From a738801edfcffcff73fb4cf3101d00341b5ff714 Mon Sep 17 00:00:00 2001
From: Alexandre Duret-Lutz <adl@lrde.epita.fr>
Date: Wed, 23 May 2018 18:34:31 +0200
Subject: [PATCH] product: optimize product with weak automata

Fixes #350.

* spot/twaalgos/product.cc: Implement this change.
* NEWS, spot/twaalgos/product.hh: Mention it.
* spot/twa/acc.cc, spot/twa/acc.hh (acc_cond::sat_mark): New method.
* tests/python/_product_weak.ipynb: New file.
* tests/Makefile.am: Add it.
* tests/python/automata.ipynb, tests/python/highlighting.ipynb,
tests/python/product.ipynb, tests/core/prodor.test: Adjust test cases.
---
 NEWS                             |     5 +
 spot/twa/acc.cc                  |    26 +-
 spot/twa/acc.hh                  |    13 +-
 spot/twaalgos/product.cc         |   220 +-
 spot/twaalgos/product.hh         |    22 +-
 tests/Makefile.am                |     1 +
 tests/core/prodor.test           |    25 +-
 tests/python/_product_weak.ipynb | 13400 +++++++++++++++++++++++++++++
 tests/python/automata.ipynb      |   636 +-
 tests/python/highlighting.ipynb  |   177 +-
 tests/python/product.ipynb       |  2428 +++---
 11 files changed, 15348 insertions(+), 1605 deletions(-)
 create mode 100644 tests/python/_product_weak.ipynb

diff --git a/NEWS b/NEWS
index c1073d13c..cee198016 100644
--- a/NEWS
+++ b/NEWS
@@ -89,6 +89,11 @@ New in spot 2.5.3.dev  (not yet released)
     They are most welcome in Python, since we used to redefine
     them every now and them.
 
+  - spot::product() and spot::product_or() learned so produce an
+    automaton with a simpler acceptance condition if one of the
+    argument is a weak automaton.  In this case the resulting
+    acceptance condition is (usually) that of the other argument.
+
   - spot::parity_product() and spot::parity_product_or() were dropped.
     The code was buggy, hard to maintain, and these functions do not
     seem to be used.
diff --git a/spot/twa/acc.cc b/spot/twa/acc.cc
index 92e339cf8..4286c06bc 100644
--- a/spot/twa/acc.cc
+++ b/spot/twa/acc.cc
@@ -1153,12 +1153,22 @@ namespace spot
   }
 
   std::pair<bool, acc_cond::mark_t>
-  acc_cond::unsat_mark() const
+  acc_cond::sat_unsat_mark(bool sat) const
   {
-    if (is_t())
-      return {false, mark_t({})};
-    if (!uses_fin_acceptance())
-      return {true, mark_t({})};
+    if (sat)
+      {
+        if (is_f())
+          return {false, mark_t({})};
+        if (!uses_fin_acceptance())
+          return {true, all_sets()};
+      }
+    else
+      {
+        if (is_t())
+          return {false, mark_t({})};
+        if (!uses_fin_acceptance())
+          return {true, mark_t({})};
+      }
 
     auto used = code_.used_sets();
     unsigned c = used.count();
@@ -1185,11 +1195,11 @@ namespace spot
     bdd res = to_bdd_rec(&code_.back(), &r[0]);
 
     if (res == bddtrue)
-      return {false, mark_t({})};
+      return {sat, mark_t({})};
     if (res == bddfalse)
-      return {true, mark_t({})};
+      return {!sat, mark_t({})};
 
-    bdd cube = bdd_satone(!res);
+    bdd cube = bdd_satone(sat ? res : !res);
     mark_t i = {};
     while (cube != bddtrue)
       {
diff --git a/spot/twa/acc.hh b/spot/twa/acc.hh
index 93e6a16f2..253cd608b 100644
--- a/spot/twa/acc.hh
+++ b/spot/twa/acc.hh
@@ -1165,10 +1165,21 @@ namespace spot
     // Return (true, m) if there exist some acceptance mark m that
     // does not satisfy the acceptance condition.  Return (false, 0U)
     // otherwise.
-    std::pair<bool, acc_cond::mark_t> unsat_mark() const;
+    std::pair<bool, acc_cond::mark_t> unsat_mark() const
+    {
+      return sat_unsat_mark(false);
+    }
+    // Return (true, m) if there exist some acceptance mark m that
+    // does satisfy the acceptance condition.  Return (false, 0U)
+    // otherwise.
+    std::pair<bool, acc_cond::mark_t> sat_mark() const
+    {
+      return sat_unsat_mark(true);
+    }
 
   protected:
     bool check_fin_acceptance() const;
+    std::pair<bool, acc_cond::mark_t> sat_unsat_mark(bool) const;
 
   public:
     static acc_code inf(mark_t mark)
diff --git a/spot/twaalgos/product.cc b/spot/twaalgos/product.cc
index df32d2c8b..56a2bcbac 100644
--- a/spot/twaalgos/product.cc
+++ b/spot/twaalgos/product.cc
@@ -40,33 +40,18 @@ namespace spot
       }
     };
 
+
+    template<typename T>
     static
-    twa_graph_ptr product_aux(const const_twa_graph_ptr& left,
-                              const const_twa_graph_ptr& right,
-                              unsigned left_state,
-                              unsigned right_state,
-                              bool and_acc)
+    void product_main(const const_twa_graph_ptr& left,
+                      const const_twa_graph_ptr& right,
+                      unsigned left_state,
+                      unsigned right_state,
+                      twa_graph_ptr res, T merge_acc)
     {
-      if (!(left->is_existential() && right->is_existential()))
-        throw std::runtime_error
-          ("product() does not support alternating automata");
       std::unordered_map<product_state, unsigned, product_state_hash> s2n;
       std::deque<std::pair<product_state, unsigned>> todo;
 
-      if (left->get_dict() != right->get_dict())
-        throw std::runtime_error("product: left and right automata should "
-                                 "share their bdd_dict");
-      auto res = make_twa_graph(left->get_dict());
-      res->copy_ap_of(left);
-      res->copy_ap_of(right);
-      auto left_num = left->num_sets();
-      auto right_acc = right->get_acceptance() << left_num;
-      if (and_acc)
-        right_acc &= left->get_acceptance();
-      else
-        right_acc |= left->get_acceptance();
-      res->set_acceptance(left_num + right->num_sets(), right_acc);
-
       auto v = new product_states;
       res->set_named_prop("product-states", v);
 
