From 0f131f2eee8e71113c6298016d94aba1e540f21b Mon Sep 17 00:00:00 2001
From: Alexandre Duret-Lutz <adl@lrde.epita.fr>
Date: Tue, 6 Sep 2022 18:05:52 +0200
Subject: [PATCH] =?UTF-8?q?product:=20B=C3=BCchi|B=C3=BCchi=3DB=C3=BCchi,?=
 =?UTF-8?q?=20CoB=C3=BCchi&CoB=C3=BCchi=3DCoB=C3=BCchi?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Improve the construction of the above constructions, saving colors.

* spot/twaalgos/product.cc: Here.
* spot/twaalgos/product.hh, NEWS: Mention it.
* tests/core/prodchain.test, tests/core/prodor.test,
tests/python/_product_weak.ipynb: Adjust.
---
 NEWS                             |    6 +-
 spot/twaalgos/product.cc         |   22 +-
 spot/twaalgos/product.hh         |   43 +-
 tests/core/prodchain.test        |   42 +-
 tests/core/prodor.test           |    8 +-
 tests/python/_product_weak.ipynb | 8468 ++++++++++++++++++++++--------
 6 files changed, 6449 insertions(+), 2140 deletions(-)

diff --git a/NEWS b/NEWS
index 5b55ebbdd..66f56e75d 100644
--- a/NEWS
+++ b/NEWS
@@ -114,10 +114,14 @@ New in spot 2.10.6.dev  (not yet released)
     to obtain a simple model checker (that returns true or false,
     without counterexample).
 
+  - product() learned that the product of two co-Büchi automata
+    is a co-Büchi automaton.  And product_or() learned that the
+    "or"-product of two Büchi automata is a Büchi automaton.
+
   - spot::parallel_policy is an object that can be passed to some
     algorithm to specify how many threads can be used if Spot has been
     compiled with --enable-pthread.  Currently, only
-    twa_graph::merge_states() support it.
+    twa_graph::merge_states() supports it.
 
  Python bindings:
 
diff --git a/spot/twaalgos/product.cc b/spot/twaalgos/product.cc
index 7fb70ddd6..243f3768c 100644
--- a/spot/twaalgos/product.cc
+++ b/spot/twaalgos/product.cc
@@ -1,5 +1,5 @@
 // -*- coding: utf-8 -*-
-// Copyright (C) 2014-2020 Laboratoire de Recherche et Développement
+// Copyright (C) 2014-2020, 2022 Laboratoire de Recherche et Développement
 // de l'Epita (LRDE).
 //
 // This file is part of Spot, a model checking library.
@@ -122,8 +122,23 @@ namespace spot
       res->copy_ap_of(left);
       res->copy_ap_of(right);
 
