ltl2tgba_fm.cc 45.3 KB
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// Copyright (C) 2008, 2009, 2010 Laboratoire de Recherche et
// Dveloppement de l'Epita (LRDE).
// Copyright (C) 2003, 2004, 2005, 2006 Laboratoire
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// 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"
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#include "ltlvisit/lunabbrev.hh"
#include "ltlvisit/nenoform.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 "ltlvisit/mark.hh"
#include "ltlvisit/tostring.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 "ltlvisit/contain.hh"
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#include "ltlvisit/consterm.hh"
#include "tgba/bddprint.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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	  i->first->destroy();
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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
	  {
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	    f = f->clone();
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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;
      }

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      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;
	}
	os << "Shared Dict:" << std::endl;
	dict->dump(os);
	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 isi->second->clone();
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	isi = dict->acc_formula_map.find(var);
	if (isi != dict->acc_formula_map.end())
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	  return isi->second->clone();
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	isi = dict->var_formula_map.find(var);
	if (isi != dict->var_formula_map.end())
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	  return isi->second->clone();
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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, multop::type op = multop::And) const
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      {
	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(op, v);
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      }

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      formula*
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      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
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      conj_bdd_to_acc(tgba_explicit_formula* a, bdd b,
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		      state_explicit_formula::transition* t)
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      {
	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(ac->clone());
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		a->add_acceptance_condition(t, ac);
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		b = high;
	      }
	    assert(b != bddfalse);
	  }
      }
    };


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    // Debugging function.
    std::ostream&
    trace_ltl_bdd(const translate_dict& d, bdd f)
    {
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      std::cerr << "Displaying BDD ";
      bdd_print_set(std::cerr, d.dict, f) << ":" << std::endl;

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      minato_isop isop(f);
      bdd cube;
      while ((cube = isop.next()) != bddfalse)
	{
	  bdd label = bdd_exist(cube, d.next_set);
	  bdd dest_bdd = bdd_existcomp(cube, d.next_set);
	  const formula* dest = d.conj_bdd_to_formula(dest_bdd);
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	  bdd_print_set(std::cerr, d.dict, label) << " => ";
	  bdd_print_set(std::cerr, d.dict, dest_bdd) << " = "
						     << to_string(dest)
						     << std::endl;
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	  dest->destroy();
	}
      return std::cerr;
    }


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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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    // Rewrite rule for rational operators.
    class ratexp_trad_visitor: public const_visitor
    {
    public:
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      ratexp_trad_visitor(translate_dict& dict, formula* to_concat = 0)
	: dict_(dict), to_concat_(to_concat)
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      {
      }

      virtual
      ~ratexp_trad_visitor()
      {
	if (to_concat_)
	  to_concat_->destroy();
      }

      bdd
      result() const
      {
	return res_;
      }

      bdd next_to_concat()
      {
	if (!to_concat_)
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	  return bddtrue;
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	int x = dict_.register_next_variable(to_concat_);
	return bdd_ithvar(x);
      }

      bdd now_to_concat()
      {
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	if (to_concat_ && to_concat_ != constant::empty_word_instance())
	  return recurse(to_concat_);

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	return bddfalse;
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      }

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

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

      void
      visit(const unop* node)
      {
	switch (node->op())
	  {
	  case unop::F:
	  case unop::G:
	  case unop::X:
	  case unop::Finish:
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	  case unop::Closure:
	  case unop::NegClosure:
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	    assert(!"not a rational operator");
	    return;
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	  case unop::Not:
	    {
	      // Not can only appear in front of constants or atomic
	      // propositions.
	      const formula* f = node->child();
	      assert(dynamic_cast<const atomic_prop*>(f)
		     || dynamic_cast<const constant*>(f));
	      res_ = !recurse(f) & next_to_concat();
	      return;
	    }
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	  }
	/* Unreachable code.  */
	assert(0);
      }
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      void
      visit(const bunop* bo)
      {
	formula* f;
	unsigned min = bo->min();
	unsigned max = bo->max();
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	assert(max > 0);

