if only_accessible_components: |
| 3411 | if new_input_alphabet is None: |
| 3412 | result.determine_alphabets() |
| 3413 | return result.accessible_components() |
| 3414 | else: |
| 3415 | return result |
| 3416 | |
| 3417 | |
| 3418 | def cartesian_product(self, other, only_accessible_components=True): |
| 3419 | """ |
| 3420 | Returns a new finite state machine, which is the cartesian |
| 3421 | product of self and other. |
| 3422 | |
| 3423 | INPUT: |
| 3424 | |
| 3425 | - ``other`` -- a finite state machine. |
| 3426 | |
| 3427 | - ``only_accessible_components`` |
| 3428 | |
| 3429 | OUTPUT: |
| 3430 | |
| 3431 | A new finite state machine, which is the cartesian product of |
| 3432 | self and other. |
| 3433 | |
| 3434 | The set of states of the new automaton is the cartesian |
| 3435 | product of the set of states of both given automata. There is |
| 3436 | a transition `((A, B), (C, D), a)` in the new automaton if |
| 3437 | there are transitions `(A, C, a)` and `(B, C, a)` in the old |
| 3438 | automata. |
| 3439 | |
| 3440 | EXAMPLES:: |
| 3441 | |
| 3442 | sage: aut1 = Automaton([('1', '2', 1), ('2', '2', 1), ('2', '2', 0)], |
| 3443 | ....: initial_states=['1'], final_states=['2'], |
| 3444 | ....: determine_alphabets=True) |
| 3445 | sage: aut2 = Automaton([('A', 'A', 1), ('A', 'B', 1), |
| 3446 | ....: ('B', 'B', 0), ('B', 'A', 0)], |
| 3447 | ....: initial_states=['A'], final_states=['B'], |
| 3448 | ....: determine_alphabets=True) |
| 3449 | sage: res = aut1.cartesian_product(aut2) |
| 3450 | sage: res.transitions() |
| 3451 | [Transition from ('1', 'A') to ('2', 'A'): 1|-, |
| 3452 | Transition from ('1', 'A') to ('2', 'B'): 1|-, |
| 3453 | Transition from ('2', 'A') to ('2', 'A'): 1|-, |
| 3454 | Transition from ('2', 'A') to ('2', 'B'): 1|-, |
| 3455 | Transition from ('2', 'B') to ('2', 'B'): 0|-, |
| 3456 | Transition from ('2', 'B') to ('2', 'A'): 0|-] |
| 3457 | """ |
| 3458 | def function(transition1, transition2): |
| 3459 | if transition1.word_in == transition2.word_in \ |
| 3460 | and transition1.word_out == transition2.word_out: |
| 3461 | return (transition1.word_in, transition1.word_out) |
| 3462 | else: |
| 3463 | raise LookupError |
| 3464 | |
| 3465 | return self.product_FiniteStateMachine( |
| 3466 | other, function, |
| 3467 | only_accessible_components = only_accessible_components) |
| 3468 | |
| 3469 | |
| 3470 | def composition(self, other, algorithm=None, |
| 3471 | only_accessible_components=True): |
| 3472 | """ |
| 3473 | Returns a new transducer which is the composition of self and |
| 3474 | other. |
| 3475 | |
| 3476 | INPUT: |
| 3477 | |
| 3478 | - ``other`` -- a transducer |
| 3479 | |
| 3480 | - ``algorithm`` -- can be one of the following |
| 3481 | |
| 3482 | - ``direct`` -- The composition is calculated directly. |
| 3483 | |
| 3484 | There can be arbitrarily many initial and final states, |
| 3485 | but the input and output labels must have length 1. |
| 3486 | |
| 3487 | |
| 3488 | WARNING: The output of other is fed into self. |
| 3489 | |
| 3490 | - ``explorative`` -- An explorative algorithm is used. |
| 3491 | |
| 3492 | At least the following restrictions apply, but are not |
| 3493 | checked: |
| 3494 | - both self and other have exactly one initial state |
| 3495 | - all input labels of transitions have length exactly 1 |
| 3496 | |
| 3497 | The input alphabet of self has to be specified. |
| 3498 | |
| 3499 | This is a very limited implementation of composition. |
| 3500 | WARNING: The output of ``other`` is fed into ``self``. |
| 3501 | |
| 3502 | If algorithm is ``None``, then the algorithm is chosen |
| 3503 | automatically (at the moment always ``direct``). |
| 3504 | |
| 3505 | OUTPUT: |
| 3506 | |
| 3507 | A new transducer. |
| 3508 | |
| 3509 | The labels of the new finite state machine are pairs |
| 3510 | of states of the original finite state machines. |
| 3511 | |
| 3512 | EXAMPLES:: |
| 3513 | |
| 3514 | sage: F = Transducer([('A', 'B', 1, 0), ('B', 'A', 0, 1)], |
| 3515 | ....: initial_states=['A', 'B'], final_states=['B'], |
| 3516 | ....: determine_alphabets=True) |
| 3517 | sage: G = Transducer([(1, 1, 1, 0), (1, 2, 0, 1), |
| 3518 | ....: (2, 2, 1, 1), (2, 2, 0, 0)], |
| 3519 | ....: initial_states=[1], final_states=[2], |
| 3520 | ....: determine_alphabets=True) |
| 3521 | sage: Hd = F.composition(G, algorithm='direct') |
| 3522 | sage: Hd.initial_states() |
| 3523 | [(1, 'B'), (1, 'A')] |
| 3524 | sage: Hd.transitions() |
| 3525 | [Transition from (1, 'B') to (1, 'A'): 1|1, |
| 3526 | Transition from (1, 'A') to (2, 'B'): 0|0, |
| 3527 | Transition from (2, 'B') to (2, 'A'): 0|1, |
| 3528 | Transition from (2, 'A') to (2, 'B'): 1|0] |
| 3529 | |
| 3530 | :: |
| 3531 | |
| 3532 | sage: F = Transducer([('A', 'B', 1, [1, 0]), ('B', 'B', 1, 1), |
| 3533 | ....: ('B', 'B', 0, 0)], |
| 3534 | ....: initial_states=['A'], final_states=['B']) |
| 3535 | sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0), |
| 3536 | ....: (2, 2, 0, 1), (2, 1, 1, 1)], |
| 3537 | ....: initial_states=[1], final_states=[1]) |
| 3538 | sage: He = G.composition(F, algorithm='explorative') |
| 3539 | sage: He.transitions() |
| 3540 | [Transition from ('A', 1) to ('B', 2): 1|0,1, |
| 3541 | Transition from ('B', 2) to ('B', 2): 0|1, |
| 3542 | Transition from ('B', 2) to ('B', 1): 1|1, |
| 3543 | Transition from ('B', 1) to ('B', 1): 0|0, |
| 3544 | Transition from ('B', 1) to ('B', 2): 1|0] |
| 3545 | |
| 3546 | Be aware that after composition, different transitions may |
| 3547 | share the same output label (same python object):: |
| 3548 | |
| 3549 | sage: F = Transducer([ ('A','B',0,0), ('B','A',0,0)], |
| 3550 | ....: initial_states=['A'], |
| 3551 | ....: final_states=['A']) |
| 3552 | sage: F.transitions()[0].word_out is F.transitions()[1].word_out |
| 3553 | False |
| 3554 | sage: G = Transducer([('C','C',0,1)],) |
| 3555 | ....: initial_states=['C'], |
| 3556 | ....: final_states=['C']) |
| 3557 | sage: H = G.composition(F) |
| 3558 | sage: H.transitions()[0].word_out is H.transitions()[1].word_out |
| 3559 | True |
| 3560 | |
| 3561 | TESTS: |
| 3562 | |
| 3563 | Due to the limitations of the two algorithms the following |
| 3564 | (examples from above, but different algorithm used) does not |
| 3565 | give a full answer or does not work |
| 3566 | |
| 3567 | In the following, ``algorithm='explorative'`` is inadequate, |
| 3568 | as ``F`` has more than one initial state:: |
| 3569 | |
| 3570 | sage: F = Transducer([('A', 'B', 1, 0), ('B', 'A', 0, 1)], |
| 3571 | ....: initial_states=['A', 'B'], final_states=['B'], |
| 3572 | ....: determine_alphabets=True) |
| 3573 | sage: G = Transducer([(1, 1, 1, 0), (1, 2, 0, 1), |
| 3574 | ....: (2, 2, 1, 1), (2, 2, 0, 0)], |
| 3575 | ....: initial_states=[1], final_states=[2], |
| 3576 | ....: determine_alphabets=True) |
| 3577 | sage: He = F.composition(G, algorithm='explorative') |
| 3578 | sage: He.initial_states() |
| 3579 | [(1, 'A')] |
| 3580 | sage: He.transitions() |
| 3581 | [Transition from (1, 'A') to (2, 'B'): 0|0, |
| 3582 | Transition from (2, 'B') to (2, 'A'): 0|1, |
| 3583 | Transition from (2, 'A') to (2, 'B'): 1|0] |
| 3584 | |
| 3585 | In the following example, ``algorithm='direct'`` is inappropriate |
| 3586 | as there are edges with output labels of length greater than 1:: |
| 3587 | |
| 3588 | sage: F = Transducer([('A', 'B', 1, [1, 0]), ('B', 'B', 1, 1), |
| 3589 | ....: ('B', 'B', 0, 0)], |
| 3590 | ....: initial_states=['A'], final_states=['B']) |
| 3591 | sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0), |
| 3592 | ....: (2, 2, 0, 1), (2, 1, 1, 1)], |
| 3593 | ....: initial_states=[1], final_states=[1]) |
| 3594 | sage: Hd = G.composition(F, algorithm='direct') |
| 3595 | """ |
| 3596 | if algorithm == None: |
| 3597 | algorithm = 'direct' |
