ltl2tgba_fm.cc 27.4 KB
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// Copyright (C) 2003, 2004, 2005, 2006 Laboratoire d'Informatique de
// Paris 6 (LIP6), dpartement Systmes Rpartis Coopratifs (SRC),
// Universit Pierre et Marie Curie.
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
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//
// This file is part of Spot, a model checking library.
//
// Spot 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 2 of the License, or
// (at your option) any later version.
//
// Spot 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 Spot; see the file COPYING.  If not, write to the Free
// Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
// 02111-1307, USA.

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#include "misc/hash.hh"
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#include "misc/bddalloc.hh"
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#include "misc/bddlt.hh"
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#include "misc/minato.hh"
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#include "ltlast/visitor.hh"
#include "ltlast/allnodes.hh"
#include "ltlvisit/lunabbrev.hh"
#include "ltlvisit/nenoform.hh"
#include "ltlvisit/destroy.hh"
#include "ltlvisit/tostring.hh"
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#include "ltlvisit/postfix.hh"
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#include "ltlvisit/apcollect.hh"
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#include <cassert>
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#include <memory>
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#include "ltl2tgba_fm.hh"
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#include "tgba/tgbaproduct.hh"
#include "tgbaalgos/gtec/gtec.hh"
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namespace spot
{
  using namespace ltl;

  namespace
  {

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    // Helper dictionary.  We represent formulae using BDDs to
    // simplify them, and then translate BDDs back into formulae.
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    //
    // The name of the variables are inspired from Couvreur's FM paper.
    //   "a" variables are promises (written "a" in the paper)
    //   "next" variables are X's operands (the "r_X" variables from the paper)
    //   "var" variables are atomic propositions.
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    class translate_dict
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    {
    public:

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      translate_dict(bdd_dict* dict)
	: dict(dict),
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	  a_set(bddtrue),
	  var_set(bddtrue),
	  next_set(bddtrue)
      {
      }

      ~translate_dict()
      {
	fv_map::iterator i;
	for (i = next_map.begin(); i != next_map.end(); ++i)
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	  destroy(i->first);
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	dict->unregister_all_my_variables(this);
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      }

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      bdd_dict* dict;

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      typedef bdd_dict::fv_map fv_map;
      typedef bdd_dict::vf_map vf_map;
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      fv_map next_map;	       ///< Maps "Next" variables to BDD variables
      vf_map next_formula_map; ///< Maps BDD variables to "Next" variables

      bdd a_set;
      bdd var_set;
      bdd next_set;

      int
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      register_proposition(const formula* f)
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      {
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	int num = dict->register_proposition(f, this);
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	var_set &= bdd_ithvar(num);
	return num;
      }

      int
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      register_a_variable(const formula* f)
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      {
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	int num = dict->register_acceptance_variable(f, this);
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	a_set &= bdd_ithvar(num);
	return num;
      }

      int
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      register_next_variable(const formula* f)
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      {
	int num;
	// Do not build a Next variable that already exists.
	fv_map::iterator sii = next_map.find(f);
	if (sii != next_map.end())
	  {
	    num = sii->second;
	  }
	else
	  {
	    f = clone(f);
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	    num = dict->register_anonymous_variables(1, this);
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	    next_map[f] = num;
	    next_formula_map[num] = f;
	  }
	next_set &= bdd_ithvar(num);
	return num;
      }

      std::ostream&
      dump(std::ostream& os) const
      {
	fv_map::const_iterator fi;
	os << "Next Variables:" << std::endl;
	for (fi = next_map.begin(); fi != next_map.end(); ++fi)
	{
	  os << "  " << fi->second << ": Next[";
	  to_string(fi->first, os) << "]" << std::endl;
	}
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	os << "Shared Dict:" << std::endl;
	dict->dump(os);
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	return os;
      }

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      formula*
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      var_to_formula(int var) const
      {
	vf_map::const_iterator isi = next_formula_map.find(var);
	if (isi != next_formula_map.end())
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	  return clone(isi->second);
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	isi = dict->acc_formula_map.find(var);
	if (isi != dict->acc_formula_map.end())
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	  return clone(isi->second);
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	isi = dict->var_formula_map.find(var);
	if (isi != dict->var_formula_map.end())
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	  return clone(isi->second);
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	assert(0);
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	// Never reached, but some GCC versions complain about
	// a missing return otherwise.
	return 0;
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      }