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	unsigned min2 = (min == 0) ? 0 : (min - 1);
	unsigned max2 =
	  (max == bunop::unbounded) ? bunop::unbounded : (max - 1);

	bunop::type op = bo->op();
	switch (op)
	  {
	  case bunop::Star:
	    f = bunop::instance(op, bo->child()->clone(), min2, max2);

	    if (to_concat_)
	      f = multop::instance(multop::Concat, f, to_concat_->clone());

	    res_ = recurse(bo->child(), f);
	    if (min == 0)
	      res_ |= now_to_concat();
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	    return;
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	  case bunop::Equal:
	    {
	      // b[=min..max] == (!b;b[=min..max]) | (b;b[=min-1..max-1])
	      // b[=0..max]   == [*0] | (!b;b[=0..max]) | (b;b[=0..max-1])
	      // Note: b[=0] == (!b)[*] is a trivial identity, so it will
	      // never occur here.

	      formula* f1 = // !b;b[min..max]
		multop::instance(multop::Concat,
				 unop::instance(unop::Not,
						bo->child()->clone()),
				 bo->clone());

	      formula* f2 = // b;b[=min-1..max-1]
		multop::instance(multop::Concat,
				 bo->child()->clone(),
				 bunop::instance(bunop::Equal,
						 bo->child()->clone(),
						 min2, max2));
	      f = multop::instance(multop::Or, f1, f2);

	      res_ = recurse_and_concat(f);

	      f->destroy();
	      if (min == 0)
		res_ |= now_to_concat();
	      return;
	    }
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	  }
	/* Unreachable code.  */
	assert(0);
      }

      void
      visit(const binop*)
      {
	assert(!"not a rational operator");
      }

      void
      visit(const automatop*)
      {
	assert(!"not a rational operator");
      }

      void
      visit(const multop* node)
      {
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	multop::type op = node->op();
	switch (op)
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	  {
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	  case multop::AndNLM:
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	  case multop::And:
	    {
	      unsigned s = node->size();
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	      if (op == multop::AndNLM)
		{
		  multop::vec* final = new multop::vec;
		  multop::vec* non_final = new multop::vec;

		  for (unsigned n = 0; n < s; ++n)
		    {
		      const formula* f = node->nth(n);
		      if (constant_term_as_bool(f))
			final->push_back(f->clone());
		      else
			non_final->push_back(f->clone());
		    }

		  if (non_final->empty())
		    {
		      delete non_final;
		      // (a* & b*);c = (a*|b*);c
		      formula* f = multop::instance(multop::Or, final);
		      res_ = recurse_and_concat(f);
		      f->destroy();
		      break;
		    }
		  if (!final->empty())
		    {
		      // let F_i be final formulae
		      //     N_i be non final formula
		      // (F_1 & ... & F_n & N_1 & ... & N_m)
		      // =   (F_1 | ... | F_n);[*] && (N_1 & ... & N_m)
		      //   | (F_1 | ... | F_n) && (N_1 & ... & N_m);[*]
		      formula* f = multop::instance(multop::Or, final);
		      formula* n = multop::instance(multop::AndNLM, non_final);
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		      formula* t = bunop::instance(bunop::Star,
						   constant::true_instance());
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		      formula* ft = multop::instance(multop::Concat,
						     f->clone(), t->clone());
		      formula* nt = multop::instance(multop::Concat,
						     n->clone(), t);
		      formula* ftn = multop::instance(multop::And, ft, n);
		      formula* fnt = multop::instance(multop::And, f, nt);
		      formula* all = multop::instance(multop::Or, ftn, fnt);
		      res_ = recurse_and_concat(all);
		      all->destroy();
		      break;
		    }
		  // No final formula.
		  // Apply same rule as &&, until we reach a point where
		  // we have final formulae.
		  delete final;
		  for (unsigned n = 0; n < s; ++n)
		    (*non_final)[n]->destroy();
		  delete non_final;
		}

	      res_ = bddtrue;
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	      for (unsigned n = 0; n < s; ++n)
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		{
		  bdd res = recurse(node->nth(n));
		  // trace_ltl_bdd(dict_, res);
		  res_ &= res;
		}
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	      //std::cerr << "Pre-Concat:" << std::endl;
	      //trace_ltl_bdd(dict_, res_);

	      if (to_concat_)
		{
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		  // have to append ";c" to all destinations.

