Opened 12 years ago
Last modified 4 years ago
#7660 needs_work defect
arithmetic with equations and inequalities confusing
Reported by:  burcin  Owned by:  burcin 

Priority:  major  Milestone:  sage8.2 
Component:  symbolics  Keywords:  inequality, solver, maxima 
Cc:  kcrisman  Merged in:  
Authors:  Burcin Erocal, Ralf Stephan  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  u/rws/76601 (Commits, GitHub, GitLab)  Commit:  37c5190e9fdc3a6195a564fd3e61f2eaa06bb892 
Dependencies:  Stopgaps: 
Description (last modified by )
Equations and relations should behave like this: equations:
(a==b) +*/ c
same as:(a==b).add_to_both_sides(c)
(a==b).subtract_from_both_sides(c)
(a==b).multiply_both_sides(c)
(a==b).divide_both_sides(c)
False
if*/0
(a==b)^c
>a^c == b^c
f(a==b)
>f(a)==f(b)
relations:
(a<b) + c
same as:(a<b).add_to_both_sides(c)
(a<b).subtract_from_both_sides(c)
(a<b) */ c
same as:a*/c > b*/c
forc
real and negative, or ifc
is assumed negativea*/c < b*/c
forc
real and positive, or ifc
is assumed positiveFalse
ifc=0
or assumed zero if
c
contains variables (and no assumptions exist about it) raiseArithmeticError: missing assumption: is ...>0?
 if
c
contains no variablesArithmeticError: multiplication of inequality with irreal
(a<b)^c
>(a<b)^c
f(a<b)
>f(a<b)
Original description:
From the following sagedevel thread:
http://groups.google.com/group/sagedevel/t/951d510c814f894f
Arithmetic with inequalities can be confusing, since Sage does nothing to keep the inequality correct
. For example:
On Thu, 10 Dec 2009 00:37:10 0800 (PST) "marik@mendelu.cz" <marik@mendelu.cz> wrote: > sage: f = x + 3 < y  2 > sage: f*(1) > x  3 < y + 2
It seems MMA doesn't apply any automatic simplification in this case:
On Thu, 10 Dec 2009 09:54:36 0800 William Stein <wstein@gmail.com> wrote: > Mathematica does something weird and formal: > > In[1]:= f := x+3 < y2; > In[3]:= f*(1) > Out[3]= (3 + x < 2 + y)
Maple acts more intuitively, though the way
formal products
are printed leaves something to be desired, IMHO:
On Thu, 10 Dec 2009 14:15:53 0800 William Stein <wstein@gmail.com> wrote: > Here is what Maple does: > > flat:release_notes wstein$ maple > \^/ Maple 13 (APPLE UNIVERSAL OSX) > ._\ /_. Copyright (c) Maplesoft, a division of Waterloo Maple > Inc. 2009 \ MAPLE / All rights reserved. Maple is a trademark of > <____ ____> Waterloo Maple Inc. >  Type ? for help. > > f := x < y; > f := x < y > > > f*(3); > 3 y < 3 x > > > f*z; > *(x < y, z) > > > f*a; > *(x < y, a)
We should multiply both sides of the inequality only if the argument is a real number (as opposed to a symbol with real domain), and invert the relation when the argument is negative.
Note that GiNaC leaves everything formal, like MMA, by default:
ginsh  GiNaC Interactive Shell (ginac V1.5.3) __, _______ Copyright (C) 19992009 Johannes Gutenberg University Mainz, (__) *  Germany. This is free software with ABSOLUTELY NO WARRANTY. ._) i N a C  You are welcome to redistribute it under certain conditions. <' For details type `warranty;'. Type ?? for a list of help topics. > f= x < y; x<y > f*1; (x<y) > f*5; 5*(x<y) > f*z; z*(x<y) > f*z; z*(x<y)
Attachments (1)
Change History (70)
comment:1 Changed 11 years ago by
 Cc kcrisman added
comment:2 Changed 11 years ago by
 Description modified (diff)
comment:3 Changed 11 years ago by
 Description modified (diff)
comment:4 Changed 11 years ago by
 Description modified (diff)
comment:5 Changed 11 years ago by
comment:6 Changed 10 years ago by
I propose the following: a*(x<y)
should not be simplified, ever, other than simplifying a
, x
, or y
individually. Are there any problems with this?
comment:7 Changed 10 years ago by
Given the ticket description details above and the discussion at #11309 and here, I think that Burcin's original proposal of
 keeping the same if we know
a
is positive  reversing if we know
a
is negative (presumably in both of these cases only ifa
is coerced to the reals successfully, as Burcin says)  (not in original but reasonable) return something like
False
for<
and=
for<=
ifa=0
?  returning something formal otherwise
seems best. It does seem reasonable to multiply by 1
, after all, and tnv is right that this should either reverse the inequality or remain formal.
comment:8 Changed 10 years ago by
It doesn't seem at all reasonable to me to distribute multiplication over a relation. When you distribute multiplication over addition, the addition expression (a+b)
lives in the same space as a
and b
individually. This is not the case with (x<y)
and x
and y
.