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      formula*
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      conj_bdd_to_formula(bdd b)
      {
	if (b == bddfalse)
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	  return constant::false_instance();
	multop::vec* v = new multop::vec;
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	while (b != bddtrue)
	  {
	    int var = bdd_var(b);
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	    formula* res = var_to_formula(var);
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	    bdd high = bdd_high(b);
	    if (high == bddfalse)
	      {
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		res = unop::instance(unop::Not, res);
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		b = bdd_low(b);
	      }
	    else
	      {
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		assert(bdd_low(b) == bddfalse);
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		b = high;
	      }
	    assert(b != bddfalse);
	    v->push_back(res);
	  }
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	return multop::instance(multop::And, v);
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      }

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      const formula*
      bdd_to_formula(bdd f)
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      {
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	if (f == bddfalse)
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	  return constant::false_instance();
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	multop::vec* v = new multop::vec;

	minato_isop isop(f);
	bdd cube;
	while ((cube = isop.next()) != bddfalse)
	  v->push_back(conj_bdd_to_formula(cube));

	return multop::instance(multop::Or, v);
      }
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      void
      conj_bdd_to_acc(tgba_explicit* a, bdd b, tgba_explicit::transition* t)
      {
	assert(b != bddfalse);
	while (b != bddtrue)
	  {
	    int var = bdd_var(b);
	    bdd high = bdd_high(b);
	    if (high == bddfalse)
	      {
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		// Simply ignore negated acceptance variables.
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		b = bdd_low(b);
	      }
	    else
	      {
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		formula* ac = var_to_formula(var);
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		if (!a->has_acceptance_condition(ac))
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		  a->declare_acceptance_condition(clone(ac));
		a->add_acceptance_condition(t, ac);
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		b = high;
	      }
	    assert(b != bddfalse);
	  }
      }
    };


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    // Gather all promises of a formula.  These are the
    // right-hand sides of U or F operators.
    class ltl_promise_visitor: public postfix_visitor
    {
    public:
      ltl_promise_visitor(translate_dict& dict)
	: dict_(dict), res_(bddtrue)
      {
      }

      virtual
      ~ltl_promise_visitor()
      {
      }

      bdd
      result() const
      {
	return res_;
      }

      using postfix_visitor::doit;

      virtual void
      doit(unop* node)
      {
	if (node->op() == unop::F)
	  res_ &= bdd_ithvar(dict_.register_a_variable(node->child()));
      }

      virtual void
      doit(binop* node)
      {
	if (node->op() == binop::U)
	  res_ &= bdd_ithvar(dict_.register_a_variable(node->second()));
      }

    private:
      translate_dict& dict_;
      bdd res_;
    };


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    // The rewrite rules used here are adapted from Jean-Michel
    // Couvreur's FM paper.
    class ltl_trad_visitor: public const_visitor
    {
    public:
      ltl_trad_visitor(translate_dict& dict)
	: dict_(dict)
      {
      }

      virtual
      ~ltl_trad_visitor()
      {
      }

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      bdd
      result() const
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      {
	return res_;
      }

      void
      visit(const atomic_prop* node)
      {
	res_ = bdd_ithvar(dict_.register_proposition(node));
      }

      void
      visit(const constant* node)
      {
	switch (node->val())
	  {
	  case constant::True:
	    res_ = bddtrue;
	    return;
	  case constant::False:
	    res_ = bddfalse;
	    return;
	  }
	/* Unreachable code.  */
	assert(0);
      }

      void
      visit(const unop* node)
      {
	switch (node->op())
	  {
	  case unop::F:
	    {
	      // r(Fy) = r(y) + a(y)r(XFy)
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	      const formula* child = node->child();
	      bdd y = recurse(child);
	      int a = dict_.register_a_variable(child);
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	      int x = dict_.register_next_variable(node);
	      res_ = y | (bdd_ithvar(a) & bdd_ithvar(x));
	      return;
	    }
	  case unop::G:
	    {
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	      // The paper suggests that we optimize GFy
	      // as
	      //   r(GFy) = (r(y) + a(y))r(XGFy)
	      // instead of
	      //   r(GFy) = (r(y) + a(y)r(XFy)).r(XGFy)
	      // but this is just a particular case
	      // of the "merge all states with the same
	      // symbolic rewriting" optimization we do later.
	      // (r(Fy).r(GFy) and r(GFy) have the same symbolic
	      // rewriting.)  Let's keep things simple here.