		  minato_isop isop(res_);
		  bdd cube;
		  res_ = bddfalse;
		  while ((cube = isop.next()) != bddfalse)
		    {
		      bdd label = bdd_exist(cube, dict_.next_set);
		      bdd dest_bdd = bdd_existcomp(cube, dict_.next_set);
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		      formula* dest =
			dict_.conj_bdd_to_formula(dest_bdd, op);
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		      formula* dest2;
		      int x;
		      if (dest == constant::empty_word_instance())
			{
			  res_ |= label & next_to_concat();
			}
		      else
			{
			  dest2 = multop::instance(multop::Concat, dest,
						   to_concat_->clone());
			  if (dest2 != constant::false_instance())
			    {
			      x = dict_.register_next_variable(dest2);
			      dest2->destroy();
			      res_ |= label & bdd_ithvar(x);
			    }
			  if (constant_term_as_bool(node))
			    res_ |= label & next_to_concat();
			}
		    }
		}
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	      if (constant_term_as_bool(node))
		res_ |= now_to_concat();
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	      break;
	    }
	  case multop::Or:
	    {
	      res_ = bddfalse;
	      unsigned s = node->size();
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	      for (unsigned n = 0; n < s; ++n)
		res_ |= recurse_and_concat(node->nth(n));
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	      break;
	    }
	  case multop::Concat:
	    {
	      multop::vec* v = new multop::vec;
	      unsigned s = node->size();
	      v->reserve(s);
	      for (unsigned n = 1; n < s; ++n)
		v->push_back(node->nth(n)->clone());
	      if (to_concat_)
		v->push_back(to_concat_->clone());
	      res_ = recurse(node->nth(0),
			     multop::instance(multop::Concat, v));
	      break;
	    }
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	  case multop::Fusion:
	    {
	      assert(node->size() >= 2);

	      // the head
	      bdd res = recurse(node->nth(0));

	      // the tail
	      multop::vec* v = new multop::vec;
	      unsigned s = node->size();
	      v->reserve(s - 1);
	      for (unsigned n = 1; n < s; ++n)
		v->push_back(node->nth(n)->clone());
	      formula* tail = multop::instance(multop::Fusion, v);
	      bdd tail_bdd;
	      bool tail_computed = false;

	      //trace_ltl_bdd(dict_, res);

	      minato_isop isop(res);
	      bdd cube;
	      res_ = bddfalse;
	      while ((cube = isop.next()) != bddfalse)
		{
		  bdd label = bdd_exist(cube, dict_.next_set);
		  bdd dest_bdd = bdd_existcomp(cube, dict_.next_set);
		  formula* dest = dict_.conj_bdd_to_formula(dest_bdd);

		  if (constant_term_as_bool(dest))
		    {
		      // The destination is a final state.  Make sure we
		      // can also exit if tail is satisfied.
		      if (!tail_computed)
			{
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			  tail_bdd = recurse_and_concat(tail);
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			  tail_computed = true;
			}
		      res_ |= label & tail_bdd;
		    }

		  if (dynamic_cast<constant*>(dest) == 0)
		    {
		      // If the destination is not a constant, it
		      // means it can have successors.  Fusion the
		      // tail and append anything to concatenate.
		      formula* dest2 = multop::instance(multop::Fusion, dest,
							tail->clone());
		      if (to_concat_)
			 dest2 = multop::instance(multop::Concat, dest2,
						 to_concat_->clone());
		      if (dest2 != constant::false_instance())
			{
			  int x = dict_.register_next_variable(dest2);
			  dest2->destroy();
			  res_ |= label & bdd_ithvar(x);
			}
		    }
		}

	      tail->destroy();
	      break;
	    }
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	  }
      }

      bdd
      recurse(const formula* f, formula* to_concat = 0)
      {
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	ratexp_trad_visitor v(dict_, to_concat);
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	f->accept(v);
	return v.result();
      }

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      bdd
      recurse_and_concat(const formula* f)
      {
	return recurse(f, to_concat_ ? to_concat_->clone() : 0);
      }
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    private:
      translate_dict& dict_;
      bdd res_;
      formula* to_concat_;
    };