There is no mathematical rationale for saying that (1)*(x<y)
is x < y
or x > y
or anything other than just (1)*(x<y)
. I'm sure many can recall when they first learned basic algebra that their teacher was careful to say "we multiply both sides of the equation by c", not "we multiply the equation by c", just as he or she was careful to say "we multiply both the numerator and denominator of the fraction by c", rather than the dangerous "we multiply the fraction by c"!
If we allow multiplication to distribute over x<y
as if it were a twocoordinate vector, do we want other similarities with vector arithmetic to come into play? Should (a+b)*(x<y)
be equal to a*(x<y) + b*(x<y)
? Now we have addition and presumably subtraction of relational expressions. What is 0*(x<y)
? What is (x<y) + (z>y)
? Should we attempt to reverse inequalities to make them match up when adding them? Do we then end up with (x+y<y+z)
, or (y+z>x+y)
? Or, do we conclude that (z>y)
is equal to (1)*(z<y)
, and so (x<y) + (z>y) == (x<y)  (z<y) == (x+z<2y)
? Going beyond vectorlike arithmetic, what happens when you add a scalar to a relational expression? What is (x<y) + 3
?
I think the most sensible answer to all the above questions is the following: we should not perform arithmetic on relational expressions; when asked to do so, we should return a purely formal expression.
If the title of this ticket disagrees with that answer, I can make another ticket to implement it, but I'm just wondering if anyone disagrees with me strongly about this.
comment:9 Changed 10 years ago by
Well, we could go to the Ginac/Mathematica way too, instead of the Maple way. But now that #11309 is on the way in, probably it's time to deal with this. FWIW, Maxima seems to go that way as well.
(%i1) x<y; (%o1) x < y (%i2) a:x<y; (%o2) x < y (%i3) a; (%o3) x < y (%i4) 1*a; (%o4)  (x < y) (%i5) b*a; (%o5) b (x < y)
Maybe Burcin has a comment; I don't have that strong of feelings, though I'm partial to
(x<y)+3 == x+3<y+3
0*(x<y) is False
but those may just be sentimental, as you suggest.
comment:10 Changed 10 years ago by
Honestly I thought the most likely response would be 0*(x<y) == 0
. False
is an even stranger result because now you have (x<y) == (1+0)*(x<y) == (x<y) + False
, so either False
is now another additive identity (on relations, anyway), thus breaking the universality of ?  ? = 0
, or False == 0
, which is another can of worms...
And if (x<y)+3 == (x+3<y+3)
, then presumably 3 == (3<3) == False
...?
None of this makes any sense, IMHO. I think it would be best to go the Mathematica/GiNaC/Maxima way, as shown in your Maxima output, rather than the Maple way.
comment:11 Changed 9 years ago by
For the record, Maxima has a share package which implements arithmetic on inequalities.
(%i1) load (ineq); (%o1) /home/robert/maxima/maximagit/maximacode/share/simplification/ineq.mac (%i2) e:a < b; (%o2) a < b (%i3) x*e; Is x positive, negative, or zero? p; (%o3) a x < b x (%i4) x*e; Is x positive, negative, or zero? n; (%o4) a x > b x (%i5) x*e; Is x positive, negative, or zero? z; (%o5) (a < b) x
I gather that's similar to what Maple does.
comment:12 Changed 9 years ago by
attachment:trac_7660inequality_arithmetic.patch comments out a few lines in _add_
, _mul_
, etc., methods of symbolic expressions to pass the arithmetic on to GiNaC directly. I don't have time to test this and verify that it returns sensible answers, however we decide to define "sensible". Please test, see what doctests fail and try to produce horrific wrong results.
Quite a few doctests and some documentation will have to change to match this new design decision. I'd like to make sure we agree on the behavior before trying to produce a clean patch. Of course, I'd appreciate help with the documentation in any case.
comment:13 Changed 8 years ago by
 Milestone changed from sage5.11 to sage5.12
comment:14 Changed 8 years ago by
 Milestone changed from sage6.1 to sage6.2
comment:15 Changed 8 years ago by
OK that patch applies cleanly with patch l
. It results in
sage: var('x') x sage: x>1 x > 1 sage: ie=_ sage: ie*1 (x > 1) sage: solve(_,x) ... RuntimeError: ECL says: THROW: The catch MACSYMAQUIT is undefined.
I refrained from starting any doctests.
comment:16 Changed 8 years ago by
 Branch set to u/rws/ticket/7660
 Modified changed from 02/17/14 15:06:42 to 02/17/14 15:06:42
comment:17 Changed 8 years ago by
 Commit set to d2801be816c9c0ea801f760b762d3c47f1b8c45b
 Keywords inequality solver maxima added
New commits:
d2801be  [mq]: trac_7660inequality_arithmetic.patch