+      auto& lacc = left->acc();
+      auto& racc = right->acc();
+
       bool leftweak = left->prop_weak().is_true();
       bool rightweak = right->prop_weak().is_true();
+
+      // The conjunction of two co-Büchi automata is a co-Büchi automaton.
+      // The disjunction of two Büchi automata is a Büchi automaton.
+      //
+      // The code to handle this case is similar to the weak_weak case,
+      // except we do not set the weak property on the result.
+      if (!leftweak
+          && !rightweak
+          && ((aop == and_acc && lacc.is_co_buchi() && racc.is_co_buchi())
+              || (aop == or_acc && lacc.is_buchi() && racc.is_buchi())))
+        goto and_cobuchi_or_buchi;
+
       // We have optimization to the standard product in case one
       // of the arguments is weak.
       if (leftweak || rightweak)
@@ -132,14 +147,13 @@ namespace spot
           // t, f, Büchi or co-Büchi.  We use co-Büchi only when
           // t and f cannot be used, and both acceptance conditions
           // are in {t,f,co-Büchi}.
-          if (leftweak && rightweak)
+          if ((leftweak && rightweak))
             {
             weak_weak:
               res->prop_weak(true);
+            and_cobuchi_or_buchi:
               acc_cond::mark_t accmark = {0};
               acc_cond::mark_t rejmark = {};
-              auto& lacc = left->acc();
-              auto& racc = right->acc();
               if ((lacc.is_co_buchi() && (racc.is_co_buchi()
                                          || racc.num_sets() == 0))
                   || (lacc.num_sets() == 0 && racc.is_co_buchi()))
diff --git a/spot/twaalgos/product.hh b/spot/twaalgos/product.hh
index 49ee9acdf..784a3cb49 100644
--- a/spot/twaalgos/product.hh
+++ b/spot/twaalgos/product.hh
@@ -1,5 +1,5 @@
 // -*- coding: utf-8 -*-
-// Copyright (C) 2014-2015, 2018-2020 Laboratoire de Recherche et
+// Copyright (C) 2014-2015, 2018-2020, 2022 Laboratoire de Recherche et
 // Développement de l'Epita (LRDE).
 //
 // This file is part of Spot, a model checking library.
@@ -37,10 +37,14 @@ 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.  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,
+  /// automata.
+  ///
+  /// As an optionmization, 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.
+  /// Similarly, the product of two co-Büchi automata will be a
+  /// co-Büchi automaton.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
@@ -64,10 +68,14 @@ 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.  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,
+  /// automata.
+  ///
+  /// As an optionmization, 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.
+  /// Similarly, the product of two co-Büchi automata will be a
+  /// co-Büchi automaton.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
@@ -89,10 +97,15 @@ 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.  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,
+  /// automata.
+  ///
+  /// As an optionmization, 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.
+  /// Similarly, the product_pr of two Büchi automata will be a
+  /// Büchi automaton.
+  ///
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
@@ -112,10 +125,14 @@ 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. 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,
+  /// automata.
+  ///
+  /// As an optionmization, 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.
+  /// Similarly, the product_pr of two Büchi automata will be a
+  /// Büchi automaton.
   ///
   /// The algorithm also defines a named property called
   /// "product-states" with type spot::product_states.  This stores
diff --git a/tests/core/prodchain.test b/tests/core/prodchain.test
index e00422148..9a9c74648 100755
--- a/tests/core/prodchain.test
+++ b/tests/core/prodchain.test
@@ -32,12 +32,12 @@ for i in *.hoa; do
   shift
 done
 shift
-if $MAX_ACCSETS -eq 32; then
+if [ $MAX_ACCSETS -eq 32 ]; then
   autfilt "$@" 2> error && exit 1
   grep 'Too many acceptance sets used' error
 fi
-autfilt -B "$@" > result
-test "127,253,508,1" = `autfilt --stats=%s,%e,%t,%a result`
+autfilt -B --low "$@" > result
+test "4,7,16,1" = `autfilt --stats=%s,%e,%t,%a result`
 
 set x
 shift
@@ -46,9 +46,37 @@ for i in *.hoa; do
   shift
 done
 shift
-if $MAX_ACCSETS -eq 32; then
-  autfilt "$@" 2> error && exit 1
+autfilt -B --low "$@" > result
+test "45,89,180,1" = `autfilt --stats=%s,%e,%t,%a result`
+
+
+set x
+shift
+for i in 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 \
+         22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42; do
+  ltl2tgba -D --cobuchi -S "{a[*$i]}<>->FGb" > $i.hoa
+done
+for i in *.hoa; do
+  set x "$@" --product $i
+  shift
+done
+shift
+autfilt -D --cobuchi --low -S "$@" > result
+test "85,170,174,1" = `autfilt --stats=%s,%e,%t,%a result`
+
+set x
+shift
+for i in *.hoa; do
+  set x "$@" --product-or $i
+  shift
+done
+shift
+if [ $MAX_ACCSETS -eq 32 ]; then
+  autfilt --cobuchi -S "$@" 2> error && exit 1
   grep 'Too many acceptance sets used' error
 fi
-autfilt -B "$@" > result
-test "45,89,180,1" = `autfilt --stats=%s,%e,%t,%a result`
+# FIXME: implement degeneralization for generalized-co-Büchi
+# autfilt --cobuchi --low -S "$@" > result
+# test "45,89,180,1" = `autfilt --stats=%s,%e,%t,%a result`
+
+true
diff --git a/tests/core/prodor.test b/tests/core/prodor.test
index 03127508b..03d8cd458 100755
--- a/tests/core/prodor.test
+++ b/tests/core/prodor.test
@@ -1,6 +1,6 @@
 #!/bin/sh
 # -*- coding: utf-8 -*-
-# Copyright (C) 2015, 2017-2018, 2021 Laboratoire de Recherche et
+# Copyright (C) 2015, 2017-2018, 2021-2022 Laboratoire de Recherche et
 # Développement de l'Epita (LRDE).
 #
 # This file is part of Spot, a model checking library.
@@ -154,8 +154,8 @@ diff por.hoa exp
 
 ltl2tgba -BDH 'GFa' > gfa.hoa
 ltl2tgba -x '!wdba-minimize' -DH 'Xb' > xb.hoa
-autfilt --product-or gfa.hoa xb.hoa -H > por.hoa
-cat por.hoa
+autfilt --product-or gfa.hoa xb.hoa -H > por2.hoa
+cat por2.hoa
 