	      // r(Gy) = r(y)r(XGy)
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	      const formula* child = node->child();
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	      int x = dict_.register_next_variable(node);
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	      bdd y = recurse(child);
	      res_ = y & bdd_ithvar(x);
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	      return;
	    }
	  case unop::Not:
	    {
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	      // r(!y) = !r(y)
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	      res_ = bdd_not(recurse(node->child()));
	      return;
	    }
	  case unop::X:
	    {
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	      // r(Xy) = Next[y]
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	      int x = dict_.register_next_variable(node->child());
	      res_ = bdd_ithvar(x);
	      return;
	    }
	  }
	/* Unreachable code.  */
	assert(0);
      }

      void
      visit(const binop* node)
      {
	bdd f1 = recurse(node->first());
	bdd f2 = recurse(node->second());

	switch (node->op())
	  {
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	    // r(f1 logical-op f2) = r(f1) logical-op r(f2)
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	  case binop::Xor:
	    res_ = bdd_apply(f1, f2, bddop_xor);
	    return;
	  case binop::Implies:
	    res_ = bdd_apply(f1, f2, bddop_imp);
	    return;
	  case binop::Equiv:
	    res_ = bdd_apply(f1, f2, bddop_biimp);
	    return;
	  case binop::U:
	    {
	      // r(f1 U f2) = r(f2) + a(f2)r(f1)r(X(f1 U f2))
	      int a = dict_.register_a_variable(node->second());
	      int x = dict_.register_next_variable(node);
	      res_ = f2 | (bdd_ithvar(a) & f1 & bdd_ithvar(x));
	      return;
	    }
	  case binop::R:
	    {
	      // r(f1 R f2) = r(f1)r(f2) + r(f2)r(X(f1 U f2))
	      int x = dict_.register_next_variable(node);
	      res_ = (f1 & f2) | (f2 & bdd_ithvar(x));
	      return;
	    }
	  }
	/* Unreachable code.  */
	assert(0);
      }

      void
      visit(const multop* node)
      {
	int op = -1;
	switch (node->op())
	  {
	  case multop::And:
	    op = bddop_and;
	    res_ = bddtrue;
	    break;
	  case multop::Or:
	    op = bddop_or;
	    res_ = bddfalse;
	    break;
	  }
	assert(op != -1);
	unsigned s = node->size();
	for (unsigned n = 0; n < s; ++n)
	  {
	    res_ = bdd_apply(res_, recurse(node->nth(n)), op);
	  }
      }

      bdd
      recurse(const formula* f)
      {
	ltl_trad_visitor v(dict_);
	f->accept(v);
	return v.result();
      }


    private:
      translate_dict& dict_;
      bdd res_;
    };

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    // Check whether a formula has a R or G operator at its top-level
    // (preceding logical operators do not count).
    class ltl_possible_fair_loop_visitor: public const_visitor
    {
    public:
      ltl_possible_fair_loop_visitor()
	: res_(false)
      {
      }

      virtual
      ~ltl_possible_fair_loop_visitor()
      {
      }

      bool
      result() const
      {
	return res_;
      }

      void
      visit(const atomic_prop*)
      {
      }

      void
      visit(const constant*)
      {
      }

      void
      visit(const unop* node)
      {
	if (node->op() == unop::G)
	  res_ = true;
      }

      void
      visit(const binop* node)
      {
	switch (node->op())
	  {
	    // r(f1 logical-op f2) = r(f1) logical-op r(f2)
	  case binop::Xor:
	  case binop::Implies:
	  case binop::Equiv:
	    node->first()->accept(*this);
	    if (!res_)
	      node->second()->accept(*this);
	    return;
	  case binop::U:
	    return;
	  case binop::R:
	    res_ = true;
	    return;
	  }
	/* Unreachable code.  */
	assert(0);
      }

      void
      visit(const multop* node)
      {
	unsigned s = node->size();
	for (unsigned n = 0; n < s && !res_; ++n)
	  {
	    node->nth(n)->accept(*this);
	  }
      }

    private:
      bool res_;
    };