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    // The rewrite rules used here are adapted from Jean-Michel
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    // Couvreur's FM paper, augmented to support rational operators.
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    class ltl_trad_visitor: public const_visitor
    {
    public:
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      ltl_trad_visitor(translate_dict& dict, bool mark_all = false,
		       bool exprop = false)
	: dict_(dict), rat_seen_(false), has_marked_(false),
	  mark_all_(mark_all), exprop_(exprop)
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      {
      }

      virtual
      ~ltl_trad_visitor()
      {
      }

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      void
      reset(bool mark_all)
      {
	rat_seen_ = false;
	has_marked_ = false;
	mark_all_ = mark_all;
      }

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      bdd
      result() const
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      {
	return res_;
      }

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      const translate_dict&
      get_dict() const
      {
	return dict_;
      }

      bool
      has_rational() const
      {
	return rat_seen_;
      }

      bool
      has_marked() const
      {
	return has_marked_;
      }

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      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;
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	  case constant::EmptyWord:
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	    assert(!"Not an LTL operator");
	    return;
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	  }
	/* Unreachable code.  */
	assert(0);
      }

      void
      visit(const unop* node)
      {
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	unop::type op = node->op();

	switch (op)
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	  {
	  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));
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	      break;
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	    }
	  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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	      break;
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	    }
	  case unop::Not:
	    {
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	      // r(!y) = !r(y)
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	      res_ = bdd_not(recurse(node->child()));
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	      break;
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	    }
	  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);
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	      break;
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	    }
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	  case unop::Closure:
	    {
	      rat_seen_ = true;
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	      if (constant_term_as_bool(node->child()))
		{
		  res_ = bddtrue;
		  return;
		}

	      ratexp_trad_visitor v(dict_);
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	      node->child()->accept(v);
	      bdd f1 = v.result();
	      res_ = bddfalse;

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	      if (exprop_)
		{
		  bdd var_set = bdd_existcomp(bdd_support(f1), dict_.var_set);
		  bdd all_props = bdd_existcomp(f1, dict_.var_set);
		  while (all_props != bddfalse)
		    {
		      bdd label = bdd_satoneset(all_props, var_set, bddtrue);
		      all_props -= label;

		      formula* dest =
			dict_.bdd_to_formula(bdd_exist(f1 & label,
						       dict_.var_set));

		      const formula* dest2;
		      if (constant_term_as_bool(dest))
			{
			  dest->destroy();
			  res_ |= label;
			}
		      else
			{
			  dest2 = unop::instance(op, dest);
			  if (dest2 == constant::false_instance())
			    continue;
			  int x = dict_.register_next_variable(dest2);
			  dest2->destroy();
			  res_ |= label & bdd_ithvar(x);
			}
		    }
		}
	      else
		{
		  minato_isop isop(f1);
		  bdd cube;
		  while ((cube = isop.next()) != bddfalse)
		    {
		      bdd label = bdd_exist(cube, dict_.next_set);
		      bdd dest_bdd = bdd_existcomp(cube, dict_.next_set);
		      formula* dest = dict_.conj_bdd_to_formula(dest_bdd);

		      const formula* dest2;
		      if (constant_term_as_bool(dest))
			{
			  dest->destroy();
			  res_ |= label;
			}
		      else
			{
			  dest2 = unop::instance(op, dest);
			  if (dest2 == constant::false_instance())
			    continue;
			  int x = dict_.register_next_variable(dest2);
			  dest2->destroy();
			  res_ |= label & bdd_ithvar(x);
			}
		    }
		}
	    }
	    break;

	  case unop::NegClosure:
	    {
	      rat_seen_ = true;
	      has_marked_ = true;
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	      if (constant_term_as_bool(node->child()))
		{
		  res_ = bddfalse;
		  return;
		}

	      ratexp_trad_visitor v(dict_);
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	      node->child()->accept(v);
	      bdd f1 = v.result();

	      // trace_ltl_bdd(dict_, f1);

	      bdd var_set = bdd_existcomp(bdd_support(f1), dict_.var_set);
	      bdd all_props = bdd_existcomp(f1, dict_.var_set);

	      res_ = !all_props &
		// stick X(1) to preserve determinism.
		bdd_ithvar(dict_.register_next_variable
			   (constant::true_instance()));

	      while (all_props != bddfalse)
		{
		  bdd label = bdd_satoneset(all_props, var_set, bddtrue);
		  all_props -= label;

		  formula* dest =
		    dict_.bdd_to_formula(bdd_exist(f1 & label,
						   dict_.var_set));

