CAR: new algorithm for paritizing

* NEWS: Mention it.
* spot/twaalgos/car.cc, spot/twaalgos/car.hh, tests/python/car.py:
New files.
* spot/twaalgos/Makefile.am, tests/Makefile.am: Add them.
* python/spot/impl.i: Include CAR.
* spot/twa/acc.cc, spot/twa/acc.hh, spot/twa/twagraph.cc,
spot/twa/twagraph.hh: Add supporting methods.

NEWS

@@ -88,6 +88,9 @@ New in spot 2.8.6.dev (not yet released)
     same transition structure (where the ..._maybe() variant would
     modify the Rabin automaton if needed).
 
+  - car() is a new variant of LAR algorithm that combines several
+    strategies for paritazing any automaton.
+
 New in spot 2.8.6 (2020-02-19)
 
   Bugs fixed:

python/spot/impl.i

@@ -162,6 +162,7 @@
 #include <spot/twaalgos/are_isomorphic.hh>
 #include <spot/twaalgos/rabin2parity.hh>
 #include <spot/twaalgos/toparity.hh>
+#include <spot/twaalgos/car.hh>
 
 #include <spot/parseaut/public.hh>
 
@@ -683,6 +684,7 @@ def state_is_accepting(self, src) -> "bool":
 %include <spot/twaalgos/are_isomorphic.hh>
 %include <spot/twaalgos/rabin2parity.hh>
 %include <spot/twaalgos/toparity.hh>
+%include <spot/twaalgos/car.hh>
 
 %pythonprepend spot::twa::dtwa_complement %{
   from warnings import warn

spot/twa/acc.cc

@@ -25,6 +25,7 @@
 #include <cctype>
 #include <cstring>
 #include <map>
+#include <numeric>
 #include <spot/twa/acc.hh>
 #include "spot/priv/bddalloc.hh"
 #include <spot/misc/minato.hh>
@@ -1059,6 +1060,38 @@ namespace spot
     return res;
   }
 
+  bool
+  acc_cond::has_parity_prefix(acc_cond& new_cond,
+                              std::vector<unsigned>& colors) const
+  {
+    return code_.has_parity_prefix(new_cond, colors);
+  }
+
+  bool
+  acc_cond::is_parity_max_equiv(std::vector<int>&permut, bool even) const
+  {
+    bool result = code_.is_parity_max_equiv(permut, 0, even);
+    int max_value = *std::max_element(std::begin(permut), std::end(permut));
+    for (unsigned i = 0; i < permut.size(); ++i)
+      if (permut[i] != -1)
+        permut[i] = max_value - permut[i];
+      else
+        permut[i] = i;
+    return result;
+  }
+
+  std::vector<unsigned>
+  acc_cond::colors_inf_conj(unsigned min_nb_colors)
+  {
+    return code_.colors_inf_conj(min_nb_colors);
+  }
+
+  std::vector<unsigned>
+  acc_cond::colors_fin_disj(unsigned min_nb_colors)
+  {
+    return code_.colors_fin_disj(min_nb_colors);
+  }
+
   bool acc_cond::is_parity(bool& max, bool& odd, bool equiv) const
   {
     unsigned sets = num_;
@@ -1250,6 +1283,110 @@ namespace spot
     return rescode;
   }
 