 cat >exp <<EOF
 HOA: v1
@@ -194,4 +194,4 @@ State: 6
 [!0] 6
 --END--
 EOF
-diff por.hoa exp
+diff por2.hoa exp
diff --git a/tests/python/_product_weak.ipynb b/tests/python/_product_weak.ipynb
index 5e4a2a4a0..e52b9b793 100644
--- a/tests/python/_product_weak.ipynb
+++ b/tests/python/_product_weak.ipynb
@@ -87,6 +87,20 @@
     "[0] 0 {0}\n",
     "--END--\n",
     "HOA: v1\n",
+    "name: \"FGd\"\n",
+    "States: 1\n",
+    "Start: 0\n",
+    "AP: 1 \"c\"\n",
+    "acc-name: co-Buchi\n",
+    "Acceptance: 1 Fin(0)\n",
+    "properties: trans-labels explicit-labels trans-acc complete\n",
+    "properties: deterministic stutter-invariant\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",
@@ -117,7 +131,7 @@
     {
      "data": {
       "text/plain": [
-       "6"
+       "7"
       ]
      },
      "execution_count": 3,
@@ -189,7 +203,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fd90c347ba0> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f26743d3720> >"
       ]
      },
      "metadata": {},
@@ -299,11 +313,11 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Pages: 1 -->\n",
-       "<svg width=\"203pt\" height=\"212pt\"\n",
-       " viewBox=\"0.00 0.00 202.59 211.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 207.8)\">\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-207.8 198.59,-207.8 198.59,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"75.8\" y=\"-173.6\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
+       "<svg width=\"203pt\" height=\"195pt\"\n",
+       " viewBox=\"0.00 0.00 202.59 195.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 191)\">\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-191 198.59,-191 198.59,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"75.8\" y=\"-171.8\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 0 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -379,12 +393,12 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Title: Fb Pages: 1 -->\n",
-       "<svg width=\"170pt\" height=\"125pt\"\n",
-       " viewBox=\"0.00 0.00 170.00 124.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 120.8)\">\n",
+       "<svg width=\"170pt\" height=\"108pt\"\n",
+       " viewBox=\"0.00 0.00 170.00 108.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 104)\">\n",
        "<title>Fb</title>\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-120.8 166,-120.8 166,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"59.5\" y=\"-86.6\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-104 166,-104 166,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"59.5\" y=\"-84.8\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 1 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -434,11 +448,11 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Pages: 1 -->\n",
-       "<svg width=\"310pt\" height=\"165pt\"\n",
-       " viewBox=\"0.00 0.00 310.39 164.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 160.8)\">\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-160.8 306.39,-160.8 306.39,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"129.69\" y=\"-126.6\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
+       "<svg width=\"310pt\" height=\"148pt\"\n",
+       " viewBox=\"0.00 0.00 310.39 148.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 144)\">\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-144 306.39,-144 306.39,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"129.69\" y=\"-124.8\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 0 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -508,11 +522,11 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Pages: 1 -->\n",
-       "<svg width=\"330pt\" height=\"324pt\"\n",
-       " viewBox=\"0.00 0.00 330.39 323.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 319.8)\">\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-319.8 326.39,-319.8 326.39,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"139.69\" y=\"-285.6\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
+       "<svg width=\"330pt\" height=\"307pt\"\n",
+       " viewBox=\"0.00 0.00 330.39 307.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 303)\">\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-303 326.39,-303 326.39,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"139.69\" y=\"-283.8\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 0 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -644,12 +658,12 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Title: Fb Pages: 1 -->\n",
-       "<svg width=\"170pt\" height=\"125pt\"\n",
-       " viewBox=\"0.00 0.00 170.00 124.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 120.8)\">\n",
+       "<svg width=\"170pt\" height=\"108pt\"\n",
+       " viewBox=\"0.00 0.00 170.00 108.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 104)\">\n",
        "<title>Fb</title>\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-120.8 166,-120.8 166,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"49.5\" y=\"-86.6\" font-family=\"Lato\" font-size=\"14.00\">[co&#45;Büchi]</text>\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-104 166,-104 166,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"49.5\" y=\"-84.8\" font-family=\"Lato\" font-size=\"14.00\">[co&#45;Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 1 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -699,11 +713,11 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Pages: 1 -->\n",
-       "<svg width=\"318pt\" height=\"167pt\"\n",
-       " viewBox=\"0.00 0.00 318.39 166.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 162.8)\">\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-162.8 314.39,-162.8 314.39,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"123.69\" y=\"-128.6\" font-family=\"Lato\" font-size=\"14.00\">[co&#45;Büchi]</text>\n",
+       "<svg width=\"318pt\" height=\"150pt\"\n",