		  // !{ Exp } is false if Exp accepts the empty word.
		  if (constant_term_as_bool(dest))
		    {
		      dest->destroy();
		      continue;
		    }

		  const formula* dest2 = unop::instance(op, dest);

		  if (dest == constant::false_instance())
		    continue;

		  int x = dict_.register_next_variable(dest2);
		  dest2->destroy();
		  res_ |= label & bdd_ithvar(x);
		}
	    }
	    break;

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	  case unop::Finish:
	    assert(!"unsupported operator");
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	    break;
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	  }
      }

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      void
      visit(const bunop*)
      {
	assert(!"Not an LTL operator");
      }

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      void
      visit(const binop* node)
      {
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	binop::type op = node->op();
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	switch (op)
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	  {
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	    // r(f1 logical-op f2) = r(f1) logical-op r(f2)
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	  case binop::Xor:
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	    {
	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
	      res_ = bdd_apply(f1, f2, bddop_xor);
	      return;
	    }
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	  case binop::Implies:
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	    {
	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
	      res_ = bdd_apply(f1, f2, bddop_imp);
	      return;
	    }
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	  case binop::Equiv:
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	    {
	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
	      res_ = bdd_apply(f1, f2, bddop_biimp);
	      return;
	    }
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	  case binop::U:
	    {
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	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
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	      // 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));
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	      break;
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	    }
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	  case binop::W:
	    {
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	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
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	      // r(f1 W f2) = r(f2) + r(f1)r(X(f1 U f2))
	      int x = dict_.register_next_variable(node);
	      res_ = f2 | (f1 & bdd_ithvar(x));
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	      break;
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	    }
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	  case binop::R:
	    {
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	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
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	      // 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));
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	      break;
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	    }
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	  case binop::M:
	    {
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	      bdd f1 = recurse(node->first());
	      bdd f2 = recurse(node->second());
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	      // r(f1 M f2) = r(f1)r(f2) + a(f1)r(f2)r(X(f1 M f2))
	      int a = dict_.register_a_variable(node->first());
	      int x = dict_.register_next_variable(node);
	      res_ = (f1 & f2) | (bdd_ithvar(a) & f2 & bdd_ithvar(x));
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	      break;
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	    }
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	  case binop::EConcatMarked:
	    has_marked_ = true;
	    /* fall through */
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	  case binop::EConcat:
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	    rat_seen_ = true;
	    {
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	      // Recognize f2 on transitions going to destinations
	      // that accept the empty word.
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	      bdd f2 = recurse(node->second());
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	      ratexp_trad_visitor v(dict_);
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	      node->first()->accept(v);
	      bdd f1 = v.result();
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	      res_ = bddfalse;
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	      if (mark_all_)
		{
		  op = binop::EConcatMarked;
		  has_marked_ = true;
		}

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	      if (exprop_)
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		{
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		  bdd var_set = bdd_existcomp(bdd_support(f1), dict_.var_set);
		  bdd all_props = bdd_existcomp(f1, dict_.var_set);
		  while (all_props != bddfalse)
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		    {
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		      bdd label = bdd_satoneset(all_props, var_set, bddtrue);
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		      all_props -= label;

		      formula* dest =
			dict_.bdd_to_formula(bdd_exist(f1 & label,
						       dict_.var_set));

		      const formula* dest2 =
			binop::instance(op, dest, node->second()->clone());

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		      if (dest2 != constant::false_instance())
			{
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			  int x = dict_.register_next_variable(dest2);
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			  dest2->destroy();
			  res_ |= label & bdd_ithvar(x);
			}
		      if (constant_term_as_bool(dest))
			res_ |= label & f2;
		    }
		}
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	      else
		{
		  minato_isop isop(f1);
		  bdd cube;
		  while ((cube = isop.next()) != bddfalse)
		    {
		      bdd label = bdd_exist(cube, dict_.next_set);
		      bdd dest_bdd = bdd_existcomp(cube, dict_.next_set);
		      formula* dest = dict_.conj_bdd_to_formula(dest_bdd);