@@ -86,10 +71,10 @@ namespace spot
         };
 
       res->set_init_state(new_state(left_state, right_state));
-      if (right_acc.is_f())
+      if (res->acc().is_f())
         // Do not bother doing any work if the resulting acceptance is
         // false.
-        return res;
+        return;
       while (!todo.empty())
         {
           auto top = todo.front();
@@ -102,10 +87,192 @@ namespace spot
                   continue;
                 auto dst = new_state(l.dst, r.dst);
                 res->new_edge(top.second, dst, cond,
-                              l.acc | (r.acc << left_num));
+                              merge_acc(l.acc, r.acc));
                 // If right is deterministic, we can abort immediately!
               }
         }
+    }
+
+    static
+    twa_graph_ptr product_aux(const const_twa_graph_ptr& left,
+                              const const_twa_graph_ptr& right,
+                              unsigned left_state,
+                              unsigned right_state,
+                              bool and_acc)
+    {
+      if (SPOT_UNLIKELY(!(left->is_existential() && right->is_existential())))
+        throw std::runtime_error
+          ("product() does not support alternating automata");
+      if (SPOT_UNLIKELY(left->get_dict() != right->get_dict()))
+        throw std::runtime_error("product: left and right automata should "
+                                 "share their bdd_dict");
+
+      auto res = make_twa_graph(left->get_dict());
+      res->copy_ap_of(left);
+      res->copy_ap_of(right);
+
+      bool leftweak = left->prop_weak().is_true();
+      bool rightweak = right->prop_weak().is_true();
+      // We have optimization to the standard product in case one
+      // of the arguments is weak.  However these optimizations
+      // are pointless if the said arguments are "t" or "f".
+      if ((leftweak || rightweak)
+          && (left->num_sets() > 0) && (right->num_sets() > 0))
+        {
+          // If both automata are weak, we can restrict the result to
+          // Büchi or co-Büchi.  We will favor Büchi unless the two
+          // operands are co-Büchi.
+          if (leftweak && rightweak)
+            {
+            weak_weak:
+              acc_cond::mark_t accmark = {0};
+              acc_cond::mark_t rejmark = {};
+              if (left->acc().is_co_buchi() && right->acc().is_co_buchi())
+                {
+                  res->set_co_buchi();
+                  std::swap(accmark, rejmark);
+                }
+              else
+                {
+                  res->set_buchi();
+                }
+              res->prop_weak(true);
+              auto& lacc = left->acc();
+              auto& racc = right->acc();
+              if (and_acc)
+                product_main(left, right, left_state, right_state, res,
+                             [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                             {
+                               if (lacc.accepting(ml) && racc.accepting(mr))
+                                 return accmark;
+                               else
+                                 return rejmark;
+                             });
+              else
+                product_main(left, right, left_state, right_state, res,
+                             [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                             {
+                               if (lacc.accepting(ml) || racc.accepting(mr))
+                                 return accmark;
+                               else
+                                 return rejmark;
+                             });
+            }
+          else if (!rightweak)
+            {
+              if (and_acc)
+                {
+                  auto rightunsatmark = right->acc().unsat_mark();
+                  if (!rightunsatmark.first)
+                    {
+                      // Left is weak.  Right was not weak, but it is
+                      // always accepting.  We can therefore pretend
+                      // that right is weak.
+                      goto weak_weak;
+                    }
+                  res->copy_acceptance_of(right);
+                  acc_cond::mark_t rejmark = rightunsatmark.second;
+                  auto& lacc = left->acc();
+                  product_main(left, right, left_state, right_state, res,
+                               [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                               {
+                                 if (lacc.accepting(ml))
+                                   return mr;
+                                 else
+                                   return rejmark;
+                               });
+                }
+              else
+                {
+                  auto rightsatmark = right->acc().sat_mark();
+                  if (!rightsatmark.first)
+                    {
+                      // Left is weak.  Right was not weak, but it is
+                      // always rejecting.  We can therefore pretend
+                      // that right is weak.
+                      goto weak_weak;
+                    }
+                  res->copy_acceptance_of(right);
+                  acc_cond::mark_t accmark = rightsatmark.second;
+                  auto& lacc = left->acc();
+                  product_main(left, right, left_state, right_state, res,
+                               [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                               {
+                                 if (!lacc.accepting(ml))
+                                   return mr;
+                                 else
+                                   return accmark;
+                               });
+
+                }
+            }
+          else
+            {
+              assert(!leftweak);
+              if (and_acc)
+                {
+                  auto leftunsatmark = left->acc().unsat_mark();
+                  if (!leftunsatmark.first)
+                    {
+                      // Right is weak.  Left was not weak, but it is
+                      // always accepting.  We can therefore pretend
+                      // that left is weak.
+                      goto weak_weak;
+                    }
+                  res->copy_acceptance_of(left);
+                  acc_cond::mark_t rejmark = leftunsatmark.second;
+                  auto& racc = right->acc();
+                  product_main(left, right, left_state, right_state, res,
+                               [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                               {
+                                 if (racc.accepting(mr))
+                                   return ml;
+                                 else
+                                   return rejmark;
+                               });
+                }
+              else
+                {
+                  auto leftsatmark = left->acc().sat_mark();
+                  if (!leftsatmark.first)
+                    {
+                      // Right is weak.  Left was not weak, but it is
+                      // always rejecting.  We can therefore pretend
+                      // that left is weak.
+                      goto weak_weak;
+                    }
+                  res->copy_acceptance_of(left);
+                  acc_cond::mark_t accmark = leftsatmark.second;
+                  auto& racc = right->acc();
+                  product_main(left, right, left_state, right_state, res,
+                               [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                               {
+                                 if (!racc.accepting(mr))
+                                   return ml;
+                                 else
+                                   return accmark;
+                               });
+
+                }
+            }
+        }
+      else // general case
+        {
+          auto left_num = left->num_sets();
+          auto right_acc = right->get_acceptance() << left_num;
+          if (and_acc)
+            right_acc &= left->get_acceptance();
+          else
+            right_acc |= left->get_acceptance();
+          res->set_acceptance(left_num + right->num_sets(), right_acc);
+
+          product_main(left, right, left_state, right_state, res,
+                       [&] (acc_cond::mark_t ml, acc_cond::mark_t mr)
+                       {
+                         return ml | (mr << left_num);
+                       });
+
+        }
 
       // The product of two non-deterministic automata could be
       // deterministic.  Likewise for non-complete automata.
@@ -147,8 +314,7 @@ namespace spot
                            unsigned left_state,
                            unsigned right_state)
   {
-    return product_aux(complete(left),
-                       complete(right),
+    return product_aux(complete(left), complete(right),
                        left_state, right_state, false);
   }
 