| 3598 | if algorithm == 'direct': |
| 3599 | return self._composition_direct_(other, only_accessible_components) |
| 3600 | elif algorithm == 'explorative': |
| 3601 | return self._composition_explorative_(other) |
| 3602 | else: |
| 3603 | raise ValueError, "Unknown algorithm %s." % (algorithm,) |
| 3604 | |
| 3605 | |
| 3606 | def _composition_direct_(self, other, only_accessible_components=True): |
| 3607 | """ |
| 3608 | See :meth:`.composition` for details. |
| 3609 | |
| 3610 | TESTS:: |
| 3611 | |
| 3612 | sage: F = Transducer([('A', 'B', 1, 0), ('B', 'A', 0, 1)], |
| 3613 | ....: initial_states=['A', 'B'], final_states=['B'], |
| 3614 | ....: determine_alphabets=True) |
| 3615 | sage: G = Transducer([(1, 1, 1, 0), (1, 2, 0, 1), |
| 3616 | ....: (2, 2, 1, 1), (2, 2, 0, 0)], |
| 3617 | ....: initial_states=[1], final_states=[2], |
| 3618 | ....: determine_alphabets=True) |
| 3619 | sage: Hd = F._composition_direct_(G) |
| 3620 | sage: Hd.initial_states() |
| 3621 | [(1, 'B'), (1, 'A')] |
| 3622 | sage: Hd.transitions() |
| 3623 | [Transition from (1, 'B') to (1, 'A'): 1|1, |
| 3624 | Transition from (1, 'A') to (2, 'B'): 0|0, |
| 3625 | Transition from (2, 'B') to (2, 'A'): 0|1, |
| 3626 | Transition from (2, 'A') to (2, 'B'): 1|0] |
| 3627 | |
| 3628 | """ |
| 3629 | def function(transition1, transition2): |
| 3630 | if transition1.word_out == transition2.word_in: |
| 3631 | return (transition1.word_in, transition2.word_out) |
| 3632 | else: |
| 3633 | raise LookupError |
| 3634 | |
| 3635 | return other.product_FiniteStateMachine( |
| 3636 | self, function, |
| 3637 | only_accessible_components=only_accessible_components) |
| 3638 | |
| 3639 | |
| 3640 | def _composition_explorative_(self, other): |
| 3641 | """ |
| 3642 | See :meth:`.composition` for details. |
| 3643 | |
| 3644 | TESTS:: |
| 3645 | |
| 3646 | sage: F = Transducer([('A', 'B', 1, [1, 0]), ('B', 'B', 1, 1), |
| 3647 | ....: ('B', 'B', 0, 0)], |
| 3648 | ....: initial_states=['A'], final_states=['B']) |
| 3649 | sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0), |
| 3650 | ....: (2, 2, 0, 1), (2, 1, 1, 1)], |
| 3651 | ....: initial_states=[1], final_states=[1]) |
| 3652 | sage: He = G._composition_explorative_(F) |
| 3653 | sage: He.transitions() |
| 3654 | [Transition from ('A', 1) to ('B', 2): 1|0,1, |
| 3655 | Transition from ('B', 2) to ('B', 2): 0|1, |
| 3656 | Transition from ('B', 2) to ('B', 1): 1|1, |
| 3657 | Transition from ('B', 1) to ('B', 1): 0|0, |
| 3658 | Transition from ('B', 1) to ('B', 2): 1|0] |
| 3659 | |
| 3660 | TODO: |
| 3661 | |
| 3662 | The explorative algorithm should be re-implemented using the |
| 3663 | process iterators of both finite state machines. |
| 3664 | """ |
| 3665 | def composition_transition(state, input): |
| 3666 | (state1, state2) = state |
| 3667 | transition1 = None |
| 3668 | for transition in other.iter_transitions(state1): |
| 3669 | if transition.word_in == [input]: |
| 3670 | transition1 = transition |
| 3671 | break |
| 3672 | if transition1 is None: |
| 3673 | raise LookupError |
| 3674 | new_state1 = transition1.to_state |
| 3675 | new_state2 = state2 |
| 3676 | output = [] |
| 3677 | for o in transition1.word_out: |
| 3678 | transition2 = None |
| 3679 | for transition in self.iter_transitions(new_state2): |
| 3680 | if transition.word_in == [o]: |
| 3681 | transition2 = transition |
| 3682 | break |
| 3683 | if transition2 is None: |
| 3684 | raise LookupError |
| 3685 | new_state2 = transition2.to_state |
| 3686 | output += transition2.word_out |
| 3687 | return ((new_state1, new_state2), output) |
| 3688 | |
| 3689 | F = other.empty_copy() |
| 3690 | new_initial_states = [(other.initial_states()[0], self.initial_states()[0])] |
| 3691 | F.add_from_transition_function(composition_transition, |
| 3692 | initial_states=new_initial_states) |
| 3693 | |
| 3694 | for state in F.states(): |
| 3695 | if all(map(lambda s: s.is_final, state.label())): |
| 3696 | state.is_final = True |
| 3697 | |
| 3698 | return F |
| 3699 | |
| 3700 | |
| 3701 | def input_projection(self): |
| 3702 | """ |
| 3703 | Returns an automaton where the output of each transition of |
| 3704 | self is deleted. |
| 3705 | |
| 3706 | INPUT: |
| 3707 | |
| 3708 | Nothing |
| 3709 | |
| 3710 | OUTPUT: |
| 3711 | |
| 3712 | An automaton. |
| 3713 | |
| 3714 | EXAMPLES:: |
| 3715 | |
| 3716 | sage: F = FiniteStateMachine([('A', 'B', 0, 1), ('A', 'A', 1, 1), |
| 3717 | ....: ('B', 'B', 1, 0)]) |
| 3718 | sage: G = F.input_projection() |
| 3719 | sage: G.transitions() |
| 3720 | [Transition from 'A' to 'B': 0|-, |
| 3721 | Transition from 'A' to 'A': 1|-, |
| 3722 | Transition from 'B' to 'B': 1|-] |
| 3723 | """ |
| 3724 | return self.projection(what='input') |
| 3725 | |
| 3726 | |
| 3727 | def output_projection(self): |
| 3728 | """ |
| 3729 | Returns a automaton where the input of each transition of self |
| 3730 | is deleted and the new input is the original output. |
| 3731 | |
| 3732 | INPUT: |
| 3733 | |
| 3734 | Nothing |
| 3735 | |
| 3736 | OUTPUT: |
| 3737 | |
| 3738 | An automaton. |
| 3739 | |
| 3740 | EXAMPLES:: |
| 3741 | |
| 3742 | sage: F = FiniteStateMachine([('A', 'B', 0, 1), ('A', 'A', 1, 1), |
| 3743 | ....: ('B', 'B', 1, 0)]) |
| 3744 | sage: G = F.output_projection() |
| 3745 | sage: G.transitions() |
| 3746 | [Transition from 'A' to 'B': 1|-, |
| 3747 | Transition from 'A' to 'A': 1|-, |
| 3748 | Transition from 'B' to 'B': 0|-] |
| 3749 | """ |
| 3750 | return self.projection(what='output') |
| 3751 | |
| 3752 | |
| 3753 | def projection(self, what='input'): |
| 3754 | """ |
| 3755 | Returns an Automaton which transition labels are the projection |
| 3756 | of the transition labels of the input. |
| 3757 | |
| 3758 | INPUT: |
| 3759 | |
| 3760 | - ``what`` -- (default: ``input``) either ``input`` or ``output``. |
| 3761 | |
| 3762 | OUTPUT: |
| 3763 | |
| 3764 | An automaton. |
| 3765 | |
| 3766 | EXAMPLES:: |
| 3767 | |
| 3768 | sage: F = FiniteStateMachine([('A', 'B', 0, 1), ('A', 'A', 1, 1), |
| 3769 | ....: ('B', 'B', 1, 0)]) |
| 3770 | sage: G = F.projection(what='output') |
| 3771 | sage: G.transitions() |
| 3772 | [Transition from 'A' to 'B': 1|-, |
| 3773 | Transition from 'A' to 'A': 1|-, |
| 3774 | Transition from 'B' to 'B': 0|-] |
| 3775 | """ |
| 3776 | new = Automaton() |
| 3777 | |
| 3778 | if what == 'input': |
| 3779 | new.input_alphabet = copy(self.input_alphabet) |
| 3780 | elif what == 'output': |
| 3781 | new.input_alphabet = copy(self.output_alphabet) |
| 3782 | else: |
| 3783 | raise NotImplementedError |
| 3784 | |
| 3785 | state_mapping = {} |
| 3786 | for state in self.iter_states(): |
| 3787 | state_mapping[state] = new.add_state(deepcopy(state)) |
| 3788 | for transition in self.iter_transitions(): |
| 3789 | if what == 'input': |
| 3790 | new_word_in = transition.word_in |
| 3791 | elif what == 'output': |
| 3792 | new_word_in = transition.word_out |
| 3793 | else: |
| 3794 | raise NotImplementedError |
| 3795 | new.add_transition((state_mapping[transition.from_state], |
| 3796 | state_mapping[transition.to_state], |
| 3797 | new_word_in, None)) |
| 3798 | return new |
| 3799 | |
| 3800 | |
| 3801 | def transposition(self): |
| 3802 | """ |
| 3803 | Returns a new finite state machine, where all transitions of the |
| 3804 | input finite state machine are reversed. |
| 3805 | |
| 3806 | INPUT: |
| 3807 | |
| 3808 | Nothing. |
| 3809 | |
| 3810 | OUTPUT: |
| 3811 | |
| 3812 | A new finite state machine. |
| 3813 | |
| 3814 | EXAMPLES:: |
| 3815 | |
| 3816 | sage: aut = Automaton([('A', 'A', 0), ('A', 'A', 1), ('A', 'B', 0)], |
| 3817 | ....: initial_states=['A'], final_states=['B']) |
| 3818 | sage: aut.transposition().transitions('B') |
| 3819 | [Transition from 'B' to 'A': 0|-] |
| 3820 | |
| 3821 | :: |
| 3822 | |
