spot/doc/org/tut04.org

Testing the equivalence of two formulas

• Shell
• Python
• C++

This page show how to test whether two LTL/PSL formulas are equal.

Shell

Using a ltlfilt you can use --equivalent-to=f to filter a list of LTL formula and retain only those equivalent to f. So this gives an easy way to test the equivalence of two formulas:

ltlfilt -f '(a U b) U a' --equivalent-to 'b U a'

(a U b) U a

Since input formula was output, it means it is equivalent. Adding -c to count the number for formula output provide a yes/no answer.

ltlfilt -c -f '(a U b) U a' --equivalent-to 'b U a'

1

Python

In Python, we can test this via a language_containment_checker object:

import spot
f = spot.formula("(a U b) U a")
g = spot.formula("b U a")
c = spot.language_containment_checker()
print(c.equal(f, g))

True

The equivalence check is done by converting the formulas $f$ and $g$ and their negation into four automata $A_f$, $A_{\lnot f}$, $A_g$, and $A_{\lnot g}$, and then making sure that $A_f\otimes A_{\lnot g}$ and $A_g\otimes A_{\lnot f}$ are empty.

We could also write this check by doing the translation and emptiness check ourselves. For instance:

import spot

def implies(f, g):
    a_f = f.translate()
    a_ng = spot.formula_Not(g).translate()
    return spot.product(a_f, a_ng).is_empty()

def equiv(f, g):
    return implies(f, g) and implies(g, f)

f = spot.formula("(a U b) U a")
g = spot.formula("b U a")
print(equiv(f, g))

True

The language_containment_checker object essentially performs the same work, but it also implements a cache to avoid translating the same formulas multiple times when it is used to test multiple equivalence.

C++

Here is a C++ translation of the first Python example.

  #include <iostream>
  #include <spot/tl/parse.hh>
  #include <spot/tl/contain.hh>

  int main()
  {
    spot::formula f = spot::parse_formula("(a U b) U a");
    spot::formula g = spot::parse_formula("b U a");
    spot::language_containment_checker c;
    std::cout << (c.equal(f, g) ? "Equivalent\n" : "Not equivalent\n");
  }

Equivalent