diff --git a/spot/twaalgos/product.hh b/spot/twaalgos/product.hh
index ef8d47374..e07aedb99 100644
--- a/spot/twaalgos/product.hh
+++ b/spot/twaalgos/product.hh
@@ -1,5 +1,5 @@
 // -*- coding: utf-8 -*-
-// Copyright (C) 2014, 2015 Laboratoire de Recherche et
+// Copyright (C) 2014, 2015, 2018 Laboratoire de Recherche et
 // Développement de l'Epita (LRDE).
 //
 // This file is part of Spot, a model checking library.
@@ -36,7 +36,10 @@ namespace spot
   /// The resulting automaton will accept the intersection of both
   /// languages and have an acceptance condition that is the
   /// conjunction of the acceptance conditions of the two input
-  /// automata.
+  /// automata.  In case one of the left or right automaton is weak,
+  /// the acceptance condition of the result is made simpler: it
+  /// usually is the acceptance condition of the other argument,
+  /// therefore avoiding the need to introduce new accepting sets.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
@@ -56,7 +59,10 @@ namespace spot
   /// languages recognized by each input automaton (with its initial
   /// state changed) and have an acceptance condition that is the
   /// conjunction of the acceptance conditions of the two input
-  /// automata.
+  /// automata.  In case one of the left or right automaton is weak,
+  /// the acceptance condition of the result is made simpler: it
+  /// usually is the acceptance condition of the other argument,
+  /// therefore avoiding the need to introduce new accepting sets.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
@@ -74,7 +80,10 @@ namespace spot
   /// The resulting automaton will accept the union of both
   /// languages and have an acceptance condition that is the
   /// disjunction of the acceptance conditions of the two input
-  /// automata.
+  /// automata.  In case one of the left or right automaton is weak,
+  /// the acceptance condition of the result is made simpler: it
+  /// usually is the acceptance condition of the other argument,
+  /// therefore avoiding the need to introduce new accepting sets.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
@@ -94,7 +103,10 @@ namespace spot
   /// recognized by each input automaton (with its initial state
   /// changed) and have an acceptance condition that is the
   /// disjunction of the acceptance conditions of the two input
-  /// automata.
+  /// automata. In case one of the left or right automaton is weak,
+  /// the acceptance condition of the result is made simpler: it
+  /// usually is the acceptance condition of the other argument,
+  /// therefore avoiding the need to introduce new accepting sets.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
diff --git a/tests/Makefile.am b/tests/Makefile.am
index dceda1b95..bba62bffa 100644
--- a/tests/Makefile.am
+++ b/tests/Makefile.am
@@ -383,6 +383,7 @@ TESTS_python = \
   python/parsetgba.py \
   python/parity.py \
   python/prodexpt.py \
+  python/_product_weak.ipynb \
   python/randgen.py \
   python/relabel.py \
   python/remfin.py \
diff --git a/tests/core/prodor.test b/tests/core/prodor.test
index cf40c57fd..c91a48d8d 100755
--- a/tests/core/prodor.test
+++ b/tests/core/prodor.test
@@ -1,6 +1,6 @@
 #!/bin/sh
 # -*- coding: utf-8 -*-
-# Copyright (C) 2015, 2017 Laboratoire de Recherche et Développement
+# Copyright (C) 2015, 2017, 2018 Laboratoire de Recherche et Développement
 # de l'Epita (LRDE).
 #
 # This file is part of Spot, a model checking library.
@@ -106,6 +106,7 @@ diff pand.hoa exp
 