| 3823 | sage: aut = Automaton([('1', '1', 1), ('1', '2', 0), ('2', '2', 0)], |
| 3824 | ....: initial_states=['1'], final_states=['1', '2']) |
| 3825 | sage: aut.transposition().initial_states() |
| 3826 | ['1', '2'] |
| 3827 | """ |
| 3828 | transposition = self.empty_copy() |
| 3829 | |
| 3830 | for state in self.states(): |
| 3831 | transposition.add_state(deepcopy(state)) |
| 3832 | |
| 3833 | for transition in self.transitions(): |
| 3834 | transposition.add_transition( |
| 3835 | transition.to_state.label(), transition.from_state.label(), |
| 3836 | transition.word_in, transition.word_out) |
| 3837 | |
| 3838 | for initial in self.initial_states(): |
| 3839 | state = transposition.state(initial.label()) |
| 3840 | if not initial.is_final: |
| 3841 | state.is_final = True |
| 3842 | state.is_initial = False |
| 3843 | |
| 3844 | for final in self.final_states(): |
| 3845 | state = transposition.state(final.label()) |
| 3846 | if not final.is_initial: |
| 3847 | state.is_final = False |
| 3848 | state.is_initial = True |
| 3849 | |
| 3850 | return transposition |
| 3851 | |
| 3852 | |
| 3853 | def split_transitions(self): |
| 3854 | """ |
| 3855 | Returns a new transducer, where all transitions in self with input |
| 3856 | labels consisting of more than one letter |
| 3857 | are replaced by a path of the corresponding length. |
| 3858 | |
| 3859 | INPUT: |
| 3860 | |
| 3861 | Nothing. |
| 3862 | |
| 3863 | OUTPUT: |
| 3864 | |
| 3865 | A new transducer. |
| 3866 | |
| 3867 | EXAMPLES:: |
| 3868 | |
| 3869 | sage: A = Transducer([('A', 'B', [1, 2, 3], 0)], |
| 3870 | ....: initial_states=['A'], final_states=['B']) |
| 3871 | sage: A.split_transitions().states() |
| 3872 | [('A', ()), ('B', ()), |
| 3873 | ('A', (1,)), ('A', (1, 2))] |
| 3874 | """ |
| 3875 | new = self.empty_copy() |
| 3876 | for state in self.states(): |
| 3877 | new.add_state(FSMState((state, ()), is_initial=state.is_initial, |
| 3878 | is_final=state.is_final)) |
| 3879 | for transition in self.transitions(): |
| 3880 | for j in range(len(transition.word_in)-1): |
| 3881 | new.add_transition(( |
| 3882 | (transition.from_state, tuple(transition.word_in[:j])), |
| 3883 | (transition.from_state, tuple(transition.word_in[:j+1])), |
| 3884 | transition.word_in[j], |
| 3885 | [])) |
| 3886 | new.add_transition(( |
| 3887 | (transition.from_state, tuple(transition.word_in[:-1])), |
| 3888 | (transition.to_state, ()), |
| 3889 | transition.word_in[-1:], |
| 3890 | transition.word_out)) |
| 3891 | return new |
| 3892 | |
| 3893 | |
| 3894 | # ************************************************************************* |
| 3895 | # simplifications |
| 3896 | # ************************************************************************* |
| 3897 | |
| 3898 | |
| 3899 | def prepone_output(self): |
| 3900 | """ |
| 3901 | Apply the following to each state `s` (except initial and |
| 3902 | final states) of the finite state machine as often as |
| 3903 | possible: |
| 3904 | |
| 3905 | If the letter a is prefix of the output label of all |
| 3906 | transitions from `s`, then remove it from all these labels and |
| 3907 | append it to all output labels of all transitions leading to |
| 3908 | `s`. |
| 3909 | |
| 3910 | We assume that the states have no output labels. |
| 3911 | |
| 3912 | INPUT: |
| 3913 | |
| 3914 | Nothing. |
| 3915 | |
| 3916 | OUTPUT: |
| 3917 | |
| 3918 | Nothing. |
| 3919 | |
| 3920 | EXAMPLES:: |
| 3921 | |
| 3922 | sage: A = Transducer([('A', 'B', 1, 1), ('B', 'B', 0, 0), ('B', 'C', 1, 0)], |
| 3923 | ....: initial_states=['A'], final_states=['C']) |
| 3924 | sage: A.prepone_output() |
| 3925 | sage: A.transitions() |
| 3926 | [Transition from 'A' to 'B': 1|1,0, |
| 3927 | Transition from 'B' to 'B': 0|0, |
| 3928 | Transition from 'B' to 'C': 1|-] |
| 3929 | |
| 3930 | :: |
| 3931 | |
| 3932 | sage: B = Transducer([('A', 'B', 0, 1), ('B', 'C', 1, [1, 1]), ('B', 'C', 0, 1)], |
| 3933 | ....: initial_states=['A'], final_states=['C']) |
| 3934 | sage: B.prepone_output() |
| 3935 | sage: B.transitions() |
| 3936 | [Transition from 'A' to 'B': 0|1,1, |
| 3937 | Transition from 'B' to 'C': 1|1, |
| 3938 | Transition from 'B' to 'C': 0|-] |
| 3939 | |
| 3940 | If initial states are not labeled as such, unexpected results may be obtained:: |
| 3941 | |
| 3942 | sage: C = Transducer([(0,1,0,0)]) |
| 3943 | sage: C.prepone_output() |
| 3944 | prepone_output: All transitions leaving state 0 have an |
| 3945 | output label with prefix 0. However, there is no inbound |
| 3946 | transition and it is not an initial state. This routine |
| 3947 | (possibly called by simplification) therefore erased this |
| 3948 | prefix from all outbound transitions. |
| 3949 | sage: C.transitions() |
| 3950 | [Transition from 0 to 1: 0|-] |
| 3951 | |
| 3952 | """ |
| 3953 | def find_common_output(state): |
| 3954 | if len(filter(lambda transition: len(transition.word_out) == 0, self.transitions(state))) > 0: |
| 3955 | return () |
| 3956 | first_letters = set(map(lambda transition: transition.word_out[0], self.transitions(state))) |
| 3957 | if len(first_letters) == 1: |
| 3958 | return (first_letters.pop(),) |
| 3959 | return () |
| 3960 | |
| 3961 | changed = 1 |
| 3962 | iteration = 0 |
| 3963 | while changed > 0: |
| 3964 | changed = 0 |
| 3965 | iteration += 1 |
| 3966 | for state in self.states(): |
| 3967 | if state.is_initial or state.is_final: |
| 3968 | continue |
| 3969 | assert len(state.word_out) == 0, \ |
| 3970 | "prepone_output assumes that all states have empty output word, but state %s has output word %s" % \ |
| 3971 | (state, state.word_out) |
| 3972 | common_output = find_common_output(state) |
| 3973 | if len(common_output) > 0: |
| 3974 | changed += 1 |
| 3975 | for transition in self.transitions(state): |
| 3976 | assert transition.word_out[0] == common_output[0] |
| 3977 | transition.word_out = transition.word_out[1:] |
| 3978 | found_inbound_transition = False |
| 3979 | for transition in self.transitions(): |
| 3980 | if transition.to_state == state: |
| 3981 | transition.word_out = transition.word_out + [common_output[0]] |
| 3982 | found_inbound_transition = True |
| 3983 | if not found_inbound_transition: |
| 3984 | print "prepone_output: All transitions leaving state %s have an output label with prefix %s. "\ |
| 3985 | "However, there is no inbound transition and it is not an initial state. "\ |
| 3986 | "This routine (possibly called by simplification) therefore erased this prefix from all "\ |
| 3987 | "outbound transitions." % (state, common_output[0]) |
| 3988 | |
| 3989 | |
| 3990 | |
| 3991 | def equivalence_classes(self): |
| 3992 | """ |
| 3993 | Returns a list of equivalence classes of states. |
| 3994 | |
| 3995 | INPUT: |
| 3996 | |
| 3997 | Nothing. |
| 3998 | |
| 3999 | OUTPUT: |
| 4000 | |
| 4001 | A list of equivalence classes of states. |
| 4002 | |
| 4003 | Two states `a` and `b` are equivalent, if and only if for each |
| 4004 | input label word_in the following holds: |
| 4005 | |
| 4006 | For paths `p_a` from `a` to `a'` with input label ``word_in`` |
| 4007 | and output label ``word_out_a`` and `p_b` from `b` to `b'` |
| 4008 | with input label ``word_in`` and output label ``word_out_b``, |
| 4009 | we have ``word_out_a=word_out_b``, `a'` and `b'` have the same |
| 4010 | output label and are both final or both non-final. |
| 4011 | |
| 4012 | The function :meth:`.equivalence_classes` returns a list of |
| 4013 | the equivalence classes to this equivalence relation. |
| 4014 | |
| 4015 | This is one step of Moore's minimization algorithm. |
| 4016 | |
| 4017 | .. SEEALSO:: |
| 4018 | |
| 4019 | :meth:`.minimization` |
| 4020 | |
| 4021 | EXAMPLES:: |
| 4022 | |
| 4023 | sage: fsm = FiniteStateMachine([("A", "B", 0, 1), ("A", "B", 1, 0), |