+  bool
+  acc_cond::acc_code::is_parity_max_equiv(std::vector<int>& permut,
+                    unsigned new_color,
+                    bool even) const
+  {
+    auto conj = top_conjuncts();
+    auto disj = top_disjuncts();
+    if (conj.size() == 1)
+    {
+      if (disj.size() == 1)
+      {
+        acc_cond::acc_code elem = conj[0];
+        if ((even && elem.back().sub.op == acc_cond::acc_op::Inf)
+          || (!even && elem.back().sub.op == acc_cond::acc_op::Fin))
+        {
+          for (auto color : disj[0][0].mark.sets())
+          {
+            if (permut[color] != -1
+                && ((unsigned) permut[color]) != new_color)
+              return false;
+            permut[color] = new_color;
+          }
+          return true;
+        }
+        return false;
+      }
+      else
+      {
+        std::sort(disj.begin(), disj.end(),
+          [](acc_code c1, acc_code c2)
+                  {
+            (void) c2;
+            return c1.back().sub.op == acc_cond::acc_op::Inf;
+          });
+        unsigned i = 0;
+        for (; i < disj.size() - 1; ++i)
+        {
+          if (disj[i].back().sub.op != acc_cond::acc_op::Inf
+            || disj[i][0].mark.count() != 1)
+            return false;
+          for (auto color : disj[i][0].mark.sets())
+          {
+            if (permut[color] != -1
+              && ((unsigned) permut[color]) != new_color)
+                        return false;
+            permut[color] = new_color;
+          }
+        }
+        if (disj[i].back().sub.op == acc_cond::acc_op::Inf)
+        {
+          if (!even || disj[i][0].mark.count() != 1)
+            return false;
+          for (auto color : disj[i][0].mark.sets())
+          {
+            if (permut[color] != -1
+              && ((unsigned) permut[color]) != new_color)
+                        return false;
+            permut[color] = new_color;
+          }
+          return true;
+        }
+        return disj[i].is_parity_max_equiv(permut, new_color + 1, even);
+      }
+    }
+    else
+      {       std::sort(conj.begin(), conj.end(),
+        [](acc_code c1, acc_code c2)
+              {
+          (void) c2;
+          return c1.back().sub.op == acc_cond::acc_op::Fin;
+        });
+      unsigned i = 0;
+      for (; i < conj.size() - 1; i++)
+      {
+        if (conj[i].back().sub.op != acc_cond::acc_op::Fin
+          || conj[i][0].mark.count() != 1)
+            return false;
+        for (auto color : conj[i][0].mark.sets())
+        {
+          if (permut[color] != -1 && permut[color != new_color])
+            return false;
+          permut[color] = new_color;
+        }
+      }
+      if (conj[i].back().sub.op == acc_cond::acc_op::Fin)
+      {
+        if (even)
+          return 0;
+        if (conj[i][0].mark.count() != 1)
+          return false;
+        for (auto color : conj[i][0].mark.sets())
+        {
+          if (permut[color] != -1 && permut[color != new_color])
+            return false;
+          permut[color] = new_color;
+        }
+        return true;
+      }
+
+      return conj[i].is_parity_max_equiv(permut, new_color + 1, even);
+    }
+  }
+
+
   namespace
   {
     template<typename Fun>

spot/twa/acc.hh

@@ -23,6 +23,8 @@
 #include <sstream>
 #include <vector>
 #include <iostream>
+#include <algorithm>
+#include <numeric>
 
 #include <spot/misc/_config.h>
 #include <spot/misc/bitset.hh>
@@ -58,12 +60,22 @@ namespace spot
   class SPOT_API acc_cond
   {
 
+  public:
+    bool
+    has_parity_prefix(acc_cond& new_acc, std::vector<unsigned>& colors) const;
+
 #ifndef SWIG
   private:
     [[noreturn]] static void report_too_many_sets();
 #endif
   public:
 
+    std::vector<unsigned>
+    colors_inf_conj(unsigned min_nb_colors);
+
+    std::vector<unsigned>
+    colors_fin_disj(unsigned min_nb_colors);
+
     /// \brief An acceptance mark
     ///
     /// This type is used to represent a set of acceptance sets.  It
@@ -95,6 +107,9 @@ namespace spot
       /// Initialize an empty mark_t.
       mark_t() = default;
 