		      if (dest == constant::empty_word_instance())
			{
			  res_ |= label & f2;
			}
		      else
			{
			  formula* dest2 = binop::instance(op, dest,
						  node->second()->clone());
			  if (dest2 != constant::false_instance())
			    {
			      int x = dict_.register_next_variable(dest2);
			      dest2->destroy();
			      res_ |= label & bdd_ithvar(x);
			    }
			  if (constant_term_as_bool(dest))
			    res_ |= label & f2;
			}
		    }
		}
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	    }
	    break;

	  case binop::UConcat:
	    {
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	      // Transitions going to destinations accepting the empty
	      // word should recognize f2, and the automaton for f1
	      // should be understood as universal.
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	      bdd f2 = recurse(node->second());
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	      ratexp_trad_visitor v(dict_);
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	      node->first()->accept(v);
	      bdd f1 = v.result();
	      res_ = bddtrue;
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	      bdd var_set = bdd_existcomp(bdd_support(f1), dict_.var_set);
	      bdd all_props = bdd_existcomp(f1, dict_.var_set);
	      while (all_props != bddfalse)
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		{

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		  bdd one_prop_set = bddtrue;
		  if (exprop_)
		    one_prop_set = bdd_satoneset(all_props, var_set, bddtrue);
		  all_props -= one_prop_set;
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		  minato_isop isop(f1 & one_prop_set);
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		  bdd cube;
		  while ((cube = isop.next()) != bddfalse)
		    {
		      bdd label = bdd_exist(cube, dict_.next_set);
		      bdd dest_bdd = bdd_existcomp(cube, dict_.next_set);
		      formula* dest = dict_.conj_bdd_to_formula(dest_bdd);
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		      formula* dest2 =
			binop::instance(op, dest, node->second()->clone());
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		      bdd udest =
			bdd_ithvar(dict_.register_next_variable(dest2));
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		      if (constant_term_as_bool(dest))
			udest &= f2;

		      dest2->destroy();

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		      res_ &= bdd_apply(label, udest, bddop_imp);
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		    }
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		}
	    }
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	    break;
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	  }
      }

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      void
      visit(const automatop*)
      {
	assert(!"unsupported operator");
      }

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      void
      visit(const multop* node)
      {
	switch (node->op())
	  {
	  case multop::And:
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	    {
	      res_ = bddtrue;
	      unsigned s = node->size();
	      for (unsigned n = 0; n < s; ++n)
		{
		  bdd res = recurse(node->nth(n));
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		  //std::cerr << "== in And (" << to_string(node->nth(n))
		  // << ")" << std::endl;
		  // trace_ltl_bdd(dict_, res);
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		  res_ &= res;
		}
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	      //std::cerr << "=== And final" << std::endl;
	      // trace_ltl_bdd(dict_, res_);
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	      break;
	    }
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	  case multop::Or:
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	    {
	      res_ = bddfalse;
	      unsigned s = node->size();
	      for (unsigned n = 0; n < s; ++n)
		res_ |= recurse(node->nth(n));
	      break;
	    }
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	  case multop::Concat:
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	  case multop::Fusion:
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	  case multop::AndNLM:
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	    assert(!"Not an LTL operator");
	    break;
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	  }
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      }

      bdd
      recurse(const formula* f)
      {
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	ltl_trad_visitor v(dict_, mark_all_, exprop_);
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	f->accept(v);
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	rat_seen_ |= v.has_rational();
	has_marked_ |= v.has_marked();
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	return v.result();
      }


    private:
      translate_dict& dict_;
      bdd res_;
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      bool rat_seen_;
      bool has_marked_;
      bool mark_all_;
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      bool exprop_;
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    };

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    // Check whether a formula has a R, W, or G operator at its
    // top-level (preceding logical operators do not count).
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    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:
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	  case binop::M:
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	    return;
	  case binop::R:
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	  case binop::W:
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	    res_ = true;
	    return;
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	  case binop::UConcat:
	  case binop::EConcat:
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	  case binop::EConcatMarked:
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	    node->second()->accept(*this);
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	    // FIXME: we might need to add Acc[1]
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	    return;
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	  }
	/* Unreachable code.  */
	assert(0);
      }