| 4024 | ....: ("B", "C", 0, 0), ("B", "C", 1, 1), |
| 4025 | ....: ("C", "D", 0, 1), ("C", "D", 1, 0), |
| 4026 | ....: ("D", "A", 0, 0), ("D", "A", 1, 1)]) |
| 4027 | sage: fsm.equivalence_classes() |
| 4028 | [['A', 'C'], ['B', 'D']] |
| 4029 | """ |
| 4030 | |
| 4031 | # Two states a and b are said to be 0-equivalent, if their output |
| 4032 | # labels agree and if they are both final or non-final. |
| 4033 | # |
| 4034 | # For some j >= 1, two states a and b are said to be j-equivalent, if |
| 4035 | # they are j-1 equivalent and if for each element letter letter_in of |
| 4036 | # the input alphabet and transitions t_a from a with input label |
| 4037 | # letter_in, output label word_out_a to a' and t_b from b with input |
| 4038 | # label letter_in, output label word_out_b to b', we have |
| 4039 | # word_out_a=word_out_b and a' and b' are j-1 equivalent. |
| 4040 | |
| 4041 | # If for some j the relations j-1 equivalent and j-equivalent |
| 4042 | # coincide, then they are equal to the equivalence relation described |
| 4043 | # in the docstring. |
| 4044 | |
| 4045 | # classes_current holds the equivalence classes of j-equivalence, |
| 4046 | # classes_previous holds the equivalence classes of j-1 equivalence. |
| 4047 | |
| 4048 | if not self.is_deterministic(): |
| 4049 | raise NotImplementedError, "Minimization via Moore's Algorithm is only implemented for deterministic finite state machines" |
| 4050 | |
| 4051 | # initialize with 0-equivalence |
| 4052 | classes_previous = [] |
| 4053 | key_0 = lambda state: (state.is_final, state.word_out) |
| 4054 | states_grouped = full_group_by(self.states(), key=key_0) |
| 4055 | classes_current = [equivalence_class for |
| 4056 | (key,equivalence_class) in states_grouped] |
| 4057 | |
| 4058 | while len(classes_current) != len(classes_previous): |
| 4059 | class_of = {} |
| 4060 | classes_previous = classes_current |
| 4061 | classes_current = [] |
| 4062 | |
| 4063 | for k in range(len(classes_previous)): |
| 4064 | for state in classes_previous[k]: |
| 4065 | class_of[state] = k |
| 4066 | |
| 4067 | key_current = lambda state: sorted( |
| 4068 | [(transition.word_in, |
| 4069 | transition.word_out, |
| 4070 | class_of[transition.to_state]) |
| 4071 | for transition in state.transitions]) |
| 4072 | |
| 4073 | for class_previous in classes_previous: |
| 4074 | states_grouped = full_group_by(class_previous, key=key_current) |
| 4075 | classes_current.extend([equivalence_class for |
| 4076 | (key,equivalence_class) in states_grouped]) |
| 4077 | |
| 4078 | return classes_current |
| 4079 | |
| 4080 | |
| 4081 | def quotient(self, classes): |
| 4082 | """ |
| 4083 | Constructs the quotient with respect to the equivalence |
| 4084 | classes. |
| 4085 | |
| 4086 | INPUT: |
| 4087 | |
| 4088 | - ``classes`` is a list of equivalence classes of states. |
| 4089 | |
| 4090 | OUTPUT: |
| 4091 | |
| 4092 | A finite state machine. |
| 4093 | |
| 4094 | Assume that `c` is a class and `s`, `s'` are states in `c`. If |
| 4095 | there is a transition from `s` to some `t` with input label |
| 4096 | ``word_in`` and output label ``word_out``, then there has to |
| 4097 | be a transition from `s'` to some `t'` with input label |
| 4098 | ``word_in`` and output label ``word_out`` such that `s'` and |
| 4099 | `t'` are states of the same class `c'`. Then there is a |
| 4100 | transition from `c` to `c'` in the quotient with input label |
| 4101 | ``word_in`` and output label ``word_out``. |
| 4102 | |
| 4103 | Non-initial states may be merged with initial states, the |
| 4104 | resulting state is an initial state. |
| 4105 | |
| 4106 | All states in a class must have the same ``is_final`` and |
| 4107 | ``word_out`` values. |
| 4108 | |
| 4109 | EXAMPLES:: |
| 4110 | |
| 4111 | sage: fsm = FiniteStateMachine([("A", "B", 0, 1), ("A", "B", 1, 0), |
| 4112 | ....: ("B", "C", 0, 0), ("B", "C", 1, 1), |
| 4113 | ....: ("C", "D", 0, 1), ("C", "D", 1, 0), |
| 4114 | ....: ("D", "A", 0, 0), ("D", "A", 1, 1)]) |
| 4115 | sage: fsmq = fsm.quotient([[fsm.state("A"), fsm.state("C")], |
| 4116 | ....: [fsm.state("B"), fsm.state("D")]]) |
| 4117 | sage: fsmq.transitions() |
| 4118 | [Transition from ('A', 'C') |
| 4119 | to ('B', 'D'): 0|1, |
| 4120 | Transition from ('A', 'C') |
| 4121 | to ('B', 'D'): 1|0, |
| 4122 | Transition from ('B', 'D') |
| 4123 | to ('A', 'C'): 0|0, |
| 4124 | Transition from ('B', 'D') |
| 4125 | to ('A', 'C'): 1|1] |
| 4126 | sage: fsmq.relabeled().transitions() |
| 4127 | [Transition from 0 to 1: 0|1, |
| 4128 | Transition from 0 to 1: 1|0, |
| 4129 | Transition from 1 to 0: 0|0, |
| 4130 | Transition from 1 to 0: 1|1] |
| 4131 | sage: fsmq1 = fsm.quotient(fsm.equivalence_classes()) |
| 4132 | sage: fsmq1 == fsmq |
| 4133 | True |
| 4134 | sage: fsm.quotient([[fsm.state("A"), fsm.state("B"), fsm.state("C"), fsm.state("D")]]) |
| 4135 | Traceback (most recent call last): |
| 4136 | ... |
| 4137 | ValueError: There is a transition Transition from 'B' to 'C': 0|0 in the original transducer, but no corresponding transition in the new transducer. |
| 4138 | """ |
| 4139 | new = self.empty_copy() |
| 4140 | state_mapping = {} |
| 4141 | |
| 4142 | # Create new states and build state_mapping |
| 4143 | for c in classes: |
| 4144 | new_state = new.add_state(tuple(c)) |
| 4145 | for state in c: |
| 4146 | state_mapping[state] = new_state |
| 4147 | |
| 4148 | # Copy data from old transducer |
| 4149 | for c in classes: |
| 4150 | new_state = state_mapping[c[0]] |
| 4151 | # copy all data from first class member |
| 4152 | new_state.is_initial = c[0].is_initial |
| 4153 | new_state.is_final = c[0].is_final |
| 4154 | new_state.word_out = c[0].word_out |
| 4155 | for transition in self.iter_transitions(c[0]): |
| 4156 | new.add_transition( |
| 4157 | from_state=new_state, |
| 4158 | to_state = state_mapping[transition.to_state], |
| 4159 | word_in = transition.word_in, |
| 4160 | word_out = transition.word_out) |
| 4161 | |
| 4162 | # check that all class members have the same information (modulo classes) |
| 4163 | for state in c: |
| 4164 | new_state.is_initial = new_state.is_initial or state.is_initial |
| 4165 | assert new_state.is_final == state.is_final, "Class %s mixes final and non-final states" % (c,) |
| 4166 | assert new_state.word_out == state.word_out, "Class %s mixes different word_out" % (c,) |
| 4167 | assert len(self.transitions(state)) == len(new.transitions(new_state)), \ |
| 4168 | "Class %s has %d outgoing transitions, but class member %s has %d outgoing transitions" % \ |
| 4169 | (c, len(new.transitions(new_state)), state, len(self.transitions(state))) |
| 4170 | for transition in self.transitions(state): |
| 4171 | try: |
| 4172 | new.transition((new_state, state_mapping[transition.to_state], transition.word_in, transition.word_out)) |
| 4173 | except LookupError: |
| 4174 | raise ValueError, "There is a transition %s in the original transducer, but no corresponding transition in the new transducer." % transition |
| 4175 | return new |
| 4176 | |
| 4177 | |
| 4178 | # ************************************************************************* |
| 4179 | # other |
| 4180 | # ************************************************************************* |
| 4181 | |
| 4182 | |
| 4183 | def graph(self, edge_labels='words_in_out'): |
| 4184 | """ |
| 4185 | Returns the graph of the finite state machine with labeled |
| 4186 | vertices and labeled edges. |
| 4187 | |
| 4188 | INPUT: |
| 4189 | |
| 4190 | - ``edge_label``: (default: ``'words_in_out'``) can be |
| 4191 | - ``'words_in_out'`` (labels will be strings ``'i|o'``) |
| 4192 | - a function with which takes as input a transition |
| 4193 | and outputs (returns) the label |
| 4194 | |
| 4195 | OUTPUT: |
| 4196 | |
| 4197 | A graph. |
| 4198 | |
| 4199 | EXAMPLES:: |
| 4200 | |