 test 2 = `autfilt -c --intersect pand.hoa gfa.hoa fgb.hoa`
 
+# Xb is weak, so the product will still be Büchi
 ltl2tgba -BDH 'GFa' > gfa.hoa
 ltl2tgba -BDH 'Xb' > xb.hoa
 autfilt --product-or gfa.hoa xb.hoa -H > por.hoa
@@ -116,14 +117,15 @@ HOA: v1
 States: 7
 Start: 0
 AP: 2 "a" "b"
-Acceptance: 2 Inf(0) | Inf(1)
+acc-name: Buchi
+Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc complete
 properties: deterministic
 --BODY--
-State: 0 {1}
+State: 0 {0}
 [0] 1
 [!0] 2
-State: 1 {1}
+State: 1 {0}
 [0&1] 3
 [!0&1] 4
 [0&!1] 5
@@ -133,13 +135,13 @@ State: 2
 [!0&1] 4
 [0&!1] 5
 [!0&!1] 6
-State: 3 {0 1}
+State: 3 {0}
 [0] 3
 [!0] 4
 State: 4 {0}
 [0] 3
 [!0] 4
-State: 5 {1}
+State: 5 {0}
 [0] 5
 [!0] 6
 State: 6
@@ -159,14 +161,15 @@ HOA: v1
 States: 7
 Start: 0
 AP: 2 "a" "b"
-Acceptance: 2 Inf(0) | Inf(1)
+acc-name: Buchi
+Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc complete
 properties: deterministic
 --BODY--
-State: 0 {0 1}
+State: 0 {0}
 [0] 1
 [!0] 2
-State: 1 {0 1}
+State: 1 {0}
 [0&1] 3
 [!0&1] 4
 [0&!1] 5
@@ -176,13 +179,13 @@ State: 2 {0}
 [!0&1] 4
 [0&!1] 5
 [!0&!1] 6
-State: 3 {0 1}
+State: 3 {0}
 [0] 3
 [!0] 4
 State: 4 {0}
 [0] 3
 [!0] 4
-State: 5 {1}
+State: 5 {0}
 [0] 5
 [!0] 6
 State: 6
diff --git a/tests/python/_product_weak.ipynb b/tests/python/_product_weak.ipynb
new file mode 100644
index 000000000..b5a233435
--- /dev/null
+++ b/tests/python/_product_weak.ipynb
@@ -0,0 +1,13400 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Test the optimization of `product()` for weak arguments."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import spot\n",
+    "spot.setup()\n",
+    "from spot.jupyter import display_inline"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "auts = [a for a in spot.automata(\"\"\"\n",
+    "HOA: v1\n",
+    "name: \"a\"\n",
+    "States: 2\n",
+    "Start: 1\n",
+    "AP: 1 \"a\"\n",
+    "acc-name: all\n",
+    "Acceptance: 0 t\n",
+    "properties: trans-labels explicit-labels state-acc deterministic\n",
+    "properties: stutter-invariant terminal\n",
+    "--BODY--\n",
+    "State: 0\n",
+    "[t] 0\n",
+    "State: 1\n",
+    "[0] 0\n",
+    "--END--\n",
+    "HOA: v1\n",
+    "name: \"Fb\"\n",
+    "States: 2\n",
+    "Start: 1\n",
+    "AP: 1 \"b\"\n",
+    "acc-name: Buchi\n",
+    "Acceptance: 1 Inf(0)\n",
+    "properties: trans-labels explicit-labels state-acc complete\n",
+    "properties: deterministic stutter-invariant terminal\n",
+    "--BODY--\n",
+    "State: 0 {0}\n",
+    "[t] 0\n",
+    "State: 1\n",
+    "[0] 0\n",
+    "[!0] 1\n",
+    "--END--\n",
+    "HOA: v1\n",
+    "name: \"Fb\"\n",
+    "States: 2\n",
+    "Start: 1\n",
+    "AP: 1 \"b\"\n",
+    "acc-name: co-Buchi\n",
+    "Acceptance: 1 Fin(0)\n",
+    "properties: trans-labels explicit-labels state-acc complete\n",
+    "properties: deterministic stutter-invariant terminal\n",
+    "--BODY--\n",
+    "State: 0 \n",
+    "[t] 0\n",
+    "State: 1 {0}\n",
+    "[0] 0\n",
+    "[!0] 1\n",
+    "--END--\n",
+    "HOA: v1\n",
+    "name: \"GFc\"\n",
+    "States: 1\n",
+    "Start: 0\n",
+    "AP: 1 \"c\"\n",
+    "acc-name: Buchi\n",
+    "Acceptance: 1 Inf(0)\n",
+    "properties: trans-labels explicit-labels trans-acc complete\n",
+    "properties: deterministic stutter-invariant\n",
+    "--BODY--\n",
+    "State: 0\n",
+    "[!0] 0\n",
+    "[0] 0 {0}\n",
+    "--END--\n",
+    "HOA: v1\n",
+    "States: 1\n",
+    "Start: 0\n",
+    "AP: 1 \"d\"\n",
+    "Acceptance: 1 Inf(0)&Fin(0)\n",
+    "--BODY--\n",
+    "State: 0\n",
+    "[!0] 0 {0}\n",
+    "[0] 0\n",
+    "--END--\n",
+    "HOA: v1\n",
+    "States: 1\n",
+    "Start: 0\n",
+    "AP: 1 \"d\"\n",
+    "Acceptance: 1 Inf(0)|Fin(0)\n",
+    "--BODY--\n",
+    "State: 0 \n",
+    "[!0] 0 {0}\n",
+    "[0] 0\n",
+    "--END--\n",
+    "\"\"\")]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "scrolled": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
+       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
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+       "<title>0&#45;&gt;0</title>\n",
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+       "</svg>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "for left in auts:\n",
+    "    for right in auts:\n",
+    "        display_inline(left, right, spot.product(left, right))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "scrolled": false
+   },
+   "outputs": [
+    {
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+       "</svg>\n",
+       "</div><div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
+       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
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+       "<text text-anchor=\"start\" x=\"209\" y=\"-136.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#000000\">)</text>\n",
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+       "<g id=\"edge1\" class=\"edge\">\n",
+       "<title>I&#45;&gt;0</title>\n",
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+       "</g>\n",
+       "<!-- 0&#45;&gt;0 -->\n",
+       "<g id=\"edge2\" class=\"edge\">\n",
+       "<title>0&#45;&gt;0</title>\n",
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+       "</g>\n",
+       "</svg>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "for left in auts:\n",
+    "    for right in auts:\n",
+    "        display_inline(left, right, spot.product_or(left, right))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/tests/python/automata.ipynb b/tests/python/automata.ipynb
index d1e14083a..327b764f0 100644
--- a/tests/python/automata.ipynb
+++ b/tests/python/automata.ipynb
@@ -8,6 +8,7 @@
    "source": [
     "from IPython.display import display\n",
     "import spot\n",
+    "from spot.jupyter import display_inline\n",
     "spot.setup()"
    ]
   },
@@ -163,7 +164,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe2c0c80f0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec60d690> >"
       ]
      },
      "execution_count": 2,
@@ -600,7 +601,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efccd80> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec57bc90> >"
       ]
      },
      "execution_count": 6,
@@ -676,7 +677,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efdc180> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec58a9f0> >"
       ]
      },
      "execution_count": 7,
@@ -759,7 +760,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efdc150> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec58ab10> >"
       ]
      },
      "execution_count": 8,
@@ -1278,7 +1279,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efee7e0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59c840> >"
       ]
      },
      "execution_count": 12,
@@ -1392,7 +1393,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efee900> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59ca20> >"
       ]
      },
      "execution_count": 13,
@@ -1523,7 +1524,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efee7b0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59cab0> >"
       ]
      },
      "execution_count": 14,
@@ -1635,7 +1636,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efee810> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59cb10> >"
       ]
      },
      "execution_count": 15,
@@ -2180,7 +2181,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efcc9c0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59c210> >"
       ]
      },
      "execution_count": 16,
@@ -2391,7 +2392,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efee2a0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59cbd0> >"
       ]
      },
      "metadata": {},
@@ -2541,7 +2542,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efcce40> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec57b9c0> >"
       ]
      },
      "metadata": {},
@@ -2713,7 +2714,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efeec90> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59ce40> >"
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+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec59cc90> >"
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@@ -3074,7 +3075,7 @@
        "</svg>\n"
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      "execution_count": 21,