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      void
      visit(const automatop*)
      {
	assert(!"unsupported operator");
      }

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      void
      visit(const bunop*)
      {
	assert(!"unsupported operator");
      }

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      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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    class formula_canonizer
    {
    public:
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      formula_canonizer(translate_dict& d,
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			bool fair_loop_approx, bdd all_promises, bool exprop)
	: v_(d, false, exprop),
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	  fair_loop_approx_(fair_loop_approx),
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	  all_promises_(all_promises),
	  d_(d)
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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()
      {
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	while (!f2b_.empty())
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	  {
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	    formula_to_bdd_map::iterator i = f2b_.begin();
	    const formula* f = i->first;
	    f2b_.erase(i);
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	    f->destroy();
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	  }
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      }

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      struct translated
      {
	bdd symbolic;
	bool has_rational:1;
	bool has_marked:1;
      };

      const translated&
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      translate(const formula* f, bool* new_flag = 0)
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      {
	// Use the cached result if available.
	formula_to_bdd_map::const_iterator i = f2b_.find(f);
	if (i != f2b_.end())
	  return i->second;

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	if (new_flag)
	  *new_flag = true;

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	// Perform the actual translation.
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	v_.reset(!has_mark(f));
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	f->accept(v_);
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	translated t;
	t.symbolic = v_.result();
	t.has_rational = v_.has_rational();
	t.has_marked = v_.has_marked();

//	std::cerr << "-----" << std::endl;
//	std::cerr << "Formula: " << to_string(f) << std::endl;
//	std::cerr << "Rational: " << t.has_rational << std::endl;
//	std::cerr << "Marked: " << t.has_marked << std::endl;
//	std::cerr << "Mark all: " << !has_mark(f) << std::endl;
//	std::cerr << "Transitions:" << std::endl;
//	trace_ltl_bdd(v_.get_dict(), t.symbolic);

	if (t.has_rational)
	  {
	    bdd res = bddfalse;

	    minato_isop isop(t.symbolic);
	    bdd cube;
	    while ((cube = isop.next()) != bddfalse)
	      {
		bdd label = bdd_exist(cube, d_.next_set);
		bdd dest_bdd = bdd_existcomp(cube, d_.next_set);
		formula* dest =
		  d_.conj_bdd_to_formula(dest_bdd);

		// Handle a Miyano-Hayashi style unrolling for
		// rational operators.  Marked nodes correspond to
		// subformulae in the Miyano-Hayashi set.
		if (simplify_mark(dest))
		  {
		    // Make the promise that we will exit marked sets.
		    int a =
		      d_.register_a_variable(constant::true_instance());
		    label &= bdd_ithvar(a);
		  }
		else
		  {
		    // We have left marked operators, but still
		    // have other rational operator to check.
		    // Start a new marked cycle.
		    formula* dest2 = mark_concat_ops(dest);
		    dest->destroy();
		    dest = dest2;
		  }
		// Note that simplify_mark may have changed dest.
		dest_bdd = bdd_ithvar(d_.register_next_variable(dest));
		dest->destroy();
		res |= label & dest_bdd;
	      }
	    t.symbolic = res;
//	    std::cerr << "Marking rewriting:" << std::endl;
//	    trace_ltl_bdd(v_.get_dict(), t.symbolic);
	  }
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	// 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))
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	      t.symbolic &= all_promises_;
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	  }

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

	return f2b_[f->clone()] = t;
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      }

      const formula*
      canonize(const formula* f)
      {
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	bool new_variable = false;
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	bdd b = translate(f, &new_variable).symbolic;
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	bdd_to_formula_map::iterator i = b2f_.find(b);
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	// Since we have just translated the formula, it is
	// necessarily in b2f_.
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	assert(i != b2f_.end());

	if (i->second != f)
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	  {
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	    // The translated bdd maps to an already seen formula.
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	    f->destroy();
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	    f = i->second->clone();
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	  }
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	return f;
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      }

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    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.
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      typedef std::map<const formula*, translated> formula_to_bdd_map;
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      formula_to_bdd_map f2b_;
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      possible_fair_loop_checker pflc_;
      bool fair_loop_approx_;
      bdd all_promises_;
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      translate_dict& d_;
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    };

  }

  typedef std