comment:18 Changed 8 years ago by
 Work issues set to fix bug
Here are the statements sent via maxima_eval("#$%s$"%statement)
:
Before:
sage: solve((x > 1),x) sage0 : (x)*(1) > 1 sage1 : solve_rat_ineq(sage0) _tmp_ : sage1 kill(sage1) kill(sage0) [[x < 1]]
After:
sage: solve((x > 1),x) sage12 : (x > 1)*(1) = 0 sage13 : x sage14 : solve(sage12,sage13) kill(sage13) kill(sage14) _tmp_ : [x>1=0] <? kill(sage12)
Apparently, Sage's last statement sent is the result itself (no idea why), and maxima then barfs because it can't digest its own output. In maxima:
(%i11) solve((x > 1)*(1) = 0,x); (%o11) [x > 1 = 0] (%i12) [x>1=0]; incorrect syntax: Found logical expression where algebraic expression expected [x>1= ^
In summary there are three problems:
 46 doctests fail in symbolic alone
 maxima can't handle the formal expressions generated via this patch, and
solve()
will fail because it uses maxima solve() and not solve_rat_ineq() (probably because the expression looks like an algebraic). However:sage: solve_ineq((x>1)*(1),[x,y]) #0: solve_rat_ineq(ineq=(x > 1)) ... TypeError: ECL says: Error executing code in Maxima: solve_rat_ineq: (x > 1) is not an inequality.
So maxima refuses to handle the formal expression generated by this patch, and this means they cannot be solved, regardless of method called. So, in addition, solve()
should be changed to do simplification of these expressions before handing them on. This special simplification avoids all issues raised in comment:10. It can be implemented after this ticket (#15906).
calculus.py:symbolic_expression_from_maxima_string()
should catch maximaRuntimeError
s fromecl.c
and rethrow them with meaningful information. (#15902)
`
comment:19 Changed 8 years ago by
 Commit changed from d2801be816c9c0ea801f760b762d3c47f1b8c45b to 39d85bf4e577abc0e9c6136c396e2d234243188c
Branch pushed to git repo; I updated commit sha1. New commits:
39d85bf  Merge branch 'ticket/7660' into tmp

