SynchronizingWordExistence.h

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/*
 * This file is part of Algorithms library toolkit.
 * Copyright (C) 2017 Jan Travnicek (jan.travnicek@fit.cvut.cz)
 
 * Algorithms library toolkit is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 
 * Algorithms library toolkit is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 
 * You should have received a copy of the GNU General Public License
 * along with Algorithms library toolkit.  If not, see <http://www.gnu.org/licenses/>.
 */
 
#pragma once
 
#include <alib/set>
#include <queue>
 
#include <automaton/FSM/DFA.h>
#include <automaton/transform/AutomataIntersectionCartesianProduct.h>
 
namespace automaton::properties {
 
class SynchronizingWordExistence {
public:
    template < class SymbolType, class StateType >
    static bool exists ( const automaton::DFA < SymbolType, StateType > & dfa );
};
 
template < class SymbolType, class StateType >
bool SynchronizingWordExistence::exists ( const automaton::DFA < SymbolType, StateType > & dfa ) {
    /*
     * It must hold that (1) <=> (2)
     * (1) DFA is synchronizing
     * (2) Foreach q1, q2 in Q, exists a word w such that \delta(q1, w) = \delta(q2, w)
     */
 
    /* Therefore the algorithm is as follows:
     * Create a parallel run of the automaton with itseld
     * All pairs (q1, q2) \in Q x Q MUST have a path to some state-pair (q, q).
     *  -> For every pair (q,q) run a BFS search and check that all pairs (q1, q2) are visited
     */
 
    automaton::DFA < SymbolType, ext::pair < StateType, StateType > > cart = automaton::transform::AutomataIntersectionCartesianProduct:: intersection ( dfa, dfa );
 
    std::queue < ext::pair < StateType, StateType > > q;
    ext::set < ext::pair < StateType, StateType > > visited;
 
    for ( const ext::pair < StateType, StateType > & state: cart.getStates ( ) ) {
        if ( state.first != state.second )
            continue;
 
        visited.insert ( state );
        q.push ( state );
    }
 
    while ( ! q.empty ( ) ) {
        const ext::pair < StateType, StateType > cstate = std::move ( q.front ( ) );
        q.pop ( );
 
        for ( const auto & transition : cart.getTransitionsToState ( cstate ) ) {
            const auto & srcState = transition.first.first;
 
            if ( visited.count ( srcState ) == 0 ) {
                visited.insert ( srcState );
                q.push ( srcState );
            }
        }
    }
 
    return visited == cart.getStates ( );
}
 
} /* namespace automaton::properties */