+      mark_t
+      apply_permutation(std::vector<unsigned> permut);
+
 #ifndef SWIG
       /// Create a mark_t from a range of set numbers.
       template<class iterator>
@@ -453,7 +468,174 @@ namespace spot
     /// provided methods instead.
     struct SPOT_API acc_code: public std::vector<acc_word>
     {
-      bool operator==(const acc_code& other) const
+
+      std::vector<unsigned>
+      colors_inf_conj(unsigned min_nb_colors = 2)
+      {
+        auto result = std::vector<unsigned>();
+        auto conj = top_conjuncts();
+        if (conj.size() != 1)
+        {
+          std::sort(conj.begin(), conj.end(),
+            [](acc_code c1, acc_code c2)
+              {
+                (void)c2;
+                return c1.back().sub.op == acc_cond::acc_op::Inf;
+              });
+          unsigned i = 0;
+          while (i < conj.size())
+          {
+            acc_cond::acc_code elem = conj[i];
+            if (elem.back().sub.op == acc_cond::acc_op::Inf)
+            {
+              if (elem[0].mark.count() == 1)
+                result.insert(result.end(), elem[0].mark.min_set() - 1);
+            } else
+              break;
+            ++i;
+          }
+          if (result.size() >= min_nb_colors)
+                return result;
+          while (i < conj.size())
+          {
+            result = conj[i].colors_inf_conj();
+            if (result.size() >= min_nb_colors)
+              return result;
+            result.clear();
+            ++i;
+          }
+        }
+        else
+        {
+          auto disj = top_disjuncts();
+          if (disj.size() > 1)
+          {
+            for (auto elem : disj)
+            {
+              result = elem.colors_inf_conj();
+              if (result.size() >= min_nb_colors)
+                return result;
+              result.clear();
+            }
+          }
+          else
+            return {};
+        }
+        return result;
+      }
+
+      std::vector<unsigned>
+      colors_fin_disj(unsigned min_nb_colors = 2)
+      {
+        auto result = std::vector<unsigned>();
+        auto disj = top_disjuncts();
+        if (disj.size() != 1)
+        {
+          std::sort(disj.begin(), disj.end(),
+            [](acc_code c1, acc_code c2)
+              {
+                (void) c2;
+                return c1.back().sub.op == acc_cond::acc_op::Fin;
+              });
+          unsigned i = 0;
+          while (i < disj.size())
+          {
+            acc_cond::acc_code elem = disj[i];
+            if (elem.back().sub.op == acc_cond::acc_op::Fin)
+            {
+              if (elem[0].mark.count() == 1)
+                result.insert(result.end(), elem[0].mark.min_set() - 1);
+            } else
+              break;
+            ++i;
+          }
+          if (result.size() >= min_nb_colors)
+                return result;
+          while (i < disj.size())
+          {
+            result = disj[i].colors_fin_disj();
+            if (result.size() >= min_nb_colors)
+              return result;
+            result.clear();
+            ++i;
+          }
+        }
+        else
+        {
+          auto disj = top_conjuncts();
+          if (disj.size() > 1)
+          {
+            for (auto elem : disj)
+            {
+              result = elem.colors_fin_disj();
+              if (result.size() >= min_nb_colors)
+                return result;
+              result.clear();
+            }
+          }
+          else
+            return {};
+        }
+        return result;
+      }
+
+      bool
+      has_parity_prefix_aux(spot::acc_cond& new_cond,
+        std::vector<unsigned>& colors, std::vector<acc_code> elements,
+        acc_cond::acc_op op) const
+      {
+        mark_t empty_m = { };
+        if (elements.size() > 2)
+        {
+          new_cond = (*this);
+          return false;
+        }
+        if (elements.size() == 2)
+        {
+          // Vaut 1 si si c'est le 2e qui est bon
+          unsigned pos = elements[1].back().sub.op == op
+                        && elements[1][0].mark.count() == 1;
+          if (!(elements[0].back().sub.op == op || pos))
+          {
+            new_cond = (*this);
+            return false;
+          }
+          if ((elements[1 - pos].used_sets() & elements[pos][0].mark)
+                != empty_m)
+          {
+            new_cond = (*this);
+            return false;
+          }
+          if (elements[pos][0].mark.count() != 1)
+          {
+            return false;
+          }
+          colors.push_back(elements[pos][0].mark.min_set() - 1);
+          elements[1 - pos].has_parity_prefix(new_cond, colors);
+          return true;
+        }
+        return false;
+      }
+
+      bool has_parity_prefix(spot::acc_cond& new_cond,
+        std::vector<unsigned>& colors) const
+      {
+        auto disj = top_disjuncts();
+        if (!
+          (has_parity_prefix_aux(new_cond, colors,
+          top_conjuncts(), acc_cond::acc_op::Fin) ||
+          has_parity_prefix_aux(new_cond, colors,
+          disj, acc_cond::acc_op::Inf)))
+            new_cond = spot::acc_cond(*this);
+        return disj.size() == 2;
+      }
+
+      bool
+      is_parity_max_equiv(std::vector<int>& permut,
+                        unsigned new_color,
+                        bool even) const;
+
+     bool operator==(const acc_code& other) const
       {
         unsigned pos = size();
         if (other.size() != pos)
@@ -1669,6 +1851,9 @@ namespace spot
     /// HOA format will be accepted.
     bool is_parity(bool& max, bool& odd, bool equiv = false) const;
 
