# -*- coding: utf-8 -*-
#+TITLE: Testing the equivalence of two formulas
#+DESCRIPTION: Code example for testing the equivalence of two LTL or PSL formulas
#+INCLUDE: setup.org
#+HTML_LINK_UP: tut.html
#+PROPERTY: header-args:sh :results verbatim :exports both
#+PROPERTY: header-args:python :results output :exports both
#+PROPERTY: header-args:C+++ :results verbatim :exports both

This page shows how to test whether two LTL/PSL formulas are
equivalent, i.e., if they denote the same languages.

* 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:

#+BEGIN_SRC sh
ltlfilt -f '(a U b) U a' --equivalent-to 'b U a'
#+END_SRC
#+RESULTS:
: (a U b) U a

Since the input formula was output, it means it is equivalent to =b U
a=.  You may want to add =-c= to count the number of formula output if
you prefer a 1/0 answer:

#+BEGIN_SRC sh
ltlfilt -c -f '(a U b) U a' --equivalent-to 'b U a'
#+END_SRC
#+RESULTS:
: 1

Or use =-q= if you only care about the exit status of =ltlfilt=: the
exist status is =0= if some formula matched, and =1= if no formula
matched.  (The effect of these =-c= and =-q= options should be
familiar to =grep= users.)

* Python

In Python, we can implement this in a number of ways.  The
easiest is to use the =spot.are_equivalent()= function.

#+BEGIN_SRC python :results output :exports both
import spot
are_eq = spot.are_equivalent("(a U b) U a", "b U a")
print("Equivalent" if are_eq else "Not equivalent")
#+END_SRC
#+RESULTS:
: Equivalent

The equivalence check is done by converting the input 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 [[file:tut10.org][the translation]] and emptiness
check ourselves.  For instance:

#+BEGIN_SRC python
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("Equivalent" if equiv(f, g) else "Not equivalent")
#+END_SRC
#+RESULTS:
: Equivalent


This can also be done via a =language_containment_checker= object:

#+BEGIN_SRC python
import spot
f = spot.formula("(a U b) U a")
g = spot.formula("b U a")
c = spot.language_containment_checker()
print("Equivalent" if c.equal(f, g) else "Not equivalent")
#+END_SRC
#+RESULTS:
: Equivalent

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
equivalences.

* C++

Here are possible C++ implementations using either =are_equivalent()=
or the =language_containment_checker=.  Note that the
=are_equivalent()= function also work with automata.

#+BEGIN_SRC C++
#include <iostream>
#include <spot/tl/parse.hh>
#include <spot/twaalgos/contains.hh>

int main()
{
  spot::formula f = spot::parse_formula("(a U b) U a");
  spot::formula g = spot::parse_formula("b U a");
  std::cout << (spot::are_equivalent(f, g) ?
                "Equivalent\n" : "Not equivalent\n");
}
#+END_SRC
#+RESULTS:
: Equivalent

#+BEGIN_SRC C++
#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");
}
#+END_SRC

#+RESULTS:
: Equivalent

#  LocalWords:  utf html args ltlfilt SRC eq lnot otimes ng iostream