@@ -3150,7 +3151,7 @@
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       ]
      },
      "execution_count": 22,
@@ -3169,123 +3170,151 @@
    "outputs": [
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+      "text/plain": [
+       "True"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "spot.formula('(a W c) & FGa').is_syntactic_persistence()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "no\n"
+     ]
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+    {
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+       "<title>2-&gt;3</title>\n",
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+       "<title>3-&gt;1</title>\n",
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        "<!-- 3&#45;&gt;3 -->\n",
-       "<g class=\"edge\" id=\"edge10\">\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"431\" y=\"-142.8638\">a &amp; !c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"441\" y=\"-127.8638\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge15\">\n",
+       "<title>3-&gt;3</title>\n",
+       "<path d=\"M440.1904,-105.3128C433.9385,-126.4983 436.875,-154.0638 449,-154.0638 459.8462,-154.0638 463.3399,-132.0063 459.4811,-112.1891\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"433\" y=\"-171.8638\">a &amp; c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"433\" y=\"-157.8638\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"449\" y=\"-157.8638\">❶</text>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"245.5\" y=\"-256.2783\">)</text>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"158.5\" y=\"-242.2783\">[gen. Büchi 2]</text>\n",
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+       "<ellipse cx=\"165.5\" cy=\"-28.4783\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"#000000\"/>\n",
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+       "<title>1-&gt;1</title>\n",
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+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"267\" y=\"-162.2783\">⓿</text>\n",
        "</g>\n",
        "<!-- 1&#45;&gt;1 -->\n",
        "<g class=\"edge\" id=\"edge6\">\n",
        "<title>1-&gt;1</title>\n",
-       "<path d=\"M285.0212,-133.916C282.679,-144.1504 285.6719,-154 294,-154 300.3762,-154 303.625,-148.2263 303.7465,-140.9268\" fill=\"none\" stroke=\"#000000\"/>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"289.5\" y=\"-171.8\">1</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"278\" y=\"-157.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"294\" y=\"-157.8\">❶</text>\n",
+       "<path d=\"M267.7325,-139.0742C262.2416,-160.3416 264.6641,-188.4783 275,-188.4783 284.2458,-188.4783 287.1594,-165.9637 283.7408,-145.9915\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"282.2675,-139.0742 286.8066,-145.2644 282.9966,-142.4974 283.7257,-145.9206 283.7257,-145.9206 283.7257,-145.9206 282.9966,-142.4974 280.6448,-146.5768 282.2675,-139.0742 282.2675,-139.0742\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"257\" y=\"-206.2783\">!a &amp; c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"259\" y=\"-192.2783\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"275\" y=\"-192.2783\">❶</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge7\">\n",
+       "<title>1-&gt;3</title>\n",
+       "<path d=\"M279.9602,-139.9118C285.1203,-154.443 294.7683,-174.21 311,-183.4783 326.4383,-192.2935 336.388,-193.6044 351,-183.4783 363.0357,-175.1376 374.471,-137.8639 381.1061,-112.293\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"382.8468,-105.4248 384.1805,-112.9841 381.9869,-108.8175 381.127,-112.2102 381.127,-112.2102 381.127,-112.2102 381.9869,-108.8175 378.0736,-111.4363 382.8468,-105.4248 382.8468,-105.4248\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"313\" y=\"-194.2783\">a &amp; !c</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge8\">\n",
+       "<title>1-&gt;3</title>\n",
+       "<path d=\"M291.8393,-129.0964C307.8181,-134.2861 332.1267,-139.2918 351,-130.4783 360.4199,-126.0794 368.2449,-117.8688 374.1666,-109.7918\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"378.3188,-103.7066 376.9753,-111.2642 376.346,-106.5977 374.3733,-109.4887 374.3733,-109.4887 374.3733,-109.4887 376.346,-106.5977 371.7713,-107.7133 378.3188,-103.7066 378.3188,-103.7066\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"315\" y=\"-153.2783\">a &amp; c</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"323\" y=\"-138.2783\">❶</text>\n",
        "</g>\n",
        "<!-- 2&#45;&gt;1 -->\n",
-       "<g class=\"edge\" id=\"edge7\">\n",
+       "<g class=\"edge\" id=\"edge9\">\n",
        "<title>2-&gt;1</title>\n",
-       "<path d=\"M208.1154,-118C224.8642,-118 249.92,-118 268.6521,-118\" fill=\"none\" stroke=\"#000000\"/>\n",
-       "<polygon fill=\"#000000\" points=\"275.7432,-118 268.7433,-121.1501 272.2432,-118 268.7432,-118.0001 268.7432,-118.0001 268.7432,-118.0001 272.2432,-118 268.7432,-114.8501 275.7432,-118 275.7432,-118\" stroke=\"#000000\"/>\n",
-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"237.5\" y=\"-136.8\">b</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"234\" y=\"-121.8\">⓿</text>\n",
+       "<path d=\"M179.3683,-40.3835C198.7033,-56.9816 233.8422,-87.1465 255.7182,-105.9259\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"261.2444,-110.6698 253.8812,-108.5004 258.5887,-108.39 255.933,-106.1102 255.933,-106.1102 255.933,-106.1102 258.5887,-108.39 257.9848,-103.7201 261.2444,-110.6698 261.2444,-110.6698\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"202\" y=\"-92.2783\">!a &amp; b</text>\n",
        "</g>\n",
        "<!-- 2&#45;&gt;2 -->\n",
-       "<g class=\"edge\" id=\"edge8\">\n",
+       "<g class=\"edge\" id=\"edge10\">\n",
        "<title>2-&gt;2</title>\n",
-       "<path d=\"M181.0212,-133.916C178.679,-144.1504 181.6719,-154 190,-154 196.3762,-154 199.625,-148.2263 199.7465,-140.9268\" fill=\"none\" stroke=\"#000000\"/>\n",
-       "<polygon fill=\"#000000\" points=\"198.9788,-133.916 202.8721,-140.5315 199.3598,-137.3952 199.7408,-140.8744 199.7408,-140.8744 199.7408,-140.8744 199.3598,-137.3952 196.6095,-141.2174 198.9788,-133.916 198.9788,-133.916\" stroke=\"#000000\"/>\n",
-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"171.5\" y=\"-172.8\">a &amp; !b</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"182\" y=\"-157.8\">⓿</text>\n",
+       "<path d=\"M156.2674,-44.0198C153.6685,-54.387 156.7461,-64.4783 165.5,-64.4783 172.2022,-64.4783 175.577,-58.563 175.6245,-51.1374\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"174.7326,-44.0198 178.7286,-50.5738 175.1678,-47.4926 175.603,-50.9654 175.603,-50.9654 175.603,-50.9654 175.1678,-47.4926 172.4775,-51.3571 174.7326,-44.0198 174.7326,-44.0198\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"147\" y=\"-83.2783\">a &amp; !b</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"157.5\" y=\"-68.2783\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 2&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge11\">\n",
+       "<title>2-&gt;3</title>\n",
+       "<path d=\"M183.594,-27.7587C217.9351,-27.0979 294.2023,-29.0444 351,-55.4783 357.8232,-58.6539 364.2983,-63.5517 369.8268,-68.5683\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"375.0175,-73.5652 367.7898,-70.9798 372.496,-71.1378 369.9744,-68.7105 369.9744,-68.7105 369.9744,-68.7105 372.496,-71.1378 372.1591,-66.4411 375.0175,-73.5652 375.0175,-73.5652\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"258\" y=\"-56.2783\">a &amp; b</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"267\" y=\"-41.2783\">⓿</text>\n",
        "</g>\n",
        "<!-- 3&#45;&gt;1 -->\n",
-       "<g class=\"edge\" id=\"edge9\">\n",
+       "<g class=\"edge\" id=\"edge12\">\n",
        "<title>3-&gt;1</title>\n",
-       "<path d=\"M203.1717,-30.6651C221.5355,-48.3226 254.9095,-80.413 275.6867,-100.3911\" fill=\"none\" stroke=\"#000000\"/>\n",
-       "<polygon fill=\"#000000\" points=\"280.9353,-105.4378 273.7061,-102.8566 278.4123,-103.0119 275.8894,-100.586 275.8894,-100.586 275.8894,-100.586 278.4123,-103.0119 278.0727,-98.3153 280.9353,-105.4378 280.9353,-105.4378\" stroke=\"#000000\"/>\n",
-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"238.5\" y=\"-99.8\">c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"226\" y=\"-85.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"242\" y=\"-85.8\">❶</text>\n",