+
+    bool is_parity_max_equiv(std::vector<int>& permut, bool even) const;
+
     /// \brief check is the acceptance condition matches one of the
     /// four type of parity acceptance defined in the HOA format.
     bool is_parity() const
@@ -1838,6 +2023,57 @@ namespace spot
       return all_;
     }
 
+    acc_cond
+    apply_permutation(std::vector<unsigned>permut)
+    {
+      return acc_cond(apply_permutation_aux(permut));
+    }
+
+    acc_code
+    apply_permutation_aux(std::vector<unsigned>permut)
+    {
+      auto conj = top_conjuncts();
+      auto disj = top_disjuncts();
+
+      if (conj.size() > 1)
+      {
+        auto transformed = std::vector<acc_code>();
+        for (auto elem : conj)
+          transformed.push_back(elem.apply_permutation_aux(permut));
+        std::sort(transformed.begin(), transformed.end());
+        auto uniq = std::unique(transformed.begin(), transformed.end());
+        auto result = std::accumulate(transformed.begin(), uniq, acc_code::t(),
+          [](acc_code c1, acc_code c2)
+              {
+                return c1 & c2;
+              });
+        return result;
+      }
+      else if (disj.size() > 1)
+      {
+        auto transformed = std::vector<acc_code>();
+        for (auto elem : disj)
+          transformed.push_back(elem.apply_permutation_aux(permut));
+        std::sort(transformed.begin(), transformed.end());
+        auto uniq = std::unique(transformed.begin(), transformed.end());
+        auto result = std::accumulate(transformed.begin(), uniq, acc_code::f(),
+          [](acc_code c1, acc_code c2)
+              {
+                return c1 | c2;
+              });
+        return result;
+      }
+      else
+      {
+        if (code_.back().sub.op == acc_cond::acc_op::Fin)
+          return fin(code_[0].mark.apply_permutation(permut));
+        if (code_.back().sub.op == acc_cond::acc_op::Inf)
+          return inf(code_[0].mark.apply_permutation(permut));
+      }
+      SPOT_ASSERT(false);
+      return {};
+    }
+
     /// \brief Check whether visiting *exactly* all sets \a inf
     /// infinitely often satisfies the acceptance condition.
     bool accepting(mark_t inf) const
@@ -2219,6 +2455,16 @@ namespace spot
   {
     return {*this};
   }
+
+  inline acc_cond::mark_t
+  acc_cond::mark_t::apply_permutation(std::vector<unsigned> permut)
+  {
+    mark_t result { };
+    for (auto color : sets())
+      if (color < permut.size())
+        result.set(permut[color]);
+    return result;
+  }
 }
 
 namespace std

spot/twa/twagraph.cc

@@ -31,6 +31,15 @@ using namespace std::string_literals;
 namespace spot
 {
 
+  void
+  twa_graph::apply_permutation(std::vector<unsigned> permut)
+  {
+    for (auto& e : edges())
+    {
+      e.acc.apply_permutation(permut);
+    }
+  }
+
   std::string twa_graph::format_state(unsigned n) const
   {
     if (is_univ_dest(n))

spot/twa/twagraph.hh

@@ -219,6 +219,9 @@ namespace spot
     mutable unsigned init_number_;
 
   public:
+
+    void apply_permutation(std::vector<unsigned> permut);
+
     twa_graph(const bdd_dict_ptr& dict)
       : twa(dict),
         init_number_(0)