    // Check whether a formula can be part of a fair loop.
    // Cache the result for efficiency.
    class possible_fair_loop_checker
    {
    public:
      bool
      check(const formula* f)
      {
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	pfl_map::const_iterator i = pfl_.find(f);
	if (i != pfl_.end())
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	  return i->second;
	ltl_possible_fair_loop_visitor v;
	f->accept(v);
	bool rel = v.result();
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	pfl_[f] = rel;
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	return rel;
      }

    private:
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      typedef Sgi::hash_map<const formula*, bool, formula_ptr_hash> pfl_map;
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      pfl_map pfl_;
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    };

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    // Keep a map of the TGBA translation of all subformulae and their
    // negations, for easy language containment check.
    class language_containment_checker
    {
      struct record_
      {
	const tgba* translation;
	typedef std::set<const record_*> incomp_map;
	incomp_map incompatible;
      };
      typedef Sgi::hash_map<const formula*,
			    record_, formula_ptr_hash> trans_map;
    public:
      language_containment_checker(bdd_dict* dict, bool exprop,
				   bool symb_merge,
				   bool branching_postponement,
				   bool fair_loop_approx)
	: dict_(dict), exprop_(exprop), symb_merge_(symb_merge),
	  branching_postponement_(branching_postponement),
	  fair_loop_approx_(fair_loop_approx)
      {
      }

      ~language_containment_checker()
      {

	while (!translated_.empty())
	  {
	    trans_map::iterator i = translated_.begin();
	    delete i->second.translation;
	    const formula* f = i->first;
	    translated_.erase(i);
	    destroy(f);
	  }
      }

      // Check whether L(l) is a subset of L(g).
      bool
      contained(const formula* l, const formula* g)
      {
	const record_* rl = register_formula_(l);
	const formula* ng = unop::instance(unop::Not, clone(g));
	const record_* rng = register_formula_(ng);
	destroy(ng);
	bool res = rl->incompatible.find(rng) != rl->incompatible.end();
	return res;
      }

      // Check whether L(l) = L(g).
      bool
      equal(const formula* l, const formula* g)
      {
	return contained(l,g) && contained(g,l);
      }

    protected:
      const record_*
      register_formula_(const formula* f)
      {
	trans_map::iterator i = translated_.find(f);
	if (i != translated_.end())
	  return &i->second;

	const tgba_explicit* e = ltl_to_tgba_fm(f, dict_, exprop_, symb_merge_,
						branching_postponement_,
						fair_loop_approx_);
	record_& r = translated_[clone(f)];
	r.translation = e;

	// Check the emptiness of the product of this formula with any
	// other registered formula.
	for (i = translated_.begin(); i != translated_.end(); ++i)
	  {
	    if (f == i->first)
	      continue;
	    const tgba* p = new tgba_product(e, i->second.translation);
	    emptiness_check* ec = couvreur99(p);
	    emptiness_check_result* ecr = ec->check();
	    if (!ecr)
	      {
		r.incompatible.insert(&i->second);
		i->second.incompatible.insert(&r);
	      }
	    else
	      delete ecr;
	    delete ec;
	    delete p;
	  }
	return &r;
      }

    private:
      /* Translation options */
      bdd_dict* dict_;
      bool exprop_;
      bool symb_merge_;
      bool branching_postponement_;
      bool fair_loop_approx_;
      /* Translation Maps */
      trans_map translated_;
    };

