Ask Your Question
1

How to simplify expression

asked 2021-09-02 18:12:33 +0200

maslbl4 gravatar image

What are some ways to simplify the following expression?

(A <= B && B <= C) || (A <= D && D <= C) || (B <= A && A <= D && D <= C) || (D >= C && A <= B && B <= C ) || (B <= A && D >= C)
edit retag flag offensive close merge delete

Comments

to try to answer, I tried the code below (which must surely not be correct) but the kernel never stops !

log = SymbolicLogic()
s = log.statement("((A <= B) | (B <= C)) | ((A <= D) & (D <= C)) | ((B <= A) & ((A <= D) & (D <= C)) | ((D >= C) & (A <= B)) & ((B <= C ) | (B <= A) & (D >= C))")
ortollj gravatar imageortollj ( 2021-09-03 08:23:44 +0200 )edit

From the manual :

"Formulas consist of the following operators:

  • & – and

  • | – or

  • ~ – not

  • ^ – xor

  • -> – if-then

  • <-> – if and only if

Operators can be applied to variables that consist of a leading letter and trailing underscores and alphanumerics. Parentheses may be used to explicitly show order of operation."

Your code uses symbols not in the above list ; no wonder that it can't compile...

More important : "A>=B" is a predicative assertion, not a proposition. First-order predicate logic is a horse of another color...

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2021-09-03 13:56:00 +0200 )edit

2 Answers

Sort by » oldest newest most voted
2

answered 2021-09-03 08:13:38 +0200

Emmanuel Charpentier gravatar image

A couple of answer elements :

  • Sage per se has not (yet) symbolic logical operators (but see Trac#31911).

  • Your (Mathematica ?) syntax is not Python : there is no such operators as && or ||...

With these limitations in mind, you can do :

sage: var("A, B, C, D")
(A, B, C, D)
sage: import sympy
sage: sympy.sympify('((A <= B) & (B <= C)) | ((A <= D) & (D <= C)) | ((B <= A) & (A <= D) & (D <= C)) | ((D >= C) & (A <= B) & (B <= C) ) | ((B <= A) & (D >= C))').simplify()
((A >= B) & (C <= D)) | ((C >= D) & (A <= D)) | ((A <= B) & (B <= C))

but this result cannot (yet) be translated in Sage.

There is some support for propositional logic in Sage, but these tools use string representation of the formulæ ; the translation to Sage symbolic expressions would require the addition of some support for symbolic logical functions (see the Trac cicket...).

FWIW :

sage: mathematica("FullSimplify[(A <= B && B <= C) || (A <= D && D <= C) || (B <= A && A <= D && D <= C) || (D >= C && A <= B && B <= C ) || (B <= A && D >= C)]")
(D >= C && B <= A) || (A <= B && B <= C) || (A <= D && D <= C)

sage: maxima_console()
;;; Loading #P"/usr/local/sage-9/local/lib/ecl-21.2.1/sb-bsd-sockets.fas"
;;; Loading #P"/usr/local/sage-9/local/lib/ecl-21.2.1/sockets.fas"
Maxima 5.45.0 https://maxima.sourceforge.io
using Lisp ECL 21.2.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) simplify((A <= B and B <= C) or (A <= D and D <= C) or (B <= A and A <= D and D <= C) or (D >= C and A <= B and B <= C ) or (B <= A and D >= C));
(%o1) simplify(((A <= B) and (B <= C)) or ((A <= D) and (D <= C))
 or ((B <= A) and (A <= D) and (D <= C))
 or ((D >= C) and (A <= B) and (B <= C)) or ((B <= A) and (D >= C)))

HTH,

edit flag offensive delete link more
2

answered 2021-09-03 09:27:47 +0200

philipp7 gravatar image

If we assume that all $A,B,C,D$ are real, we can use cylindrical algebraic decomposition (CAD) to simplify such an expression (in general, CAD can handle logical formulas involving polynomial (in)equalities). Sage has an interface to QEPCAD (QEPCAD itself, however, I believe is not shipped with sage, so has to be installed) which is one of the most popular implementations of CAD. Here is how QEPCAD simplifies your expression:

sage: from sage.interfaces.qepcad import qepcad_formula, qepcad, qformula
sage: var("A,B,C,D")
sage: qf = qepcad_formula
sage: expr = qf.or_([qf.and_([A<=B, B<=C]), qf.and_([A<=D, D<=C]), qf.and_([B<=A, A<=D, D<=C]), qf.and_([D>=C, A<=B, B<=C]), qf.and_([B<=A, D>=C])])
sage: qepcad(expr)
[ B - A >= 0 /\ C - B >= 0 ] \/ [ D - A >= 0 /\ D - C <= 0 ] \/ [ B - A <= 0 /\ D - C >= 0 ]
edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2021-09-02 18:12:33 +0200

Seen: 112 times

Last updated: Sep 03 '21