spot/twaalgos/Makefile.am

@@ -59,6 +59,7 @@ twaalgos_HEADERS = \
   isunamb.hh \
   isweakscc.hh \
   langmap.hh \
+  car.hh \
   lbtt.hh \
   ltl2taa.hh \
   ltl2tgba_fm.hh \
@@ -128,6 +129,7 @@ libtwaalgos_la_SOURCES = \
   isunamb.cc \
   isweakscc.cc \
   langmap.cc \
+  car.cc \
   lbtt.cc \
   ltl2taa.cc \
   ltl2tgba_fm.cc \

spot/twaalgos/car.hh

@@ -0,0 +1,59 @@
+// -*- coding: utf-8 -*-
+// Copyright (C) 2012-2019 Laboratoire de Recherche
+// et Développement de l'Epita (LRDE).
+//
+// 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 3 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 this program.  If not, see <http://www.gnu.org/licenses/>.
+
+#pragma once
+
+#include <spot/twa/twagraph.hh>
+
+namespace spot
+{
+/// \ingroup twa_acc_transform
+/// \brief Take an automaton with any acceptance condition and return an
+/// equivalent parity automaton.
+///
+/// The parity condition of the returned automaton is max even or
+/// max odd.
+/// If \a search_ex is true, when we move several elements, we
+/// try to find an order  such that the new permutation already exists.
+/// If \a partial_degen is true, we apply a partial degeneralization to remove
+// occurences of Fin | Fin and Inf & Inf.
+/// If \a scc_acc_clean is true, we remove for each SCC the colors that don't
+// appear.
+/// If \a parity_equiv is true, we check if there exists a permutations of
+// colors such that the acceptance
+/// condition is a partity condition.
+/// If \a use_last is true, we use the most recent state when looking for an
+// existing state.
+/// If \a pretty_print is true, we give a name to the states describing the
+// state of the aut_ and the permutation.
+
+  SPOT_API twa_graph_ptr
+  remove_false_transitions(const twa_graph_ptr a);
+
+  SPOT_API twa_graph_ptr
+  car(const twa_graph_ptr &aut,
+    bool search_ex = true,
+    bool partial_degen = true,
+    bool scc_acc_clean = true,
+    bool parity_equiv = true,
+    bool use_last = true,
+    bool parity_prefix = true,
+    bool rabin_to_buchi = true,
+    bool pretty_print = false);
+} // namespace spot

tests/Makefile.am

@@ -372,6 +372,7 @@ TESTS_python = \
   python/bdditer.py \
   python/bddnqueen.py \
   python/bugdet.py \
+  python/car.py \
   python/complement_semidet.py \
   python/declenv.py \
   python/decompose_scc.py \