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    class formula_canonizer
    {
    public:
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      formula_canonizer(translate_dict& d,
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			bool fair_loop_approx, bdd all_promises,
			language_containment_checker* lcc)
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	: v_(d),
	  fair_loop_approx_(fair_loop_approx),
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	  all_promises_(all_promises),
	  lcc_(lcc)
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      {
	// For cosmetics, register 1 initially, so the algorithm will
	// not register an equivalent formula first.
	b2f_[bddtrue] = constant::true_instance();
      }
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      ~formula_canonizer()
      {
661
	while (!f2b_.empty())
662
	  {
663
664
665
666
	    formula_to_bdd_map::iterator i = f2b_.begin();
	    const formula* f = i->first;
	    f2b_.erase(i);
	    destroy(f);
667
	  }
668
669
670
      }

      bdd
671
      translate(const formula* f, bool* new_flag = 0)
672
673
674
675
676
677
      {
	// Use the cached result if available.
	formula_to_bdd_map::const_iterator i = f2b_.find(f);
	if (i != f2b_.end())
	  return i->second;

678
679
680
	if (new_flag)
	  *new_flag = true;

681
682
683
	// Perform the actual translation.
	f->accept(v_);
	bdd res = v_.result();
684
685
686
687
688
689
690
691
692
693
694
695

	// Apply the fair-loop approximation if requested.
	if (fair_loop_approx_)
	  {
	    // If the source cannot possibly be part of a fair
	    // loop, make all possible promises.
	    if (fair_loop_approx_
		&& f != constant::true_instance()
		&& !pflc_.check(f))
	      res &= all_promises_;
	  }

696
697
698
699
700
701
702
703
704
705
706
	f2b_[clone(f)] = res;

	// Register the reverse mapping if it is not already done.
	if (b2f_.find(res) == b2f_.end())
	  b2f_[res] = f;
	return res;
      }

      const formula*
      canonize(const formula* f)
      {
707
708
	bool new_variable = false;
	bdd b = translate(f, &new_variable);
709
710

	bdd_to_formula_map::iterator i = b2f_.find(b);
711
712
	// Since we have just translated the formula, it is
	// necessarily in b2f_.
713
714
715
	assert(i != b2f_.end());

	if (i->second != f)
716
	  {
717
	    // The translated bdd maps to an already seen formula.
718
719
	    destroy(f);
	    f = clone(i->second);
720
	  }
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
	else if (new_variable && lcc_)
	  {
	    // It's a new bdd for a new formula.  Let's see if we can
	    // find an equivalent formula with language containment
	    // checks.
	    for (formula_to_bdd_map::const_iterator j = f2b_.begin();
		 j != f2b_.end(); ++j)
	      if (f != j->first && lcc_->equal(f, j->first))
		{
		  f2b_[f] = j->second;
		  i->second = j->first;
		  destroy(f);
		  f = clone(i->second);
		  break;
		}
	  }
737
	return f;
738
739
      }

740
741
742
743
744
745
746
747
748
749
750
751
752
    private:
      ltl_trad_visitor v_;
      // Map each formula to its associated bdd.  This speed things up when
      // the same formula is translated several times, which especially
      // occurs when canonize() is called repeatedly inside exprop.
      typedef std::map<bdd, const formula*, bdd_less_than> bdd_to_formula_map;
      bdd_to_formula_map b2f_;
      // Map a representation of successors to a canonical formula.
      // We do this because many formulae (such as `aR(bRc)' and
      // `aR(bRc).(bRc)') are equivalent, and are trivially identified
      // by looking at the set of successors.
      typedef std::map<const formula*, bdd> formula_to_bdd_map;
      formula_to_bdd_map f2b_;
753
754
755
756

      possible_fair_loop_checker pflc_;
      bool fair_loop_approx_;
      bdd all_promises_;
757
      language_containment_checker* lcc_;
758
759
760
761
762
    };

  }

  typedef std::map<bdd, bdd, bdd_less_than> prom_map;
763
  typedef Sgi::hash_map<const formula*, prom_map, formula_ptr_hash> dest_map;
764
765
766
767

  static void
  fill_dests(translate_dict& d, dest_map& dests, bdd label, const formula* dest)
  {
768
    bdd conds = bdd_existcomp(label, d.var_set);
769
770
    bdd promises = bdd_existcomp(label, d.a_set);