| 4201 | sage: from sage.combinat.finite_state_machine import FSMState |
| 4202 | sage: A = FSMState('A') |
| 4203 | sage: T = Transducer() |
| 4204 | sage: T.graph() |
| 4205 | Digraph on 0 vertices |
| 4206 | sage: T.add_state(A) |
| 4207 | 'A' |
| 4208 | sage: T.graph() |
| 4209 | Digraph on 1 vertex |
| 4210 | sage: T.add_transition(('A', 'A', 0, 1)) |
| 4211 | Transition from 'A' to 'A': 0|1 |
| 4212 | sage: T.graph() |
| 4213 | Looped digraph on 1 vertex |
| 4214 | """ |
| 4215 | if edge_labels == 'words_in_out': |
| 4216 | label_fct = lambda t:t._in_out_label_() |
| 4217 | elif hasattr(edge_labels, '__call__'): |
| 4218 | label_fct = edge_labels |
| 4219 | else: |
| 4220 | raise TypeError, 'Wrong argument for edge_labels.' |
| 4221 | |
| 4222 | graph_data = [] |
| 4223 | isolated_vertices = [] |
| 4224 | for state in self.iter_states(): |
| 4225 | transitions = state.transitions |
| 4226 | if len(transitions) == 0: |
| 4227 | isolated_vertices.append(state.label()) |
| 4228 | for t in transitions: |
| 4229 | graph_data.append((t.from_state.label(), t.to_state.label(), |
| 4230 | label_fct(t))) |
| 4231 | |
| 4232 | G = DiGraph(graph_data) |
| 4233 | G.add_vertices(isolated_vertices) |
| 4234 | return G |
| 4235 | |
| 4236 | |
| 4237 | digraph = graph |
| 4238 | |
| 4239 | |
| 4240 | def plot(self): |
| 4241 | """ |
| 4242 | Plots a graph of the finite state machine with labeled |
| 4243 | vertices and labeled edges. |
| 4244 | |
| 4245 | INPUT: |
| 4246 | |
| 4247 | Nothing. |
| 4248 | |
| 4249 | OUTPUT: |
| 4250 | |
| 4251 | A plot of the graph of the finite state machine. |
| 4252 | |
| 4253 | TESTS:: |
| 4254 | |
| 4255 | sage: FiniteStateMachine([('A', 'A', 0)]).plot() |
| 4256 | """ |
| 4257 | return self.graph(edge_labels='words_in_out').plot() |
| 4258 | |
| 4259 | |
| 4260 | def predecessors(self, state, valid_input=None): |
| 4261 | """ |
| 4262 | Lists all predecessors of a state. |
| 4263 | |
| 4264 | INPUT: |
| 4265 | |
| 4266 | - ``state`` -- the state from which the predecessors should be |
| 4267 | listed. |
| 4268 | |
| 4269 | - ``valid_input`` -- If ``valid_input`` is a list, then we |
| 4270 | only consider transitions whose input labels are contained |
| 4271 | in ``valid_input``. ``state`` has to be a :class:`FSMState` |
| 4272 | (not a label of a state). If input labels of length larger |
| 4273 | than `1` are used, then ``valid_input`` has to be a list of |
| 4274 | lists. |
| 4275 | |
| 4276 | OUTPUT: |
| 4277 | |
| 4278 | A list of states. |
| 4279 | |
| 4280 | EXAMPLES:: |
| 4281 | |
| 4282 | sage: A = Transducer([('I', 'A', 'a', 'b'), ('I', 'B', 'b', 'c'), |
| 4283 | ....: ('I', 'C', 'c', 'a'), ('A', 'F', 'b', 'a'), |
| 4284 | ....: ('B', 'F', ['c', 'b'], 'b'), ('C', 'F', 'a', 'c')], |
| 4285 | ....: initial_states=['I'], final_states=['F']) |
| 4286 | sage: A.predecessors(A.state('A')) |
| 4287 | ['A', 'I'] |
| 4288 | sage: A.predecessors(A.state('F'), valid_input=['b', 'a']) |
| 4289 | ['F', 'C', 'A', 'I'] |
| 4290 | sage: A.predecessors(A.state('F'), valid_input=[['c', 'b'], 'a']) |
| 4291 | ['F', 'C', 'B'] |
| 4292 | """ |
| 4293 | if valid_input != None: |
| 4294 | valid_list = list() |
| 4295 | for input in valid_input: |
| 4296 | input_list = input |
| 4297 | if not isinstance(input_list, list): |
| 4298 | input_list = [input] |
| 4299 | valid_list.append(input_list) |
| 4300 | valid_input = valid_list |
| 4301 | |
| 4302 | unhandeled_direct_predecessors = {s:[] for s in self.states() } |
| 4303 | for t in self.transitions(): |
| 4304 | if valid_input is None or t.word_in in valid_input: |
| 4305 | unhandeled_direct_predecessors[t.to_state].append(t.from_state) |
| 4306 | done = [] |
| 4307 | open = [state] |
| 4308 | while len(open) > 0: |
| 4309 | s = open.pop() |
| 4310 | candidates = unhandeled_direct_predecessors[s] |
| 4311 | if candidates is not None: |
| 4312 | open.extend(candidates) |
| 4313 | unhandeled_direct_predecessors[s] = None |
| 4314 | done.append(s) |
| 4315 | return(done) |
| 4316 | |
| 4317 | |
| 4318 | #***************************************************************************** |
| 4319 | |
| 4320 | |
| 4321 | def is_Automaton(FSM): |
| 4322 | """ |
| 4323 | Tests whether or not ``FSM`` inherits from :class:`Automaton`. |
| 4324 | |
| 4325 | TESTS:: |
| 4326 | |
| 4327 | sage: from sage.combinat.finite_state_machine import is_FiniteStateMachine, is_Automaton |
| 4328 | sage: is_Automaton(FiniteStateMachine()) |
| 4329 | False |
| 4330 | sage: is_Automaton(Automaton()) |
| 4331 | True |
| 4332 | sage: is_FiniteStateMachine(Automaton()) |
| 4333 | True |
| 4334 | """ |
| 4335 | return isinstance(FSM, Automaton) |
| 4336 | |
| 4337 | |
| 4338 | class Automaton(FiniteStateMachine): |
| 4339 | """ |
| 4340 | This creates an automaton, which is a special type of a finite |
| 4341 | state machine. |
| 4342 | |
| 4343 | See class :class:`FiniteStateMachine` for more information. |
| 4344 | |
| 4345 | TESTS:: |
| 4346 | |
| 4347 | sage: Automaton() |
| 4348 | finite state machine with 0 states |
| 4349 | """ |
| 4350 | |
| 4351 | def _latex_transition_label_(self, transition, format_function=latex): |
| 4352 | r""" |
| 4353 | Returns the proper transition label. |
| 4354 | |
| 4355 | INPUT: |
| 4356 | |
| 4357 | - ``transition`` - a transition |
| 4358 | |
| 4359 | - ``format_function'' - a function formatting the labels |
| 4360 | |
| 4361 | OUTPUT: |
| 4362 | |
| 4363 | A string. |
| 4364 | |
| 4365 | EXAMPLES:: |
| 4366 | |
| 4367 | sage: F = Automaton([('A', 'B', 1)]) |
| 4368 | sage: F._latex_() |
| 4369 | '\\begin{tikzpicture}[auto]\n\\node[state] (v0) at (3.000000,0.000000) {\\text{\\texttt{A}}}\n;\\node[state] (v1) at (-3.000000,0.000000) {\\text{\\texttt{B}}}\n;\\path[->] (v0) edge node {$\\left[1\\right]$} (v1);\n\\end{tikzpicture}' |
| 4370 | |
| 4371 | TESTS:: |
| 4372 | |
| 4373 | sage: F = Automaton([('A', 'B', 0, 1)]) |
| 4374 | sage: t = F.transitions()[0] |
| 4375 | sage: F._latex_transition_label_(t) |
| 4376 | \left[0\right] |
| 4377 | """ |
| 4378 | return format_function(transition.word_in) |
| 4379 | |
| 4380 | def determinisation(self): |
| 4381 | """ |
| 4382 | Returns a deterministic automaton which accepts the same input |
| 4383 | words as the original one. |
| 4384 | |
| 4385 | INPUT: |
| 4386 | |
| 4387 | Nothing. |
| 4388 | |
| 4389 | OUTPUT: |
| 4390 | |
| 4391 | A new automaton, which is deterministic. |
| 4392 | |
| 4393 | The labels of the states of the new automaton are frozensets of |
| 4394 | states of ``self``. |
| 4395 | |
| 4396 | The input alphabet must be specified. It is restricted to nice |
| 4397 | cases: input words have to have length at most `1`. |
| 4398 | |
| 4399 | EXAMPLES:: |
| 4400 | |
| 4401 | sage: aut = Automaton([('A', 'A', 0), ('A', 'B', 1), ('B', 'B', 1)], |
| 4402 | ....: initial_states=['A'], final_states=['B']) |
| 4403 | sage: aut.determinisation().transitions() |
| 4404 | [Transition from frozenset(['A']) |
| 4405 | to frozenset(['A']): 0|-, |
| 4406 | Transition from frozenset(['A']) |
| 4407 | to frozenset(['B']): 1|-, |
| 4408 | Transition from frozenset(['B']) |
| 4409 | to frozenset([]): 0|-, |
| 4410 | Transition from frozenset(['B']) |
| 4411 | to frozenset(['B']): 1|-, |
| 4412 | Transition from frozenset([]) |
| 4413 | to frozenset([]): 0|-, |
| 4414 | Transition from frozenset([]) |
| 4415 | to frozenset([]): 1|-] |
| 4416 | |
| 4417 | :: |
| 4418 | |
| 4419 | sage: A = Automaton([('A', 'A', 1), ('A', 'A', 0), ('A', 'B', 1), |
| 4420 | ....: ('B', 'C', 0), ('C', 'C', 1), ('C', 'C', 0)], |
| 4421 | ....: initial_states=['A'], final_states=['C']) |
| 4422 | sage: A.determinisation().states() |
| 4423 | [frozenset(['A']), frozenset(['A', 'B']), |
| 4424 | frozenset(['A', 'C']), frozenset(['A', 'C', 'B'])] |
| 4425 | |
| 4426 | TESTS: |
| 4427 | |
