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Here is logical formula:

$$\bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} \bigvee_{j=1}^{9}~p(i,j,n)$$

To use it in Mathematica I use that code:

X=Array[p[#1,#3,#2]&,{9,9,9}];a=Apply[And,Apply[And,Apply[Or,X,{2}],{1}],{0}];
Y=Array[p,{9,9,9}];b=Apply[And,Apply[And,Apply[Or,Transpose[Y,{1,3,2}],{2}],{1}],{0}];
c = 0

Two methods are used above. In the first one, I am putting the result in a variable and with the second method in b variable.

Now:

MatchQ[a,b]

True

MatchQ[a,c]

False

So, I conclude that the output of a and b are the same.

Now I want to check the logical equivalence of a and b.

I tried:

TrueQ[Equivalent[a, b]]

True

TrueQ[Equivalent[a, c]]

False

TrueQ[Equal[a,b]]

True

TrueQ[Equal[a,c]]

False

SameQ[a,b]

True

SameQ[a,c]

False


Questions:

  1. What is called the content of a and b? Logical formula? Function(s)? Normal expression? Object? Other?

  2. Are the all ways of checking the logical equivalence of a and b that I have used above right?

  3. Are there any other possible ways to check the logical equivalence of a and b that I have not used above?

  4. Is there one best way to check the logical equivalence of a and b and if yes which one and why its is best one?

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1). Maybe, this would be referred to as "Boolean expression" in Mathematica speech.

2.) SameQ is problematic as it does not respect the logical content:

hamlet = Or[be, Not[be]];
a = SameQ[hamlet, True]
b= Equivalent[hamlet, True]

False

be || ! be

So using these in later computions may lead to wrong results:

Simplify[a]
Simplify[b]

False

True

The TrueQ/Equal combo is similarly dangerous as TrueQ will return False whenever it gets a symbolic expression as arguments that does not evaluate immediately to True. Better not use it here.

Moreover, Equivalent may be able to perform some simplifications that Equal cannot perform because using Equivalent implies that its arguments are Boolean expressions.

3.) Some more simplifications are performed by TautologyQ:

TautologyQ[Equivalent[hamlet, True]]

True

4.) In life, there is usually never one best way to do anything. For example,SameQ is a very inexpensive test and if it evaluates to True, you are done. Equivalent and TautologyQ have to perform actual computations, taking a bit longer.

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  • $\begingroup$ Here TautologyQ[Equivalent[hamlet, True]] as far as I understand first is evaluates Equivalent[hamlet, True] and only after TautologyQ[]. TautologyQ[] receives the only final result of Equivalent[] which can only be True orFalse. In that case, how TautologyQ[] can perform more simplifications? What more simplifications can be done on True orFalse? .... $\endgroup$ – vasili111 Sep 17 '18 at 12:21
  • $\begingroup$ .... Or maybe TautologyQ[] forces Equivalent[] to perform more simplifications before Equivalent[] evaluates and returns the result? In that case, TautologyQ[] is only here to force Equivalent[] to make additional simplifications and nothing more? $\endgroup$ – vasili111 Sep 17 '18 at 12:21
  • $\begingroup$ For example, TautologicalQ might search the subexpressions for short-circuits. Equivalent alone probably does not do that for performance reasons. $\endgroup$ – Henrik Schumacher Sep 17 '18 at 12:25
  • $\begingroup$ So, TautologyQ[] is only here to force Equivalent[] to make additional simplifications before Equivalent[] actually evaluates and nothing more? $\endgroup$ – vasili111 Sep 17 '18 at 12:29
  • $\begingroup$ I am even not sure whether Equivalent performs any but the very simplest simplifications at all. Working seldomly with these things and not being a developer, I have few further knowledge about these things. All I can say: The general strategy about simplification in Mathematica is so that the user has to actively ask for more expensive simplifications. (Compare, e.g. Simplify and FullSimplify and the fact that they have further options that can be tuned.) $\endgroup$ – Henrik Schumacher Sep 17 '18 at 12:38

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