771
772
773
774
775
776
777
778
779
780
781
782
783
    dest_map::iterator i = dests.find(dest);
    if (i == dests.end())
      {
	dests[dest][promises] = conds;
      }
    else
      {
	i->second[promises] |= conds;
	destroy(dest);
      }
  }


784
  tgba_explicit*
785
  ltl_to_tgba_fm(const formula* f, bdd_dict* dict,
786
		 bool exprop, bool symb_merge, bool branching_postponement,
787
		 bool fair_loop_approx, const atomic_prop_set* unobs,
788
		 int reduce_ltl, bool containment_checks)
789
  {
790
791
    symb_merge |= containment_checks;

792
793
794
795
    // Normalize the formula.  We want all the negations on
    // the atomic propositions.  We also suppress logic
    // abbreviations such as <=>, =>, or XOR, since they
    // would involve negations at the BDD level.
796
797
798
    formula* f1 = unabbreviate_logic(f);
    formula* f2 = negative_normal_form(f1);
    destroy(f1);
799

800
801
802
803
804
805
806
807
    // Simplify the formula, if requested.
    if (reduce_ltl)
      {
	formula* tmp = reduce(f2, reduce_ltl);
	destroy(f2);
	f2 = tmp;
      }

808
809
810
    typedef std::set<const formula*, formula_ptr_less_than> set_type;
    set_type formulae_seen;
    set_type formulae_to_translate;
811

812
    translate_dict d(dict);
813

814
815
    // Compute the set of all promises that can possibly occurre
    // inside the formula.
816
    bdd all_promises = bddtrue;
817
    if (fair_loop_approx || unobs)
818
819
820
821
822
823
      {
	ltl_promise_visitor pv(d);
	f2->accept(pv);
	all_promises = pv.result();
      }

824
825
826
827
828
829
    language_containment_checker lcc(dict, exprop, symb_merge,
				     branching_postponement,
				     fair_loop_approx);

    formula_canonizer fc(d, fair_loop_approx, all_promises,
			 containment_checks ? &lcc : 0);
830

831
832
833
834
835
836
837
838
839
840
    // These are used when atomic propositions are interpreted as
    // events.  There are two kinds of events: observable events are
    // those used in the formula, and unobservable events or other
    // events that can occur at anytime.  All events exclude each
    // other.
    bdd observable_events = bddfalse;
    bdd unobservable_events = bddfalse;
    if (unobs)
      {
	bdd neg_events = bddtrue;
841
842
	std::auto_ptr<atomic_prop_set> aps(atomic_prop_collect(f));
	for (atomic_prop_set::const_iterator i = aps->begin();
843
844
845
846
847
848
849
850
	     i != aps->end(); ++i)
	  {
	    int p = d.register_proposition(*i);
	    bdd pos = bdd_ithvar(p);
	    bdd neg = bdd_nithvar(p);
	    observable_events = (observable_events & neg) | (neg_events & pos);
	    neg_events &= neg;
	  }
851
	for (atomic_prop_set::const_iterator i = unobs->begin();
852
853
854
855
856
857
858
859
860
861
862
863
864
	     i != unobs->end(); ++i)
	  {
	    int p = d.register_proposition(*i);
	    bdd pos = bdd_ithvar(p);
	    bdd neg = bdd_nithvar(p);
	    unobservable_events = ((unobservable_events & neg)
				   | (neg_events & pos));
	    observable_events &= neg;
	    neg_events &= neg;
	  }
      }
    bdd all_events = observable_events | unobservable_events;

865
866
867
868
869
870
871
872
873
874
    formulae_seen.insert(f2);
    formulae_to_translate.insert(f2);

    tgba_explicit* a = new tgba_explicit(dict);

    a->set_init_state(to_string(f2));

    while (!formulae_to_translate.empty())
      {
	// Pick one formula.
875
	const formula* f = *formulae_to_translate.begin();
876
877
878
	formulae_to_translate.erase(formulae_to_translate.begin());