| 4428 | This is from #15078, comment 13. |
| 4429 | |
| 4430 | :: |
| 4431 | |
| 4432 | sage: D = {'A': [('A', 'a'), ('B', 'a'), ('A', 'b')], |
| 4433 | ....: 'C': [], 'B': [('C', 'b')]} |
| 4434 | sage: auto = Automaton(D, initial_states=['A'], final_states=['C']) |
| 4435 | sage: auto.is_deterministic() |
| 4436 | False |
| 4437 | sage: auto.process(list('aaab')) |
| 4438 | (False, 'A', []) |
| 4439 | sage: auto.states() |
| 4440 | ['A', 'C', 'B'] |
| 4441 | sage: auto.determinisation() |
| 4442 | finite state machine with 3 states |
| 4443 | """ |
| 4444 | for transition in self.transitions(): |
| 4445 | assert len(transition.word_in) <= 1, "%s has input label of length > 1, which we cannot handle" % (transition,) |
| 4446 | |
| 4447 | epsilon_successors = {} |
| 4448 | direct_epsilon_successors = {} |
| 4449 | for state in self.states(): |
| 4450 | direct_epsilon_successors[state] = set(map(lambda t:t.to_state, |
| 4451 | filter(lambda transition: len(transition.word_in) == 0, |
| 4452 | self.transitions(state) |
| 4453 | ) |
| 4454 | ) |
| 4455 | ) |
| 4456 | epsilon_successors[state] = set([state]) |
| 4457 | |
| 4458 | old_count_epsilon_successors = 0 |
| 4459 | count_epsilon_successors = len(epsilon_successors) |
| 4460 | |
| 4461 | while old_count_epsilon_successors < count_epsilon_successors: |
| 4462 | old_count_epsilon_successors = count_epsilon_successors |
| 4463 | count_epsilon_successors = 0 |
| 4464 | for state in self.states(): |
| 4465 | for direct_successor in direct_epsilon_successors[state]: |
| 4466 | epsilon_successors[state] = epsilon_successors[state].union(epsilon_successors[direct_successor]) |
| 4467 | count_epsilon_successors += len(epsilon_successors[state]) |
| 4468 | |
| 4469 | |
| 4470 | def set_transition(states, letter): |
| 4471 | result = set() |
| 4472 | for state in states: |
| 4473 | for transition in self.transitions(state): |
| 4474 | if transition.word_in == [letter]: |
| 4475 | result.add(transition.to_state) |
| 4476 | result = result.union(*map(lambda s:epsilon_successors[s], result)) |
| 4477 | return (frozenset(result), []) |
| 4478 | |
| 4479 | result = self.empty_copy() |
| 4480 | new_initial_states = [frozenset([state for state in self.initial_states()])] |
| 4481 | result.add_from_transition_function(set_transition, |
| 4482 | initial_states=new_initial_states) |
| 4483 | |
| 4484 | for state in result.states(): |
| 4485 | if any(map(lambda s: s.is_final, state.label())): |
| 4486 | state.is_final = True |
| 4487 | |
| 4488 | |
| 4489 | return result |
| 4490 | |
| 4491 | |
| 4492 | def minimization(self, algorithm=None): |
| 4493 | """ |
| 4494 | Returns the minimization of the input automaton as a new automaton. |
| 4495 | |
| 4496 | INPUT: |
| 4497 | |
| 4498 | - ``algorithm`` -- Either Moore's algorithm is used (default |
| 4499 | or ``algorithm='Moore'``), or Brzozowski's algorithm when |
| 4500 | ``algorithm='Brzozowski'``. |
| 4501 | |
| 4502 | OUTPUT: |
| 4503 | |
| 4504 | A new automaton. |
| 4505 | |
| 4506 | The resulting automaton is deterministic and has a minimal |
| 4507 | number of states. |
| 4508 | |
| 4509 | EXAMPLES:: |
| 4510 | |
| 4511 | sage: A = Automaton([('A', 'A', 1), ('A', 'A', 0), ('A', 'B', 1), |
| 4512 | ....: ('B', 'C', 0), ('C', 'C', 1), ('C', 'C', 0)], |
| 4513 | ....: initial_states=['A'], final_states=['C']) |
| 4514 | sage: B = A.minimization(algorithm='Brzozowski') |
| 4515 | sage: B.transitions(B.states()[1]) |
| 4516 | [Transition from frozenset([frozenset(['A', 'C', 'B']), |
| 4517 | frozenset(['C', 'B']), frozenset(['A', 'C'])]) to |
| 4518 | frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']), |
| 4519 | frozenset(['A', 'C']), frozenset(['C'])]): 0|-, |
| 4520 | Transition from frozenset([frozenset(['A', 'C', 'B']), |
| 4521 | frozenset(['C', 'B']), frozenset(['A', 'C'])]) to |
| 4522 | frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']), |
| 4523 | frozenset(['A', 'C'])]): 1|-] |
| 4524 | sage: len(B.states()) |
| 4525 | 3 |
| 4526 | sage: C = A.minimization(algorithm='Brzozowski') |
| 4527 | sage: C.transitions(C.states()[1]) |
| 4528 | [Transition from frozenset([frozenset(['A', 'C', 'B']), |
| 4529 | frozenset(['C', 'B']), frozenset(['A', 'C'])]) to |
| 4530 | frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']), |
| 4531 | frozenset(['A', 'C']), frozenset(['C'])]): 0|-, |
| 4532 | Transition from frozenset([frozenset(['A', 'C', 'B']), |
| 4533 | frozenset(['C', 'B']), frozenset(['A', 'C'])]) to |
| 4534 | frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']), |
| 4535 | frozenset(['A', 'C'])]): 1|-] |
| 4536 | sage: len(C.states()) |
| 4537 | 3 |
| 4538 | |
| 4539 | :: |
| 4540 | |
| 4541 | sage: aut = Automaton([('1', '2', 'a'), ('2', '3', 'b'), |
| 4542 | ....: ('3', '2', 'a'), ('2', '1', 'b'), |
| 4543 | ....: ('3', '4', 'a'), ('4', '3', 'b')], |
| 4544 | ....: initial_states=['1'], final_states=['1']) |
| 4545 | sage: min = aut.minimization(algorithm='Brzozowski') |
| 4546 | sage: [len(min.states()), len(aut.states())] |
| 4547 | [3, 4] |
| 4548 | sage: min = aut.minimization(algorithm='Moore') |
| 4549 | Traceback (most recent call last): |
| 4550 | ... |
| 4551 | NotImplementedError: Minimization via Moore's Algorithm is only |
| 4552 | implemented for deterministic finite state machines |
| 4553 | """ |
| 4554 | if algorithm is None or algorithm == "Moore": |
| 4555 | return self._minimization_Moore_() |
| 4556 | elif algorithm == "Brzozowski": |
| 4557 | return self._minimization_Brzozowski_() |
| 4558 | else: |
| 4559 | raise NotImplementedError, "Algorithm '%s' is not implemented. Choose 'Moore' or 'Brzozowski'" % algorithm |
| 4560 | |
| 4561 | |
| 4562 | def _minimization_Brzozowski_(self): |
| 4563 | """ |
| 4564 | Returns a minimized automaton by using Brzozowski's algorithm. |
| 4565 | |
| 4566 | See also :meth:`.minimization`. |
| 4567 | |
| 4568 | TESTS:: |
| 4569 | |
| 4570 | sage: A = Automaton([('A', 'A', 1), ('A', 'A', 0), ('A', 'B', 1), |
| 4571 | ....: ('B', 'C', 0), ('C', 'C', 1), ('C', 'C', 0)], |
| 4572 | ....: initial_states=['A'], final_states=['C']) |
| 4573 | sage: B = A._minimization_Brzozowski_() |
| 4574 | sage: len(B.states()) |
| 4575 | 3 |
| 4576 | """ |
| 4577 | return self.transposition().determinisation().transposition().determinisation() |
| 4578 | |
| 4579 | |
| 4580 | def _minimization_Moore_(self): |
| 4581 | """ |
| 4582 | Returns a minimized automaton by using Brzozowski's algorithm. |
| 4583 | |
| 4584 | See also :meth:`.minimization`. |
| 4585 | |
| 4586 | TESTS:: |
| 4587 | |
| 4588 | sage: aut = Automaton([('1', '2', 'a'), ('2', '3', 'b'), |
| 4589 | ....: ('3', '2', 'a'), ('2', '1', 'b'), |
| 4590 | ....: ('3', '4', 'a'), ('4', '3', 'b')], |
| 4591 | ....: initial_states=['1'], final_states=['1']) |
| 4592 | sage: min = aut._minimization_Moore_() |
| 4593 | Traceback (most recent call last): |
| 4594 | ... |
| 4595 | NotImplementedError: Minimization via Moore's Algorithm is only |
| 4596 | implemented for deterministic finite state machines |
| 4597 | """ |
| 4598 | return self.quotient(self.equivalence_classes()) |
| 4599 | |
| 4600 | |
| 4601 | #***************************************************************************** |
| 4602 | |
| 4603 | |
| 4604 | def is_Transducer(FSM): |
| 4605 | """ |
| 4606 | Tests whether or not ``FSM`` inherits from :class:`Transducer`. |
| 4607 | |
| 4608 | TESTS:: |
| 4609 | |
| 4610 | sage: from sage.combinat.finite_state_machine import is_FiniteStateMachine, is_Transducer |
| 4611 | sage: is_Transducer(FiniteStateMachine()) |
| 4612 | False |
| 4613 | sage: is_Transducer(Transducer()) |
| 4614 | True |
| 4615 | sage: is_FiniteStateMachine(Transducer()) |
| 4616 | True |
| 4617 | """ |
| 4618 | return isinstance(FSM, Transducer) |
| 4619 | |
| 4620 | |