+       "<path d=\"M369.6229,-92.9087C350.7682,-98.8008 320.5243,-108.252 299.3164,-114.8795\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"292.5,-117.0096 298.2418,-111.915 295.8407,-115.9656 299.1814,-114.9216 299.1814,-114.9216 299.1814,-114.9216 295.8407,-115.9656 300.121,-117.9282 292.5,-117.0096 292.5,-117.0096\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"311\" y=\"-114.2783\">!a &amp; !c</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;1 -->\n",
+       "<g class=\"edge\" id=\"edge13\">\n",
+       "<title>3-&gt;1</title>\n",
+       "<path d=\"M372.8693,-75.8011C366.5789,-71.3346 358.8309,-66.7663 351,-64.4783 333.9357,-59.4925 326.5358,-55.8361 311,-64.4783 297.9795,-71.7213 289.0654,-85.8382 283.3646,-98.3307\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"280.5224,-105.05 280.3484,-97.3758 281.886,-101.8265 283.2495,-98.603 283.2495,-98.603 283.2495,-98.603 281.886,-101.8265 286.1506,-99.8302 280.5224,-105.05 280.5224,-105.05\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"313\" y=\"-83.2783\">!a &amp; c</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"323\" y=\"-68.2783\">❶</text>\n",
        "</g>\n",
        "<!-- 3&#45;&gt;3 -->\n",
-       "<g class=\"edge\" id=\"edge10\">\n",
+       "<g class=\"edge\" id=\"edge14\">\n",
        "<title>3-&gt;3</title>\n",
-       "<path d=\"M181.0212,-33.916C178.679,-44.1504 181.6719,-54 190,-54 196.3762,-54 199.625,-48.2263 199.7465,-40.9268\" fill=\"none\" stroke=\"#000000\"/>\n",
-       "<polygon fill=\"#000000\" points=\"198.9788,-33.916 202.8721,-40.5315 199.3598,-37.3952 199.7408,-40.8744 199.7408,-40.8744 199.7408,-40.8744 199.3598,-37.3952 196.6095,-41.2174 198.9788,-33.916 198.9788,-33.916\" stroke=\"#000000\"/>\n",
-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"172\" y=\"-71.8\">a &amp; !c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"174\" y=\"-57.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"190\" y=\"-57.8\">❶</text>\n",
+       "<path d=\"M382.0012,-104.8882C381.0681,-114.5662 382.7344,-123.4783 387,-123.4783 390.1326,-123.4783 391.8633,-118.6719 392.1921,-112.2856\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"391.9988,-104.8882 395.3307,-111.8035 392.0903,-108.387 392.1818,-111.8858 392.1818,-111.8858 392.1818,-111.8858 392.0903,-108.387 389.0329,-111.9682 391.9988,-104.8882 391.9988,-104.8882\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"369\" y=\"-142.2783\">a &amp; !c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"379\" y=\"-127.2783\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge15\">\n",
+       "<title>3-&gt;3</title>\n",
+       "<path d=\"M379.4634,-104.0742C373.769,-125.3416 376.2813,-153.4783 387,-153.4783 396.5883,-153.4783 399.6098,-130.9637 396.0645,-110.9915\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"394.5366,-104.0742 399.1223,-110.23 395.2915,-107.4918 396.0464,-110.9094 396.0464,-110.9094 396.0464,-110.9094 395.2915,-107.4918 392.9706,-111.5889 394.5366,-104.0742 394.5366,-104.0742\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"371\" y=\"-171.2783\">a &amp; c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"371\" y=\"-157.2783\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"387\" y=\"-157.2783\">❶</text>\n",
        "</g>\n",
        "</g>\n",
-       "</svg>"
+       "</svg></div>"
       ],
       "text/plain": [
-       "<IPython.core.display.SVG object>"
+       "<IPython.core.display.HTML object>"
       ]
      },
      "metadata": {},
@@ -3563,10 +3648,13 @@
    "source": [
     "# Using +1 in the display options is a convient way to shift the \n",
     "# set numbers in the output, as an aid in reading the product.\n",
-    "a1 = spot.translate('a W c'); display(a1.show('.t'))\n",
-    "a2 = spot.translate('a U b'); display(a2.show('.t+1'))\n",
-    "# the product should display pairs of states, unless asked not to (using 1).\n",
-    "p = spot.product(a1, a2); display(p.show('.t')); display(p.show('.t1'))"
+    "a1 = spot.translate('GF(a <-> Xa)')\n",
+    "print(a1.prop_weak())\n",
+    "a2 = spot.translate('a U b & GFc')\n",
+    "display_inline(a1.show('.t'), a2.show('.t+1'))\n",
+    "# the product should display pairs of states, unless asked not to (using '1').\n",
+    "p = spot.product(a1, a2)\n",
+    "display_inline(p.show('.t'), p.show('.t1'), per_row=2)"
    ]
   },
   {
@@ -3578,7 +3666,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [
     {
@@ -3643,7 +3731,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1effb4b0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec52abd0> >"
       ]
      },
      "metadata": {},
@@ -3667,7 +3755,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 26,
    "metadata": {},
    "outputs": [
     {
@@ -3743,10 +3831,10 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efeec90> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec52ae40> >"
       ]
      },
-     "execution_count": 25,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3764,7 +3852,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 27,
    "metadata": {},
    "outputs": [
     {
@@ -3816,10 +3904,10 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1efee030> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec52ab10> >"
       ]
      },
-     "execution_count": 26,
+     "execution_count": 27,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3837,7 +3925,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 28,
    "metadata": {},
    "outputs": [
     {
@@ -3905,7 +3993,7 @@
        "<IPython.core.display.SVG object>"
       ]
      },
-     "execution_count": 27,
+     "execution_count": 28,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3923,7 +4011,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 29,
    "metadata": {},
    "outputs": [
     {
@@ -3980,10 +4068,10 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1effbe10> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec53da50> >"
       ]
      },
-     "execution_count": 28,
+     "execution_count": 29,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4001,7 +4089,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 30,
    "metadata": {},
    "outputs": [
     {
@@ -4066,7 +4154,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1effb4b0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec52abd0> >"
       ]
      },
      "metadata": {},
@@ -4134,7 +4222,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1effb4b0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec52abd0> >"
       ]
      },
      "metadata": {},
@@ -4159,7 +4247,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 31,
    "metadata": {},
    "outputs": [
     {
@@ -4224,10 +4312,10 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fbe1effb4b0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f16ec52abd0> >"
       ]
      },
-     "execution_count": 30,
+     "execution_count": 31,
      "metadata": {},
      "output_type": "execute_result"
     }
diff --git a/tests/python/highlighting.ipynb b/tests/python/highlighting.ipynb
index 69b332f70..370aab3c6 100644
--- a/tests/python/highlighting.ipynb
+++ b/tests/python/highlighting.ipynb
@@ -240,7 +240,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f65e4c3f0c0> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0128edca50> >"
       ]
      },
      "execution_count": 4,
@@ -352,7 +352,7 @@
        "</svg>\n"
       ],
       "text/plain": [
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@@ -1711,7 +1697,7 @@
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@@ -1808,7 +1794,7 @@
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@@ -1976,7 +1962,7 @@
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@@ -2144,7 +2130,7 @@
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@@ -2307,7 +2293,7 @@
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-       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f65e4b7cb70> >"
+       "<spot.impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0128e6dc30> >"
       ]
      },
      "metadata": {},
@@ -2771,6 +2757,13 @@
     "spot.highlight_languages(aut)\n",
     "aut.show('.bas')"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