tests/python/car.py

@@ -0,0 +1,157 @@
+#!/usr/bin/python3
+# -*- mode: python; coding: utf-8 -*-
+# Copyright (C) 2018, 2019 Laboratoire de Recherche et Développement de
+# l'EPITA.
+#
+# 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 3 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 this program.  If not, see <http://www.gnu.org/licenses/>.
+
+import spot
+import time
+
+a = spot.automaton("""
+HOA: v1
+name: "(FGp0 & ((XFp0 & F!p1) | F(Gp1 & XG!p0))) | G(F!p0 & (XFp0 | F!p1) &
+F(Gp1 | G!p0))"
+States: 14
+Start: 0
+AP: 2 "p1" "p0"
+Acceptance: 6 (Fin(0) & Fin(1)) | ((Fin(4)|Fin(5)) & (Inf(2)&Inf(3)))
+properties: trans-labels explicit-labels trans-acc complete
+properties: deterministic
+--BODY--
+State: 0
+[!0] 1
+[0] 2
+State: 1
+[!0&!1] 1 {0 1 2 3 5}
+[0&!1] 3
+[!0&1] 4
+[0&1] 5
+State: 2
+[0&!1] 2 {1}
+[!0&1] 4
+[!0&!1] 6
+[0&1] 7
+State: 3
+[0&!1] 3 {1 3}
+[!0&1] 4
+[!0&!1] 6 {0 1 2 3 5}
+[0&1] 8
+State: 4
+[!0&!1] 4 {1 2 3 5}
+[!0&1] 4 {2 4 5}
+[0&!1] 5 {1 3}
+[0&1] 5 {4}
+State: 5
+[!0&1] 4 {2 4 5}
+[0&!1] 5 {1 3}
+[0&1] 8 {2 4}
+[!0&!1] 9 {1 2 3 5}
+State: 6
+[0&!1] 3 {1 3}
+[!0&1] 4
+[0&1] 5
+[!0&!1] 10
+State: 7
+[!0&1] 4
+[0&!1] 7 {1 3}
+[!0&!1] 11
+[0&1] 12 {0 4}
+State: 8
+[!0&1] 4 {2 4 5}
+[0&1] 5 {4}
+[0&!1] 8 {1 3}
+[!0&!1] 11 {1 3 5}
+State: 9
+[!0&1] 4 {2 4 5}
+[0&!1] 5 {1 3}
+[0&1] 5 {4}
+[!0&!1] 11 {1 3 5}
+State: 10
+[!0&1] 4
+[0&1] 8
+[!0&!1] 10 {0 1 2 3 5}
+[0&!1] 13 {1 2 3}
+State: 11
+[!0&1] 4 {2 4 5}
+[0&!1] 8 {1 2 3}
+[0&1] 8 {2 4}
+[!0&!1] 11 {1 2 3 5}
+State: 12
+[!0&1] 4
+[0&1] 7 {0 2 4}
+[!0&!1] 9
+[0&!1] 12 {1 3}
+State: 13
+[!0&1] 4
+[0&1] 5
+[!0&!1] 10 {0 1 3 5}
+[0&!1] 13 {1 3}
+--END--""")
+scc_ = spot.scc_info(a)
+spot.partial_degeneralize(scc_.split_on_sets(2, [])[0])
+
+
+a = spot.automaton("""
+HOA: v1
+States: 2
+Start: 0
+AP: 2 "p0" "p1"
+Acceptance: 5 (Inf(0)&Inf(1)) | ((Fin(2)|Fin(3)) & Fin(4))
+--BODY--
+State: 0
+[!0 & 1] 0 {2 3}
+[!0 & !1] 0 {3}
+[0] 1
+State: 1
+[0&1] 1 {1 2 4}
+[0&!1] 1 {4}
+[!0&1] 1 {0 1 2 3}
+[!0&!1] 1 {0 3}
+--END--""")
+p = spot.car(a, False, False, False, False, False, False, True)
+assert spot.are_equivalent(a, p)
+a = spot.automaton("""
+HOA: v1 States: 6 Start: 0 AP: 2 "p0" "p1" Acceptance: 6 Inf(5) |
+Fin(2) | Inf(1) | (Inf(0) & Fin(3)) | Inf(4) properties: trans-labels
+explicit-labels trans-acc --BODY-- State: 0 [0&1] 2 {4 5} [0&1] 4 {0 4}
+[!0&!1] 3 {3 5} State: 1 [0&!1] 3 {1 5} [!0&!1] 5 {0 1} State: 2 [!0&!1]
+0 {0 3} [0&!1] 1 {0} State: 3 [!0&1] 4 {1 2 3} [0&1] 3 {3 4 5} State:
+4 [!0&!1] 1 {2 4} State: 5 [!0&1] 4 --END--
+""")
+p = spot.car(a, False, False, False, False, False, False, True)
+if (not p.num_states() == 8):
+    print(p.num_states())
+assert spot.are_equivalent(a, p)
+
+for f in spot.randltl(15, 1000):
+    print(f)
+    d = spot.translate(f, "det", "G")
+    p = spot.car(d, False, False, False, False, False, False, True)
+    p2 = spot.car(d, False, False, False, False, False, False, True)
+    if(not spot.are_equivalent(p, d)):
+        print(f)
+        assert(False)
+
+for f in spot.randltl(5, 10000):
+    print(f)
+    n = spot.translate(f)
+    p = spot.car(n, False, False, False, False, False, False, True)
+    assert(spot.are_equivalent(n, p))
+
+a = spot.translate('!(GFa -> (GFb & GF(!b & !Xb)))', 'gen', 'det')
+b = spot.car(a, False, False, False, False, False, False, True)
+assert spot.are_equivalent(a, b)