comment:20 Changed 8 years ago by
 Commit changed from 39d85bf4e577abc0e9c6136c396e2d234243188c to ea75f5b519907a28b00f1d8d79469f80cc128015
Branch pushed to git repo; I updated commit sha1. New commits:
ea75f5b  fix relational.add/sub/mul/div_both_sides(); fix doctests

comment:21 Changed 8 years ago by
 Status changed from new to needs_review
 Work issues fix bug deleted
comment:22 Changed 8 years ago by
 Commit changed from ea75f5b519907a28b00f1d8d79469f80cc128015 to de2f0a08b18ce8255a6544ce35ce52ac1f55a231
Branch pushed to git repo; I updated commit sha1. New commits:
de2f0a0  Merge branch 'develop' (6.2.beta4) into ticket/7660

comment:23 Changed 8 years ago by
 Commit changed from de2f0a08b18ce8255a6544ce35ce52ac1f55a231 to 5d934153cf7bea5f985b2ce63113652af45aa2be
comment:24 Changed 8 years ago by
 Milestone changed from sage6.2 to sage6.3
comment:25 Changed 7 years ago by
 Milestone changed from sage6.3 to sage6.4
comment:26 followup: ↓ 28 Changed 7 years ago by
 Status changed from needs_review to needs_work
Please use python3 compatible raise TypeError
syntax.
comment:27 Changed 7 years ago by
 Commit changed from 5d934153cf7bea5f985b2ce63113652af45aa2be to 30cc86077172f9e57c6b5df06b40158e17e2320d
comment:28 in reply to: ↑ 26 Changed 7 years ago by
Replying to aapitzsch:
Please use python3 compatible
raise TypeError
syntax.
Done. Still, doctests fail in expression.pyx
.
comment:29 Changed 7 years ago by
This came up again in http://ask.sagemath.org/question/25217/howtodooperationsthatchangearelation/
comment:30 Changed 7 years ago by
I'm just now looking at this ticket and personally I don't like to lose the ability to compute with relations. I think that the fact that (a == b)^2
returns a^2 == b^2
is a feature, not a bug. I'd say it's up to the user to ensure that the operation makes sense.
Would it be possible to keep the old behaviour for equalities ==
at least?
comment:31 followup: ↓ 32 Changed 7 years ago by
 Description modified (diff)
 Summary changed from arithmetic with inequalities confusing to arithmetic with equations and inequalities confusing
Let's start this fresh. To summarize, what's wanted is the following:
equations:
(a==b) +*/ c
same as:(a==b).add_to_both_sides(c)
(a==b).subtract_from_both_sides(c)
(a==b).multiply_both_sides(c)
(a==b).divide_both_sides(c)
False
if*/0
(a==b)^c
>a^c == b^c
f(a==b)
>f(a==b)
relations:
(a<b) + c
same as:(a<b).add_to_both_sides(c)
(a<b).subtract_from_both_sides(c)
(a<b) */ c
same as:a*/c > b*/c
forc
real and negative, or ifc
is assumed negativea*/c < b*/c
forc
real and positive, or ifc
is assumed positiveFalse
ifc=0
(a<b)^c
>(a<b)^c
f(a<b)
>f(a<b)
Question: Real or: if coerced to the reals successfully? A followup extension would be like Maxima's ineq package, i.e., ask before doing a sign switch.
comment:32 in reply to: ↑ 31 ; followup: ↓ 33 Changed 7 years ago by
Replying to rws:
f(a==b)
>f(a==b)
What does f(a==b)
even mean? I would go for f(a) == f(b)
.
relations:
(a<b) */ c
same as:
a*/c > b*/c
forc
real and negative, or ifc
is assumed negativea*/c < b*/c
forc
real and positive, or ifc
is assumed positiveFalse
ifc=0
What if neither of the above conditions is true? raise ArithmeticError
in that case?
comment:33 in reply to: ↑ 32 ; followup: ↓ 34 Changed 7 years ago by
Replying to jdemeyer:
(a<b) */ c
What if neither of the above conditions is true?
raise ArithmeticError
in that case?
Not if c
contains variables. Is there even a way of determining this?
comment:34 in reply to: ↑ 33 ; followup: ↓ 35 Changed 7 years ago by
Replying to rws:
Not if
c
contains variables.
What would you propose that x * (y < z)
answers then (where x
, y
and z
are variables)?
comment:35 in reply to: ↑ 34 Changed 7 years ago by
Replying to jdemeyer:
What would you propose that
x * (y < z)
answers then (wherex
,y
andz
are variables)?
Since this ticket will check for assumptions, if there is none we should raise an ArithmeticError
demanding one.
comment:36 Changed 7 years ago by
So my statement to raise ArithmeticError
if neither condition is true still remains.
comment:37 Changed 7 years ago by
 Description modified (diff)
comment:38 Changed 7 years ago by
 Description modified (diff)
comment:39 in reply to: ↑ description Changed 7 years ago by
Replying to burcin:
 if
c
contains variables (and no assumptions exist about it) raiseArithmeticError: missing assumption: is ...>0?
 if
c
contains no variablesArithmeticError: multiplication of inequality with irreal
I think it will be easier if these are just one case.
comment:40 Changed 7 years ago by
 Branch changed from u/rws/ticket/7660 to public/7660
comment:41 Changed 7 years ago by
 Commit changed from 30cc86077172f9e57c6b5df06b40158e17e2320d to 14ede75c5f56554ef9e61c094d3bb4e971280314
 Status changed from needs_work to needs_review
Since the f()
stuff requires changes in function base classes, and assumptions are also an involvement, I'll leave both to followup tickets. Please review.
New commits:
855fba8  7660: power of inequality no longer distributes over sides