diff --git a/tests/python/product.ipynb b/tests/python/product.ipynb
index cfdd6e714..afe02367e 100644
--- a/tests/python/product.ipynb
+++ b/tests/python/product.ipynb
@@ -39,68 +39,53 @@
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+       "<!-- 3&#45;&gt;2 -->\n",
+       "<g id=\"edge19\" class=\"edge\">\n",
+       "<title>3&#45;&gt;2</title>\n",
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+       "<!-- 3&#45;&gt;2 -->\n",
+       "<g id=\"edge20\" class=\"edge\">\n",
+       "<title>3&#45;&gt;2</title>\n",
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+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
+       "<g id=\"edge21\" class=\"edge\">\n",
+       "<title>3&#45;&gt;3</title>\n",
+       "<path fill=\"none\" stroke=\"#000000\" d=\"M218.73,-124.6692C228.9166,-124.8767 237.3834,-123.3203 237.3834,-120 237.3834,-117.5098 232.6209,-116.0117 225.9421,-115.5059\"/>\n",
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+       "<text text-anchor=\"start\" x=\"237.3834\" y=\"-123.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#000000\">a &amp; !b &amp; !c</text>\n",
+       "<text text-anchor=\"start\" x=\"262.3834\" y=\"-108.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
+       "<g id=\"edge22\" class=\"edge\">\n",
+       "<title>3&#45;&gt;3</title>\n",
+       "<path fill=\"none\" stroke=\"#000000\" d=\"M217.5241,-126.8825C251.6983,-131.7668 303.3834,-129.4727 303.3834,-120 303.3834,-111.1564 258.3348,-108.5695 224.5437,-112.2394\"/>\n",
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+       "<text text-anchor=\"start\" x=\"303.3834\" y=\"-123.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#000000\">a &amp; b &amp; !c</text>\n",
+       "<text text-anchor=\"start\" x=\"318.3834\" y=\"-109.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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        "</svg>\n",
@@ -416,7 +420,7 @@
     }
    ],
    "source": [
-    "a1 = spot.translate('X(a W b)')\n",
+    "a1 = spot.translate('FGa')\n",
     "a2 = spot.translate('G(Fc U b)')\n",
     "prod = spot.product(a1, a2)\n",
     "display_inline(a1, a2, prod)"
@@ -447,68 +451,53 @@
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+       "<g id=\"edge1\" class=\"edge\">\n",
+       "<title>I&#45;&gt;0</title>\n",
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-       "<g id=\"node4\" class=\"node\">\n",
-       "<title>2</title>\n",
+       "<!-- 1 -->\n",
+       "<g id=\"node3\" class=\"node\">\n",
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-       "<!-- 0&#45;&gt;2 -->\n",
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+       "<g id=\"edge4\" class=\"edge\">\n",
+       "<title>1&#45;&gt;1</title>\n",
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@@ -604,204 +593,223 @@
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+       "<text text-anchor=\"start\" x=\"318.3834\" y=\"-109.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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@@ -2215,7 +2238,7 @@
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      "output_type": "stream",
      "text": [
-      "yes\n"
+      "maybe\n"
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+       "<!-- 1&#45;&gt;0 -->\n",
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+       "<title>1-&gt;0</title>\n",
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+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"104.5\" y=\"-183.8\">❶</text>\n",
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+       "<!-- 1&#45;&gt;0 -->\n",
+       "<g class=\"edge\" id=\"edge9\">\n",
+       "<title>1-&gt;0</title>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"95.5\" y=\"-152.8\">b &amp; c</text>\n",
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+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"112.5\" y=\"-138.8\">❷</text>\n",
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+       "<!-- 1&#45;&gt;1 -->\n",
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+       "<title>1-&gt;1</title>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"163.5\" y=\"-200.8\">!b &amp; !c</text>\n",
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+       "<!-- 1&#45;&gt;1 -->\n",
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+       "<title>1-&gt;1</title>\n",
+       "<path d=\"M174.3645,-176.4784C167.5265,-194.3259 170.7383,-215 184,-215 195.604,-215 199.5136,-199.1714 195.7287,-183.2578\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"165.5\" y=\"-233.8\">b &amp; !c</text>\n",
+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"176\" y=\"-218.8\">❷</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;2 -->\n",
+       "<g class=\"edge\" id=\"edge12\">\n",
+       "<title>1-&gt;2</title>\n",
+       "<path d=\"M200.7923,-167.9597C210.6397,-171.8629 223.3897,-176.6129 235,-180 297.7638,-198.3104 315.9837,-204.8879 381,-198 418.9684,-193.9776 428.4243,-190.9542 465,-180 469.1154,-178.7675 473.4255,-177.2751 477.5751,-175.7301\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"484.3444,-173.1169 478.9486,-178.5765 481.0793,-174.3774 477.8141,-175.6379 477.8141,-175.6379 477.8141,-175.6379 481.0793,-174.3774 476.6797,-172.6993 484.3444,-173.1169 484.3444,-173.1169\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"319\" y=\"-218.8\">a &amp; !b &amp; c</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"342\" y=\"-203.8\">❶</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;2 -->\n",
+       "<g class=\"edge\" id=\"edge13\">\n",
+       "<title>1-&gt;2</title>\n",
+       "<path d=\"M202.4134,-161.2904C256.3791,-162.1416 414.2289,-164.6314 475.9001,-165.6041\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"482.9307,-165.715 475.8819,-168.7541 479.4312,-165.6597 475.9316,-165.6045 475.9316,-165.6045 475.9316,-165.6045 479.4312,-165.6597 475.9813,-162.4549 482.9307,-165.715 482.9307,-165.715\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"321\" y=\"-181.8\">a &amp; b &amp; c</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"334\" y=\"-167.8\">❶</text>\n",
+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"350\" y=\"-167.8\">❷</text>\n",
+       "</g>\n",
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-       "<g class=\"edge\" id=\"edge7\">\n",
+       "<g class=\"edge\" id=\"edge14\">\n",
        "<title>1-&gt;3</title>\n",
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-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"344\" y=\"-212.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"360\" y=\"-212.8\">❶</text>\n",
-       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"376\" y=\"-212.8\">❷</text>\n",
+       "<path d=\"M200.4219,-153.296C232.1597,-138.3857 298.9958,-106.8751 301,-105 319.8277,-87.385 333.318,-60.8712 341.3175,-41.745\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"235\" y=\"-138.8\">a &amp; !b &amp; !c</text>\n",
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-       "<!-- 2&#45;&gt;1 -->\n",
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-       "<title>2-&gt;1</title>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"208\" y=\"-140.8\">a &amp; !b &amp; c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"223\" y=\"-126.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"239\" y=\"-126.8\">❶</text>\n",
+       "<!-- 1&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge15\">\n",
+       "<title>1-&gt;3</title>\n",
+       "<path d=\"M189.9265,-144.0001C197.5893,-124.1867 212.6594,-91.545 235,-71 249.0067,-58.1191 296.661,-38.4218 326.0487,-27.018\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"237\" y=\"-89.8\">a &amp; b &amp; !c</text>\n",
+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"260\" y=\"-74.8\">❷</text>\n",
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        "<!-- 2&#45;&gt;2 -->\n",
-       "<g class=\"edge\" id=\"edge9\">\n",
+       "<g class=\"edge\" id=\"edge16\">\n",
        "<title>2-&gt;2</title>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"275\" y=\"-180.8\">a &amp; !b &amp; !c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"300\" y=\"-165.8\">⓿</text>\n",