+       " viewBox=\"0.00 0.00 318.39 150.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 146)\">\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-146 314.39,-146 314.39,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"123.69\" y=\"-126.8\" font-family=\"Lato\" font-size=\"14.00\">[co&#45;Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 0 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -774,11 +788,11 @@
        "<!-- Generated by graphviz version 2.43.0 (0)\n",
        " -->\n",
        "<!-- Pages: 1 -->\n",
-       "<svg width=\"330pt\" height=\"324pt\"\n",
-       " viewBox=\"0.00 0.00 330.39 323.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
-       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 319.8)\">\n",
-       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-319.8 326.39,-319.8 326.39,4 -4,4\"/>\n",
-       "<text text-anchor=\"start\" x=\"139.69\" y=\"-285.6\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
+       "<svg width=\"330pt\" height=\"307pt\"\n",
+       " viewBox=\"0.00 0.00 330.39 307.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 303)\">\n",
+       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-303 326.39,-303 326.39,4 -4,4\"/>\n",
+       "<text text-anchor=\"start\" x=\"139.69\" y=\"-283.8\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
        "<!-- I -->\n",
        "<!-- 0 -->\n",
        "<g id=\"node2\" class=\"node\">\n",
@@ -1127,6 +1141,226 @@
      "metadata": {},
      "output_type": "display_data"
     },
+    {
+     "data": {
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     "# exeptions from Jupyter client.   Halving the loop helped for some times, but then the timeout\n",
-    "# came back.   So we do one left automaton at at time.\n",
+    "# came back.   So we do one left automaton at a time.\n",
     "left = auts[0]\n",
     "display(left)\n",
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+      "text/plain": [
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+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "left = auts[4]\n",
+    "display(left)\n",
+    "for right in auts:\n",
+    "    display_inline(right, spot.product(left, right), spot.product_or(left, right))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {
+    "scrolled": false
+   },
    "outputs": [
     {
      "data": {
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+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "left = auts[4]\n",
+    "display(left)\n",
+    "for right in auts:\n",
+    "    display_inline(right, spot.product_xor(left, right), spot.product_xnor(left, right))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {
+    "scrolled": false
+   },
    "outputs": [
     {
      "data": {
@@ -14037,7 +17671,7 @@
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     }
    ],
    "source": [
-    "left = auts[5]\n",
+    "left = auts[6]\n",
     "display(left)\n",
     "for right in auts:\n",
-    "    display_inline(right, spot.product_xor(left, right), spot.product_xnor(left, right))"
+    "    display_inline(right, spot.product(left, right), spot.product_or(left, right))"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
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+       "<!-- I -->\n",
+       "<!-- 0 -->\n",
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+       "<!-- I&#45;&gt;0 -->\n",
+       "<g id=\"edge1\" class=\"edge\">\n",
+       "<title>I&#45;&gt;0</title>\n",
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+       "<title>0&#45;&gt;0</title>\n",
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+       "<title>0&#45;&gt;0</title>\n",
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+       "<text text-anchor=\"start\" x=\"53.4\" y=\"-75.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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+       "<title>0&#45;&gt;0</title>\n",
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+       "<text text-anchor=\"start\" x=\"43.4\" y=\"-120.8\" font-family=\"Lato\" font-size=\"14.00\">a &amp; !b</text>\n",
+       "<text text-anchor=\"start\" x=\"53.4\" y=\"-105.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
+       "</g>\n",
+       "<!-- 0&#45;&gt;0 -->\n",
+       "<g id=\"edge5\" class=\"edge\">\n",
+       "<title>0&#45;&gt;0</title>\n",
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+       "<text text-anchor=\"start\" x=\"45.4\" y=\"-150.8\" font-family=\"Lato\" font-size=\"14.00\">a &amp; b</text>\n",
+       "<text text-anchor=\"start\" x=\"53.4\" y=\"-135.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
+       "</g>\n",
+       "</g>\n",
+       "</svg>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "a1 = spot.translate('GFa')\n",
+    "a2 = spot.translate('GFb')\n",
+    "display_inline(spot.product(a1,a2), spot.product_or(a1, a2))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
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+       "</svg>\n",
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+       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
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+       "<text text-anchor=\"start\" x=\"71.5\" y=\"-105.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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+       "<!-- 0&#45;&gt;0 -->\n",
+       "<g id=\"edge5\" class=\"edge\">\n",
+       "<title>0&#45;&gt;0</title>\n",
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+       "<text text-anchor=\"start\" x=\"63.5\" y=\"-135.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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+       "</g>\n",
+       "</g>\n",
+       "</svg>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "a1 = spot.dualize(a1)\n",
+    "a2 = spot.dualize(a2)\n",
+    "display_inline(spot.product(a1,a2), spot.product_or(a1, a2))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -16662,7 +20908,7 @@
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    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.3rc1"
+   "version": "3.10.6"
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