47251f2  7660: switch inequality sign with negative reals, False if zero; provide original behaviour in mul/div_both_sides()

efeab1c  7660: same with division

14ede75  7660: add errors; fix some doctests

comment:42 followup: ↓ 43 Changed 7 years ago by
Why convert to RR
? I would simply use
if right == 0: ... elif right >= 0: ... elif right <= 0: ... else: raise ArithmeticError(...)
This handles nonconstant symbolic expressions also.
comment:43 in reply to: ↑ 42 Changed 7 years ago by
Replying to jdemeyer:
Why convert to
RR
? I would simply useif right == 0: ... elif right >= 0: ... elif right <= 0: ... else: raise ArithmeticError(...)This handles nonconstant symbolic expressions also.
Seems we have to open a ticket?
sage: def f(ex): if ex==0: print 'zero' elif ex<0: print 'minus' else: print 'else' ....: sage: ex=I sage: f(ex) minus sage: bool(I<0) True sage: bool(I>0) True
comment:44 Changed 7 years ago by
 Commit changed from 14ede75c5f56554ef9e61c094d3bb4e971280314 to d89485d979ab0e51bd1b8b2e1ea463878b4d6ad5
comment:45 Changed 7 years ago by
 Commit changed from d89485d979ab0e51bd1b8b2e1ea463878b4d6ad5 to 17c4e8ad25114508ca1f722f66b497b594bca595
Branch pushed to git repo; I updated commit sha1. New commits:
17c4e8a  Merge branch 'develop' into t/7660/public/7660

comment:46 Changed 7 years ago by
 Description modified (diff)
comment:47 Changed 7 years ago by
 Milestone changed from sage6.4 to sage6.7
comment:48 Changed 7 years ago by
 Commit changed from 17c4e8ad25114508ca1f722f66b497b594bca595 to 463a676f26b010d4aa9079ecdccc88f20490f8d3
Branch pushed to git repo; I updated commit sha1. New commits:
463a676  Merge branch 'develop' into t/7660/public/7660