	// Translate it into a BDD to simplify it.
879
	bdd res = fc.translate(f);
880

881
882
883
884
885
886
887
888
	// Handle exclusive events.
	if (unobs)
	  {
	    res &= observable_events;
	    int n = d.register_next_variable(f);
	    res |= unobservable_events & bdd_ithvar(n) & all_promises;
	  }

889
890
	std::string now = to_string(f);

891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
	// We used to factor only Next and A variables while computing
	// prime implicants, with
	//    minato_isop isop(res, d.next_set & d.a_set);
	// in order to obtain transitions with formulae of atomic
	// proposition directly, but unfortunately this led to strange
	// factorizations.  For instance f U g was translated as
	//     r(f U g) = g + a(g).r(X(f U g)).(f + g)
	// instead of just
	//     r(f U g) = g + a(g).r(X(f U g)).f
	// Of course both formulae are logically equivalent, but the
	// latter is "more deterministic" than the former, so it should
	// be preferred.
	//
	// Therefore we now factor all variables.  This may lead to more
	// transitions than necessary (e.g.,  r(f + g) = f + g  will be
906
907
908
	// coded as two transitions), but we later merge all transitions
	// with same source/destination and acceptance conditions.  This
	// is the goal of the `dests' hash.
909
	//
910
	// Note that this is still not optimal.  For instance it is
911
	// better to encode `f U g' as
912
	//     r(f U g) = g + a(g).r(X(f U g)).f.!g
913
914
915
916
	// because that leads to a deterministic automaton.  In order
	// to handle this, we take the conditions of any transition
	// going to true (it's `g' here), and remove it from the other
	// transitions.
917
918
919
	//
	// In `exprop' mode, considering all possible combinations of
	// outgoing propositions generalizes the above trick.
920
	dest_map dests;
921

922
	// Compute all outgoing arcs.
923
924
925
926
927

	// If EXPROP is set, we will refine the symbolic
	// representation of the successors for all combinations of
	// the atomic properties involved in the formula.
	// VAR_SET is the set of these properties.
928
	bdd var_set = bdd_existcomp(bdd_support(res), d.var_set);
929
930
931
932
933
	// ALL_PROPS is the combinations we have yet to consider.
	// We used to start with `all_props = bddtrue', but it is
	// more efficient to start with the set of all satisfiable
	// variables combinations.
	bdd all_props = bdd_existcomp(res, d.var_set);
934
	while (all_props != bddfalse)
935
	  {
936
937
938
	    bdd one_prop_set =
	      exprop ? bdd_satoneset(all_props, var_set, bddtrue) : bddtrue;
	    all_props -= one_prop_set;
939

940
	    typedef std::map<bdd, const formula*, bdd_less_than> succ_map;
941
942
	    succ_map succs;

943
944
945
	    minato_isop isop(res & one_prop_set);
	    bdd cube;
	    while ((cube = isop.next()) != bddfalse)
946
	      {
947
		bdd label = bdd_exist(cube, d.next_set);
948
		bdd dest_bdd = bdd_existcomp(cube, d.next_set);
949
950
		const formula* dest = d.conj_bdd_to_formula(dest_bdd);

951
952
953
954
955
956
957
958
959
960
961
		// Simplify the formula, if requested.
		if (reduce_ltl)
		  {
		    formula* tmp = reduce(dest, reduce_ltl);
		    destroy(dest);
		    dest = tmp;
		    // Ignore the arc if the destination reduces to false.
		    if (dest == constant::false_instance())
		      continue;
		  }

962
963
964
965
		// If we already know a state with the same
		// successors, use it in lieu of the current one.
		if (symb_merge)
		  dest = fc.canonize(dest);
966

967
968
969
970
971
972
		// If we are not postponing the branching, we can
		// declare the outgoing transitions immediately.
		// Otherwise, we merge transitions with identical
		// label, and declare the outgoing transitions in a
		// second loop.
		if (!branching_postponement)
973
		  {
974
		    fill_dests(d, dests, label, dest);
975
976
977
		  }
		else
		  {
978
979
980
981
		    succ_map::iterator si = succs.find(label);
		    if (si == succs.end())
		      succs[label] = dest;
		    else
982
983
984
985
		      si->second =
			multop::instance(multop::Or,
					 const_cast<formula*>(si->second),
					 const_cast<formula*>(dest));
986
987
		  }
	      }
988
989
990
	    if (branching_postponement)
	      for (succ_map::const_iterator si = succs.begin();
		   si != succs.end(); ++si)
991
		fill_dests(d, dests, si->first, si->second);
992
	  }
993