| 4621 | class Transducer(FiniteStateMachine): |
| 4622 | """ |
| 4623 | This creates a transducer, which is a special type of a finite |
| 4624 | state machine. |
| 4625 | |
| 4626 | See class :class:`FiniteStateMachine` for more information. |
| 4627 | |
| 4628 | TESTS:: |
| 4629 | |
| 4630 | sage: Transducer() |
| 4631 | finite state machine with 0 states |
| 4632 | """ |
| 4633 | |
| 4634 | def _latex_transition_label_(self, transition, format_function=latex): |
| 4635 | r""" |
| 4636 | Returns the proper transition label. |
| 4637 | |
| 4638 | INPUT: |
| 4639 | |
| 4640 | - ``transition`` - a transition |
| 4641 | |
| 4642 | - ``format_function'' - a function formatting the labels |
| 4643 | |
| 4644 | OUTPUT: |
| 4645 | |
| 4646 | A string. |
| 4647 | |
| 4648 | sage: F = Transducer([('A', 'B', 1, 2)]) |
| 4649 | sage: F._latex_() |
| 4650 | '\\begin{tikzpicture}[auto]\n\\node[state] (v0) at (3.000000,0.000000) {\\text{\\texttt{A}}}\n;\\node[state] (v1) at (-3.000000,0.000000) {\\text{\\texttt{B}}}\n;\\path[->] (v0) edge node {$\\left[1\\right] \\mid \\left[2\\right]$} (v1);\n\\end{tikzpicture}' |
| 4651 | |
| 4652 | TESTS:: |
| 4653 | |
| 4654 | sage: F = Transducer([('A', 'B', 0, 1)]) |
| 4655 | sage: t = F.transitions()[0] |
| 4656 | sage: F._latex_transition_label_(t) |
| 4657 | \left[0\right] \mid \left[1\right] |
| 4658 | """ |
| 4659 | return (format_function(transition.word_in) + "\\mid" |
| 4660 | + format_function(transition.word_out)) |
| 4661 | |
| 4662 | |
| 4663 | def simplification(self): |
| 4664 | """ |
| 4665 | Returns a simplified transducer. |
| 4666 | |
| 4667 | INPUT: |
| 4668 | |
| 4669 | Nothing. |
| 4670 | |
| 4671 | OUTPUT: |
| 4672 | |
| 4673 | A new transducer. |
| 4674 | |
| 4675 | This function simplifies a transducer by Moore's algorithm, |
| 4676 | first moving common output labels of transitions leaving a |
| 4677 | state to output labels of transitions entering the state |
| 4678 | (cf. :meth:`.prepone_output`). |
| 4679 | |
| 4680 | The resulting transducer implements the same function as the |
| 4681 | original transducer. |
| 4682 | |
| 4683 | EXAMPLES:: |
| 4684 | |
| 4685 | sage: fsm = Transducer([("A", "B", 0, 1), ("A", "B", 1, 0), |
| 4686 | ....: ("B", "C", 0, 0), ("B", "C", 1, 1), |
| 4687 | ....: ("C", "D", 0, 1), ("C", "D", 1, 0), |
| 4688 | ....: ("D", "A", 0, 0), ("D", "A", 1, 1)]) |
| 4689 | sage: fsms = fsm.simplification() |
| 4690 | sage: fsms |
| 4691 | finite state machine with 2 states |
| 4692 | sage: fsms.transitions() |
| 4693 | [Transition from ('A', 'C') |
| 4694 | to ('B', 'D'): 0|1, |
| 4695 | Transition from ('A', 'C') |
| 4696 | to ('B', 'D'): 1|0, |
| 4697 | Transition from ('B', 'D') |
| 4698 | to ('A', 'C'): 0|0, |
| 4699 | Transition from ('B', 'D') |
| 4700 | to ('A', 'C'): 1|1] |
| 4701 | sage: fsms.relabeled().transitions() |
| 4702 | [Transition from 0 to 1: 0|1, |
| 4703 | Transition from 0 to 1: 1|0, |
| 4704 | Transition from 1 to 0: 0|0, |
| 4705 | Transition from 1 to 0: 1|1] |
| 4706 | """ |
| 4707 | fsm = deepcopy(self) |
| 4708 | fsm.prepone_output() |
| 4709 | return fsm.quotient(fsm.equivalence_classes()) |
| 4710 | |
| 4711 | |
| 4712 | #***************************************************************************** |
| 4713 | |
| 4714 | |
| 4715 | def is_FSMProcessIterator(PI): |
| 4716 | """ |
| 4717 | Tests whether or not ``PI`` inherits from :class:`FSMProcessIterator`. |
| 4718 | |
| 4719 | TESTS:: |
| 4720 | |
| 4721 | sage: from sage.combinat.finite_state_machine import is_FSMProcessIterator, FSMProcessIterator |
| 4722 | sage: is_FSMProcessIterator(FSMProcessIterator(FiniteStateMachine())) |
| 4723 | Traceback (most recent call last): |
| 4724 | ... |
| 4725 | ValueError: No state is initial. |
| 4726 | """ |
| 4727 | return isinstance(PI, FSMProcessIterator) |
| 4728 | |
| 4729 | |
| 4730 | class FSMProcessIterator: |
| 4731 | """ |
| 4732 | This class is for processing an input string on a finite state |
| 4733 | machine. |
| 4734 | |
| 4735 | An instance of this class is generated when |
| 4736 | :meth:`FiniteStateMachine.process` or |
| 4737 | :meth:`FiniteStateMachine.iter_process` of the finite state |
| 4738 | machine is invoked. It behaves like an iterator which, in each |
| 4739 | step, takes one letter of the input and runs (one step on) the |
| 4740 | finite state machine with this input. More precisely, in each |
| 4741 | step, the process iterator takes an outgoing transition of the |
| 4742 | current state, whose input label equals the input letter of the |
| 4743 | tape. The output label of the transition, if present, is written |
| 4744 | on the output tape. |
| 4745 | |
| 4746 | INPUT: |
| 4747 | |
| 4748 | - ``fsm`` -- The finite state machine on which the input should be |
| 4749 | processed. |
| 4750 | |
| 4751 | - ``input_tape`` -- The input tape. It can be anything that is |
| 4752 | iterable. |
| 4753 | |
| 4754 | - ``initial_state`` -- The initial state in which the machine |
| 4755 | starts. If this is ``None``, the unique inital state of the |
| 4756 | finite state machine is takes. If there are several, an error is |
| 4757 | reported. |
| 4758 | |
| 4759 | The process (iteration) stops if there are no more input letters |
| 4760 | on the tape. In this case a StopIteration exception is thrown. As |
| 4761 | result the following attributes are available: |
| 4762 | |
| 4763 | - ``accept_input`` -- Is True if the reached state is a final state. |
| 4764 | |
| 4765 | - ``current_state`` -- The current/reached state in the process. |
| 4766 | |
| 4767 | - ``output_tape`` -- The written output. |
| 4768 | |
| 4769 | Current values of those attribtes (except ``accept_input``) are |
| 4770 | (also) available during the iteration. |
| 4771 | |
| 4772 | OUTPUT: |
| 4773 | |
| 4774 | An iterator. |
| 4775 | |
| 4776 | EXAMPLES: |
| 4777 | |
| 4778 | The following transducer reads binary words and outputs a word, |
| 4779 | where blocks of ones are replaced by just a single one. Further |
| 4780 | only words that end with a zero are accepted. |
| 4781 | |
| 4782 | :: |
| 4783 | |
| 4784 | sage: T = Transducer({'A': [('A', 0, 0), ('B', 1, None)], |
| 4785 | ....: 'B': [('B', 1, None), ('A', 0, [1, 0])]}, |
| 4786 | ....: initial_states=['A'], final_states=['A']) |
| 4787 | sage: input = [1, 1, 0, 0, 1, 0, 1, 1, 1, 0] |
| 4788 | sage: T.process(input) |
| 4789 | (True, 'A', [1, 0, 0, 1, 0, 1, 0]) |
| 4790 | |
| 4791 | The function :meth:`FiniteStateMachine.process` created a new |
| 4792 | ``FSMProcessIterator``. We can do that manually, too, and get full |
| 4793 | access to the iteration process:: |
| 4794 | |
| 4795 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 4796 | sage: it = FSMProcessIterator(T, input_tape=input) |
| 4797 | sage: for _ in it: |
| 4798 | ....: print (it.current_state, it.output_tape) |
| 4799 | ('B', []) |
| 4800 | ('B', []) |
| 4801 | ('A', [1, 0]) |
| 4802 | ('A', [1, 0, 0]) |
| 4803 | ('B', [1, 0, 0]) |
| 4804 | ('A', [1, 0, 0, 1, 0]) |
| 4805 | ('B', [1, 0, 0, 1, 0]) |
| 4806 | ('B', [1, 0, 0, 1, 0]) |
| 4807 | ('B', [1, 0, 0, 1, 0]) |
| 4808 | ('A', [1, 0, 0, 1, 0, 1, 0]) |
| 4809 | sage: it.accept_input |
| 4810 | True |
| 4811 | """ |
| 4812 | def __init__(self, fsm, input_tape=None, initial_state=None): |
| 4813 | """ |
| 4814 | See :class:`FSMProcessIterator` for more information. |
| 4815 | |
| 4816 | EXAMPLES:: |
| 4817 | |
| 4818 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 4819 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 4820 | ....: initial_states=['A'], final_states=['A']) |
| 4821 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 4822 | sage: for _ in it: |
| 4823 | ....: pass |
| 4824 | sage: it.output_tape |
| 4825 | [1, 0] |
| 4826 | """ |
| 4827 | self.fsm = fsm |
| 4828 | if initial_state is None: |