comment:49 Changed 7 years ago by
 Milestone changed from sage6.7 to sage6.8
comment:50 Changed 6 years ago by
 Commit changed from 463a676f26b010d4aa9079ecdccc88f20490f8d3 to 32f923d9489e7538f9bf36e2a297cce348b7bbb2
Branch pushed to git repo; I updated commit sha1. New commits:
32f923d  Merge branch 'develop' into t/7660/public/7660

comment:51 Changed 6 years ago by
 Commit changed from 32f923d9489e7538f9bf36e2a297cce348b7bbb2 to 7be89303312e8a879cfed49f25091e5848adda43
Branch pushed to git repo; I updated commit sha1. New commits:
7be8930  Merge branch 'develop' into t/7660/public/7660

comment:52 Changed 6 years ago by
 Status changed from needs_review to needs_work
 Work issues set to rebase
comment:53 followup: ↓ 54 Changed 6 years ago by
@aapitzsch Are you willing to review this issue?
comment:54 in reply to: ↑ 53 Changed 6 years ago by
comment:55 Changed 5 years ago by
 Commit changed from 7be89303312e8a879cfed49f25091e5848adda43 to 60ac7182b892db0f5e670bb089e8b72e12c1ba1e
Branch pushed to git repo; I updated commit sha1. New commits:
60ac718  Merge branch 'develop' into t/7660/public/7660

comment:56 Changed 5 years ago by
 Milestone changed from sage6.8 to sage7.4
 Status changed from needs_work to needs_review
 Work issues rebase deleted
comment:57 Changed 5 years ago by
 Commit changed from 60ac7182b892db0f5e670bb089e8b72e12c1ba1e to aebaf17f072f7c06e55d2025a4279c5c78ec7a48
Branch pushed to git repo; I updated commit sha1. New commits:
aebaf17  missing change due to incomplete merge conflict

comment:58 Changed 5 years ago by
 Status changed from needs_review to needs_work
Relevant doctest fail in src/sage/misc/sageinspect.py
comment:59 Changed 5 years ago by
 Commit changed from aebaf17f072f7c06e55d2025a4279c5c78ec7a48 to 2edba4d8277d04329315a95c7c07ebc5dc69529a
comment:60 Changed 5 years ago by
 Milestone changed from sage7.4 to sage7.6
 Status changed from needs_work to needs_review
comment:62 Changed 5 years ago by
 Commit changed from 2edba4d8277d04329315a95c7c07ebc5dc69529a to 352c685e2446d6fcbbb908d0282109fcab1795c3
Branch pushed to git repo; I updated commit sha1. New commits:
352c685  Merge branch 'develop' into t/7660/public/7660

comment:63 Changed 5 years ago by
 Status changed from needs_work to needs_review
comment:65 Changed 5 years ago by
In the course of this ticket already 12 merge commits were added. The branch has 8 so I'm doing a squash.
comment:66 Changed 5 years ago by
 Branch changed from public/7660 to u/rws/76601
comment:67 Changed 5 years ago by
 Commit changed from 352c685e2446d6fcbbb908d0282109fcab1795c3 to bb1aae09cd6e1ff08126d928af3cb1f4fdfb36f6
 Milestone changed from sage7.6 to sage8.1
 Status changed from needs_work to needs_review
New commits:
bb1aae0  7660: fix arithmetic with equations and inequalities

comment:68 Changed 4 years ago by
 Milestone changed from sage8.1 to sage8.2
 Status changed from needs_review to needs_work
does not apply
comment:69 Changed 4 years ago by
 Commit changed from bb1aae09cd6e1ff08126d928af3cb1f4fdfb36f6 to 37c5190e9fdc3a6195a564fd3e61f2eaa06bb892
Branch pushed to git repo; I updated commit sha1. New commits:
37c5190  Merge branch 'develop' into t/7660/76601

this is quite dangerous (I encountered it as a bug in a project). Hopefully someone will try to fix this soon. Could it be related to this http://trac.sagemath.org/sage_trac/ticket/11309 ?