994
	// Check for an arc going to 1 (True).  Register it first, that
995
996
997
	// way it will be explored before the other during the model
	// checking.
	dest_map::const_iterator i = dests.find(constant::true_instance());
998
999
1000
1001
	// COND_FOR_TRUE is the conditions of the True arc, so when
	// can remove them from all other arcs.  It might sounds that
	// this is not needed when exprop is used, but in fact it is
	// complementary.
1002
1003
	//
	// Consider
1004
	//   f = r(X(1) R p) = p.(1 + r(X(1) R p))
1005
	// with exprop the two outgoing arcs would be
1006
1007
        //         p                  p
	//     f ----> 1       f ----------> 1
1008
1009
	//
	// where in fact we could output
1010
1011
        //         p
	//     f ----> 1
1012
	//
1013
	// because there is no point in looping on f if we can go to 1.
1014
	bdd cond_for_true = bddfalse;
1015
1016
	if (i != dests.end())
	  {
1017
	    // When translating LTL for an event-based logic with
1018
1019
	    // unobservable events, the 1 state should accept all events,
	    // even unobservable events.
1020
1021
	    if (unobs && f == constant::true_instance())
	      cond_for_true = all_events;
1022
1023
1024
1025
1026
	    else
	      {
		// There should be only one transition going to 1 (true) ...
		assert(i->second.size() == 1);
		prom_map::const_iterator j = i->second.begin();
1027
1028
1029
		// ... and it is not expected to make any promises (unless
		// fair loop approximations are used).
		assert(fair_loop_approx || j->first == bddtrue);
1030
1031
		cond_for_true = j->second;
	      }
1032
1033
	    tgba_explicit::transition* t =
	      a->create_transition(now, constant::true_instance()->val_name());
1034
	    a->add_condition(t, d.bdd_to_formula(cond_for_true));
1035
1036
1037
1038
1039
	  }
	// Register other transitions.
	for (i = dests.begin(); i != dests.end(); ++i)
	  {
	    const formula* dest = i->first;
1040
1041
1042
1043
	    // The cond_for_true optimization can cause some
	    // transitions to be removed.  So we have to remember
	    // whether a formula is actually reachable.
	    bool reachable = false;
1044

1045
1046
1047
1048
1049
1050
	    if (dest != constant::true_instance())
	      {
		std::string next = to_string(dest);
		for (prom_map::const_iterator j = i->second.begin();
		     j != i->second.end(); ++j)
		  {
1051
1052
1053
		    bdd cond = j->second - cond_for_true;
		    if (cond == bddfalse) // Skip false transitions.
		      continue;
1054
1055
		    tgba_explicit::transition* t =
		      a->create_transition(now, next);
1056
		    a->add_condition(t, d.bdd_to_formula(cond));
1057
		    d.conj_bdd_to_acc(a, j->first, t);
1058
		    reachable = true;
1059
1060
		  }
	      }
1061
1062
1063
1064
1065
1066
1067
	    else
	      {
		// "1" is reachable.
		reachable = true;
	      }
	    if (reachable
		&& formulae_seen.find(dest) == formulae_seen.end())
1068
1069
1070
1071
1072
1073
	      {
		formulae_seen.insert(dest);
		formulae_to_translate.insert(dest);
	      }
	    else
	      {
1074
		destroy(dest);
1075
1076
1077
1078
1079
	      }
	  }
      }

    // Free all formulae.
1080
    for (std::set<const formula*>::iterator i = formulae_seen.begin();
1081
	 i != formulae_seen.end(); ++i)
1082
      destroy(*i);
1083

1084
1085
    // Turn all promises into real acceptance conditions.
    a->complement_all_acceptance_conditions();
1086
1087
1088
1089
    return a;
  }

}