| 4829 | fsm_initial_states = self.fsm.initial_states() |
| 4830 | try: |
| 4831 | self.current_state = fsm_initial_states[0] |
| 4832 | except IndexError: |
| 4833 | raise ValueError, "No state is initial." |
| 4834 | if len(fsm_initial_states) > 1: |
| 4835 | raise ValueError, "Several initial states." |
| 4836 | else: |
| 4837 | self.current_state = initial_state |
| 4838 | self.output_tape = [] |
| 4839 | if input_tape is None: |
| 4840 | self._input_tape_iter_ = iter([]) |
| 4841 | else: |
| 4842 | if hasattr(input_tape, '__iter__'): |
| 4843 | self._input_tape_iter_ = iter(input_tape) |
| 4844 | else: |
| 4845 | raise ValueError, "Given input tape is not iterable." |
| 4846 | |
| 4847 | def __iter__(self): |
| 4848 | """ |
| 4849 | Returns ``self``. |
| 4850 | |
| 4851 | TESTS:: |
| 4852 | |
| 4853 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 4854 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 4855 | ....: initial_states=['A'], final_states=['A']) |
| 4856 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 4857 | sage: id(it) == id(iter(it)) |
| 4858 | True |
| 4859 | """ |
| 4860 | return self |
| 4861 | |
| 4862 | def next(self): |
| 4863 | """ |
| 4864 | Makes one step in processing the input tape. |
| 4865 | |
| 4866 | INPUT: |
| 4867 | |
| 4868 | Nothing. |
| 4869 | |
| 4870 | OUTPUT: |
| 4871 | |
| 4872 | It returns the taken transition. A ``StopIteration`` exception is |
| 4873 | thrown when there is nothing more to read. |
| 4874 | |
| 4875 | EXAMPLES:: |
| 4876 | |
| 4877 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 4878 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 4879 | ....: initial_states=['A'], final_states=['A']) |
| 4880 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 4881 | sage: it.next() |
| 4882 | Transition from 'A' to 'A': 0|1 |
| 4883 | sage: it.next() |
| 4884 | Transition from 'A' to 'A': 1|0 |
| 4885 | sage: it.next() |
| 4886 | Traceback (most recent call last): |
| 4887 | ... |
| 4888 | StopIteration |
| 4889 | """ |
| 4890 | if hasattr(self, 'accept_input'): |
| 4891 | raise StopIteration |
| 4892 | try: |
| 4893 | # process current state |
| 4894 | transition = None |
| 4895 | try: |
| 4896 | transition = self.current_state.hook( |
| 4897 | self.current_state, self) |
| 4898 | except AttributeError: |
| 4899 | pass |
| 4900 | self.write_word(self.current_state.word_out) |
| 4901 | |
| 4902 | # get next |
| 4903 | if not isinstance(transition, FSMTransition): |
| 4904 | next_word = [] |
| 4905 | found = False |
| 4906 | |
| 4907 | try: |
| 4908 | while not found: |
| 4909 | next_word.append(self.read_letter()) |
| 4910 | try: |
| 4911 | transition = self.get_next_transition( |
| 4912 | next_word) |
| 4913 | found = True |
| 4914 | except ValueError: |
| 4915 | pass |
| 4916 | except StopIteration: |
| 4917 | # this means input tape is finished |
| 4918 | if len(next_word) > 0: |
| 4919 | self.accept_input = False |
| 4920 | raise StopIteration |
| 4921 | |
| 4922 | # process transition |
| 4923 | try: |
| 4924 | transition.hook(transition, self) |
| 4925 | except AttributeError: |
| 4926 | pass |
| 4927 | self.write_word(transition.word_out) |
| 4928 | |
| 4929 | # go to next state |
| 4930 | self.current_state = transition.to_state |
| 4931 | |
| 4932 | except StopIteration: |
| 4933 | # this means, either input tape is finished or |
| 4934 | # someone has thrown StopIteration manually (in one |
| 4935 | # of the hooks) |
| 4936 | if not self.current_state.is_final: |
| 4937 | self.accept_input = False |
| 4938 | if not hasattr(self, 'accept_input'): |
| 4939 | self.accept_input = True |
| 4940 | raise StopIteration |
| 4941 | |
| 4942 | # return |
| 4943 | return transition |
| 4944 | |
| 4945 | def read_letter(self): |
| 4946 | """ |
| 4947 | Reads a letter from the input tape. |
| 4948 | |
| 4949 | INPUT: |
| 4950 | |
| 4951 | Nothing. |
| 4952 | |
| 4953 | OUTPUT: |
| 4954 | |
| 4955 | A letter. |
| 4956 | |
| 4957 | Exception ``StopIteration`` is thrown if tape has reached |
| 4958 | the end. |
| 4959 | |
| 4960 | EXAMPLES:: |
| 4961 | |
| 4962 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 4963 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 4964 | ....: initial_states=['A'], final_states=['A']) |
| 4965 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 4966 | sage: it.read_letter() |
| 4967 | 0 |
| 4968 | """ |
| 4969 | return self._input_tape_iter_.next() |
| 4970 | |
| 4971 | def write_letter(self, letter): |
| 4972 | """ |
| 4973 | Writes a letter on the output tape. |
| 4974 | |
| 4975 | INPUT: |
| 4976 | |
| 4977 | - ``letter`` -- the letter to be written. |
| 4978 | |
| 4979 | OUTPUT: |
| 4980 | |
| 4981 | Nothing. |
| 4982 | |
| 4983 | EXAMPLES:: |
| 4984 | |
| 4985 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 4986 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 4987 | ....: initial_states=['A'], final_states=['A']) |
| 4988 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 4989 | sage: it.write_letter(42) |
| 4990 | sage: it.output_tape |
| 4991 | [42] |
| 4992 | """ |
| 4993 | self.output_tape.append(letter) |
| 4994 | |
| 4995 | def write_word(self, word): |
| 4996 | """ |
| 4997 | Writes a word on the output tape. |
| 4998 | |
| 4999 | INPUT: |
| 5000 | |
| 5001 | - ``word`` -- the word to be written. |
| 5002 | |
| 5003 | OUTPUT: |
| 5004 | |
| 5005 | Nothing. |
| 5006 | |
| 5007 | EXAMPLES:: |
| 5008 | |
| 5009 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 5010 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 5011 | ....: initial_states=['A'], final_states=['A']) |
| 5012 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 5013 | sage: it.write_word([4, 2]) |
| 5014 | sage: it.output_tape |
| 5015 | [4, 2] |
| 5016 | """ |
| 5017 | for letter in word: |
| 5018 | self.write_letter(letter) |
| 5019 | |
| 5020 | def get_next_transition(self, word_in): |
| 5021 | """ |
| 5022 | Returns the next transition according to ``word_in``. It is |
| 5023 | assumed that we are in state ``self.current_state``. |
| 5024 | |
| 5025 | INPUT: |
| 5026 | |
| 5027 | - ``word_in`` -- the input word. |
| 5028 | |
| 5029 | OUTPUT: |
| 5030 | |
| 5031 | The next transition according to ``word_in``. It is assumed |
| 5032 | that we are in state ``self.current_state``. |
| 5033 | |
| 5034 | EXAMPLES:: |
| 5035 | |
| 5036 | sage: from sage.combinat.finite_state_machine import FSMProcessIterator |
| 5037 | sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]}, |
| 5038 | ....: initial_states=['A'], final_states=['A']) |
| 5039 | sage: it = FSMProcessIterator(inverter, input_tape=[0, 1]) |
| 5040 | sage: it.get_next_transition([0]) |
| 5041 | Transition from 'A' to 'A': 0|1 |
| 5042 | """ |
| 5043 | for transition in self.current_state.transitions: |
| 5044 | if transition.word_in == word_in: |
| 5045 | return transition |
| 5046 | raise ValueError |
| 5047 | |
| 5048 | |
| 5049 | #***************************************************************************** |
| 5050 | |
| 5051 | |
| 5052 | def setup_latex_preamble(): |
| 5053 | """ |
| 5054 | This function adds the package ``tikz`` with support for automata |
| 5055 | to the preamble of Latex so that the finite state machines can be |
| 5056 | drawn nicely. |
| 5057 | |
| 5058 | INPUT: |
| 5059 | |
| 5060 | Nothing. |
| 5061 | |
| 5062 | OUTPUT: |
| 5063 | |
| 5064 | Nothing. |
| 5065 | |
| 5066 | TESTS:: |
| 5067 | |
| 5068 | sage: from sage.combinat.finite_state_machine import setup_latex_preamble |
| 5069 | sage: setup_latex_preamble() |
| 5070 | """ |
| 5071 | latex.add_package_to_preamble_if_available('tikz') |
| 5072 | latex.add_to_preamble('\\usetikzlibrary{automata}') |
| 5073 | |
| 5074 | |
| 5075 | #***************************************************************************** |