+       "<path d=\"M494.9752,-183.0373C493.7456,-192.8579 495.7539,-202 501,-202 504.9346,-202 507.0479,-196.8576 507.3399,-190.1433\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"470\" y=\"-219.8\">a &amp; !b &amp; c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"485\" y=\"-205.8\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"501\" y=\"-205.8\">❶</text>\n",
+       "</g>\n",
+       "<!-- 2&#45;&gt;2 -->\n",
+       "<g class=\"edge\" id=\"edge17\">\n",
+       "<title>2-&gt;2</title>\n",
+       "<path d=\"M492.0632,-181.6856C484.8726,-202.4825 487.8516,-230 501,-230 512.7617,-230 516.3859,-207.9809 511.8726,-188.45\" fill=\"none\" stroke=\"#000000\"/>\n",
+       "<polygon fill=\"#000000\" points=\"509.9368,-181.6856 514.8912,-187.5487 510.8998,-185.0505 511.8628,-188.4154 511.8628,-188.4154 511.8628,-188.4154 510.8998,-185.0505 508.8343,-189.2821 509.9368,-181.6856 509.9368,-181.6856\" stroke=\"#000000\"/>\n",
+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"484\" y=\"-247.8\">a &amp; b</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"477\" y=\"-233.8\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"493\" y=\"-233.8\">❶</text>\n",
+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"509\" y=\"-233.8\">❷</text>\n",
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        "<!-- 2&#45;&gt;3 -->\n",
-       "<g class=\"edge\" id=\"edge10\">\n",
-       "<title>2-&gt;3</title>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"417\" y=\"-177.8\">b &amp; c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"410\" y=\"-163.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"426\" y=\"-163.8\">❶</text>\n",
-       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"442\" y=\"-163.8\">❷</text>\n",
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-       "<!-- 4 -->\n",
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-       "<title>4</title>\n",
-       "<g id=\"a_node6\"><a xlink:title=\"2,1\">\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"434\" y=\"-14.3\">4</text>\n",
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-       "<!-- 2&#45;&gt;4 -->\n",
-       "<g class=\"edge\" id=\"edge11\">\n",
-       "<title>2-&gt;4</title>\n",
-       "<path d=\"M321.9656,-114.0295C344.4825,-94.7293 388.8694,-56.6833 414.4967,-34.7171\" fill=\"none\" stroke=\"#000000\"/>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"349.5\" y=\"-111.8\">b &amp; !c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"352\" y=\"-97.8\">⓿</text>\n",
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-       "<!-- 3&#45;&gt;3 -->\n",
-       "<g class=\"edge\" id=\"edge12\">\n",
-       "<title>3-&gt;3</title>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"541.5\" y=\"-175.8\">!b &amp; c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"544\" y=\"-161.8\">⓿</text>\n",
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-       "<title>3-&gt;3</title>\n",
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-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"552\" y=\"-189.8\">❶</text>\n",
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-       "<g class=\"edge\" id=\"edge14\">\n",
-       "<title>3-&gt;4</title>\n",
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-       "<!-- 4&#45;&gt;3 -->\n",
-       "<g class=\"edge\" id=\"edge15\">\n",
-       "<title>4-&gt;3</title>\n",
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-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"484\" y=\"-122.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"500\" y=\"-122.8\">❶</text>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"483\" y=\"-85.8\">b &amp; c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"476\" y=\"-71.8\">⓿</text>\n",
-       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"492\" y=\"-71.8\">❶</text>\n",
-       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"508\" y=\"-71.8\">❷</text>\n",
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-       "<!-- 4&#45;&gt;4 -->\n",
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-       "<title>4-&gt;4</title>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"413.5\" y=\"-72.8\">!b &amp; !c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"426\" y=\"-57.8\">⓿</text>\n",
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-       "<title>4-&gt;4</title>\n",
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-       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"415.5\" y=\"-101.8\">b &amp; !c</text>\n",
-       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"418\" y=\"-87.8\">⓿</text>\n",
-       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"434\" y=\"-87.8\">❷</text>\n",
+       "<title>2-&gt;3</title>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"399\" y=\"-89.8\">a &amp; !b &amp; !c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"424\" y=\"-74.8\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;2 -->\n",
+       "<g class=\"edge\" id=\"edge19\">\n",
+       "<title>3-&gt;2</title>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"401\" y=\"-33.8\">a &amp; !b &amp; c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"416\" y=\"-19.8\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"432\" y=\"-19.8\">❶</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;2 -->\n",
+       "<g class=\"edge\" id=\"edge20\">\n",
+       "<title>3-&gt;2</title>\n",
+       "<path d=\"M355.9434,-35.035C363.2891,-53.7228 377.5133,-83.3956 399,-101 423.0284,-120.6869 439.3403,-106.4924 465,-124 473.2182,-129.6072 480.6802,-137.554 486.6524,-145.0286\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"403\" y=\"-141.8\">a &amp; b &amp; c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"408\" y=\"-127.8\">⓿</text>\n",
+       "<text fill=\"#ff4da0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"424\" y=\"-127.8\">❶</text>\n",
+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"440\" y=\"-127.8\">❷</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge21\">\n",
+       "<title>3-&gt;3</title>\n",
+       "<path d=\"M342.793,-34.6641C341.1914,-44.625 343.5938,-54 350,-54 354.8047,-54 357.3572,-48.7266 357.6575,-41.8876\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"317\" y=\"-72.8\">a &amp; !b &amp; !c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"342\" y=\"-57.8\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
+       "<g class=\"edge\" id=\"edge22\">\n",
+       "<title>3-&gt;3</title>\n",
+       "<path d=\"M339.6258,-32.9823C330.4462,-54.4094 333.9043,-84 350,-84 364.461,-84 368.721,-60.1148 362.78,-39.7239\" fill=\"none\" stroke=\"#000000\"/>\n",
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+       "<text fill=\"#000000\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"319\" y=\"-101.8\">a &amp; b &amp; !c</text>\n",
+       "<text fill=\"#1f78b4\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"334\" y=\"-87.8\">⓿</text>\n",
+       "<text fill=\"#ff7f00\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"350\" y=\"-87.8\">❷</text>\n",
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@@ -2518,11 +2557,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0: (1, 0)\n",
-      "1: (0, 0)\n",
-      "2: (0, 1)\n",
-      "3: (2, 0)\n",
-      "4: (2, 1)\n"
+      "0: (0, 0)\n",
+      "1: (0, 1)\n",
+      "2: (1, 0)\n",
+      "3: (1, 1)\n"
      ]
     }
    ],
@@ -2551,7 +2589,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "409 µs ± 6.68 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
+      "264 µs ± 2.11 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
      ]
     }
    ],
@@ -2568,7 +2606,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "7.02 µs ± 98.5 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
+      "4.9 µs ± 45.2 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
      ]
     }
    ],
@@ -2584,6 +2622,22 @@
     "\n",
     "Despite that speed difference, Python can be useful to prototype an algorithm before implementing it in C++."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# The case of weak automata\n",
+    "\n",
+    "Finally, note that `spot.product()` actually does a bit more: it has some specializations for when one of the argument of the product is marked as weak (as indicated by `prop_weak()`).  In this case, the resulting acceptance condition can be simplified a bit."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {