From dfb75632ba50f2243abb12147986fa73079a9db6 Mon Sep 17 00:00:00 2001
From: Philipp Schlehuber-Caissier <philipp@lrde.epita.fr>
Date: Thu, 24 Mar 2022 09:45:33 +0100
Subject: [PATCH] Update merge_states

Current implementation of merge_states fails
on certain self-loops.
Updated implementation to take them into
account and use a hashbased implementation
to speed up calculations.
Moreover, merge_states() is now aware
of "state-player", just like defrag_states_

* spot/twa/twagraph.cc: Here
* spot/twaalgos/game.cc: Fix odd cycle for sink
* spot/twaalgos/synthesis.cc: Adapt split_det pipeline
* tests/python/_synthesis.ipynb: Tests
---
 spot/twa/twagraph.cc          |  413 ++-
 spot/twaalgos/game.cc         |   11 +-
 spot/twaalgos/synthesis.cc    |    7 +-
 tests/python/_synthesis.ipynb | 4541 ++++++++++++++++++++++++++++++++-
 4 files changed, 4919 insertions(+), 53 deletions(-)

diff --git a/spot/twa/twagraph.cc b/spot/twa/twagraph.cc
index 051514550..b11ca12c5 100644
--- a/spot/twa/twagraph.cc
+++ b/spot/twa/twagraph.cc
@@ -21,13 +21,89 @@
 #include <spot/twa/twagraph.hh>
 #include <spot/tl/print.hh>
 #include <spot/misc/bddlt.hh>
+#include <spot/misc/timer.hh>
 #include <spot/twa/bddprint.hh>
 #include <spot/misc/escape.hh>
+#include <spot/priv/robin_hood.hh>
 #include <vector>
 #include <deque>
 
 using namespace std::string_literals;
 
+namespace
+{
+  using namespace spot;
+    // If LAST is false,
+    // it is guaranteed that there will be another src state
+    template<bool SPE, bool LAST>
+    void treat(std::vector<std::array<unsigned, 4>>& e_idx,
+                const twa_graph::graph_t::edge_vector_t& e_vec,
+                std::vector<unsigned>& e_chain,
+                std::vector<bool>& use_for_hash,
+                unsigned& idx,
+                unsigned s,
+                unsigned n_e)
+    {
+      assert(s < e_idx.size());
+      assert(idx < e_vec.size());
+      assert(e_chain.size() == e_vec.size());
+
+      //std::cout << s << "; " << idx << std::endl;
+
+      // Check if this state has outgoing transitions
+      if (s != e_vec[idx].src)
+        // Nothing to do
+        {
+          assert(!LAST);
+          return;
+        }
+
+      auto& s_idx = e_idx[s];
+      s_idx[0] = idx;
+
+      // helper
+      unsigned sub_idx[] = {-1u, -1u};
+
+      // All transitions of this state
+      while (true)
+        {
+          assert(idx < e_vec.size() + LAST);
+          if constexpr (!LAST)
+            {
+              if (e_vec[idx].src != s)
+                break;
+            }
+          else
+            {
+              if (idx == n_e)
+                break;
+            }
+
+          // Argh so many ifs
+          unsigned which = e_vec[idx].src == e_vec[idx].dst;
+          if (sub_idx[which] == -1u)
+            {
+              // First non-selflooping
+              sub_idx[which] = idx;
+              s_idx[1u+which] = idx;
+            }
+          else
+            {
+              // Continue the chained list
+              e_chain[sub_idx[which]] = idx;
+              sub_idx[which] = idx;
+            }
+          ++idx;
+        }
+      s_idx[3] = idx;
+
+      // Check if self-loops appeared
+      // If so -> do not use for hash
+      if constexpr (!SPE)
+        use_for_hash[s] = s_idx[2] == -1u;
+    }
+}
+
 namespace spot
 {
 
@@ -306,68 +382,287 @@ namespace spot
         return true;
       if (lhs.acc > rhs.acc)
         return false;
+      // compare with id?
       if (bdd_less_than_stable lt; lt(lhs.cond, rhs.cond))
         return true;
       if (rhs.cond != lhs.cond)
         return false;
-      // The destination must be sorted last
-      // for our self-loop optimization to work.
       return lhs.dst < rhs.dst;
     });
     g_.chain_edges_();
 
+    const auto n_states = num_states();
+
+    // Edges are nicely chained and there are no erased edges
+    // -> We can work with the edge_vector
+
+    // Check if it is a game <-> "state-player" is defined
+    // if so, the graph alternates between env and player vertices,
+    // so there are, by definition, no self-loops
+    auto sp = get_named_prop<std::vector<bool>>("state-player");
+    const auto spe = (bool) sp;
+
+    // The hashing is a bit delicat: We may only use the dst
+    // if it has no self-loop
+    auto use_for_hash = spe ? std::vector<bool>()
+                            : std::vector<bool>(n_states);
+
+    const auto& e_vec = edge_vector();
+    const auto n_edges = e_vec.size();
+
+    // For each state we need 4 indices of the edge vector
+    // [first, first_non_sfirst_selflooplfloop, first_selfloop, end]
+    // The init value makes sure nothing is done for dead end states
+    auto e_idx =
+      std::vector<std::array<unsigned, 4>>(n_states, {-1u, -1u,
+                                                      -1u, -1u});
+    // Like a linked list holding the non-selfloop and selfloop transitions
+    auto e_chain = std::vector<unsigned>(e_vec.size(), -1u);
+
+    unsigned idx = 1;
+
+    // Edges are sorted with repected to src first
+    const unsigned n_high = e_vec.back().src;
+    if (spe)
+      for (auto s = 0u; s < n_high; ++s)
+        treat<true, false>(e_idx, e_vec, e_chain,
+                           use_for_hash, idx, s, n_edges);
+    else
+      for (auto s = 0u; s < n_high; ++s)
+        treat<false, false>(e_idx, e_vec, e_chain,
+                            use_for_hash, idx, s, n_edges);
+    // Last one
+    if (spe)
+      treat<true, true>(e_idx, e_vec, e_chain,
+                        use_for_hash, idx, n_high, n_edges);
+    else
+      treat<false, true>(e_idx, e_vec, e_chain,
+                         use_for_hash, idx, n_high, n_edges);
+
+    assert(idx == e_vec.size() && "Something went wrong during indexing");
+
+    auto n_players = 0u;
+    if (sp)
+      n_players = std::accumulate(sp->begin(), sp->end(), 0u);
+
+    // Represents which states share a hash
+    // Head is in the unordered_map,
+    // hash_linked_list is like a linked list structure
+    // of false pointers
+
+    auto hash_linked_list = std::vector<unsigned>(n_states, -1u);
+    auto s_to_hash = std::vector<size_t>(n_states, 0);
+    auto env_map =
+      robin_hood::unordered_flat_map<size_t,
+                                     std::pair<unsigned, unsigned>>();
+    auto player_map =
+      robin_hood::unordered_flat_map<size_t,
+                                     std::pair<unsigned, unsigned>>();
+    env_map.reserve(n_states - n_players);
+    player_map.reserve(n_players);
+
+    // Sadly we need to loop the edges twice since we have
+    // to check for self-loops before hashing
+
+    auto emplace = [&hash_linked_list](auto& m, auto h, auto s)
+      {
+        auto it = m.find(h);
+        if (it == m.end())
+          m.emplace(h, std::make_pair(s, s));
+        else
+          {
+            // "tail"
+            auto idx = it->second.second;
+            assert(idx < s && "Must be monotone");
+            hash_linked_list[idx] = s;
+            it->second.second = s;
+          }
+      };
+
+    // Hash all states
+    constexpr auto SHIFT = sizeof(size_t)/2 * CHAR_BIT;
+    for (auto s = 0u; s != n_states; ++s)
+      {
+        auto h = fnv<size_t>::init;
+        const auto e = e_idx[s][3];
+        for (auto i = e_idx[s][0]; i != e; ++i)
+          {
+            // If size_t has 8byte and unsigned has 4byte
+            // then this works fine, otherwise there might be more collisions
+            size_t hh = spe || use_for_hash[e_vec[i].dst]
+                          ? e_vec[i].dst
+                          : fnv<unsigned>::init;
+            hh <<= SHIFT;
+            hh += e_vec[i].cond.id();
+            h ^= hh;
+            h *= fnv<size_t>::prime;
+            h ^= e_vec[i].acc.hash();
+            h *= fnv<size_t>::prime;
+          }
+        s_to_hash[s] = h;
+        if (spe && (*sp)[s])
+          emplace(player_map, h, s);
+        else
+          emplace(env_map, h, s);
+      }
+    // All states that might possible be merged share the same hash
+    // Info hash coll
+    //std::cout << "Hash collission rate pre merge: "
+    //          << ((env_map.size()+player_map.size())/((float)n_states))
+    //          << '\n';
+
+    // Check whether we can merge two states
+    // and takes into account the self-loops
+    auto state_equal = [&](unsigned s1, unsigned s2)
+      {
+        auto edge_data_comp = [](const auto& lhs,
+                                 const auto& rhs)
+          {
+            if (lhs.acc < rhs.acc)
+              return true;
+            if (lhs.acc > rhs.acc)
+              return false;
+            // todo compare with id
+            if (bdd_less_than_stable lt; lt(lhs.cond, rhs.cond))
+              return true;
+            return false;
+          };
+
+
+        static auto checked1 = std::vector<char>();
+        static auto checked2 = std::vector<char>();
+
+        auto [i1, nsl1, sl1, e1] = e_idx[s1];
+        auto [i2, nsl2, sl2, e2] = e_idx[s2];
+
+        if ((e2-i2) != (e1-i1))
+          return false; // Different number of outgoing trans
+
+        // checked1/2 is one element larger than necessary
+        // the last element is always false
+        // and acts like a nulltermination
+        checked1.resize(e1-i1+1);
+        std::fill(checked1.begin(), checked1.end(), false);
+        checked2.resize(e2-i2+1);
+        std::fill(checked2.begin(), checked2.end(), false);
+
+        // Try to match self-loops
+        // Not entirely sure when this helps exactly
+        while ((sl1 < e1) & (sl2 < e2))
+          {
+            // Like a search in ordered array
+            if (e_vec[sl1].data() == e_vec[sl2].data())
+              {
+                // Matched
+                checked1[sl1 - i1] = true; //never touches last element
+                checked2[sl2 - i2] = true;
+                // Advance both
+                sl1 = e_chain[sl1];
+                sl2 = e_chain[sl2];
+              }
+            else if (edge_data_comp(e_vec[sl1].data(),
+                                    e_vec[sl2].data()))
+              // sl1 needs to advance
+              sl1 = e_chain[sl1];
+            else
+              // sl2 needs to advance
+              sl2 = e_chain[sl2];
+          }
+
+        // If there are no non-self-loops, in s1
+        // Check if all have been correctly treated
+        if ((nsl1 > e1)
+            && std::all_of(checked1.begin(), checked1.end(),
+                           [](const auto& e){return e; }))
+          return true;
+
+        // The remaining edges need to match exactly
+        auto idx1 = i1;
+        auto idx2 = i2;
+        while (((idx1 < e1) & (idx2 < e2)))
+          {
+            // More efficient version?
+            // Skip checked edges
+            // Last element serves as break
+            for (; checked1[idx1 - i1]; ++idx1)
+              {
+              }
+            for (; checked2[idx2 - i2]; ++idx2)
+              {
+              }
+            // If one is out of bounds, so is the other
+            if (idx1 == e1)
+              {
+                assert(idx2 == e2);
+                break;
+              }
+
+
+            if  ((e_vec[idx1].dst != e_vec[idx2].dst)
+                || !(e_vec[idx1].data() == e_vec[idx2].data()))
+              return false;
+
+            // Advance
+            ++idx1;
+            ++idx2;
+          }
+        // All edges have bee paired
+        return true;
+      };
+
     const unsigned nb_states = num_states();
     std::vector<unsigned> remap(nb_states, -1U);
+
     for (unsigned i = 0; i != nb_states; ++i)
       {
-        auto out1 = out(i);
-        for (unsigned j = 0; j != i; ++j)
+        auto j = spe && (*sp)[i] ? player_map.at(s_to_hash[i]).first
+                                 : env_map.at(s_to_hash[i]).first;
+        for (; j<i; j=hash_linked_list[j])
           {
-            auto out2 = out(j);
-            if (std::equal(out1.begin(), out1.end(), out2.begin(), out2.end(),
-                           [](const edge_storage_t& a,
-                              const edge_storage_t& b)
-                           { return ((a.dst == b.dst
-                                      || (a.dst == a.src && b.dst == b.src))
-                                     && a.data() == b.data()); }))
-            {
-              remap[i] = (remap[j] != -1U) ? remap[j] : j;
+            if (state_equal(j, i))
+              {
+                remap[i] = (remap[j] != -1U) ? remap[j] : j;
 
-              // Because of the special self-loop tests we use above,
-              // it's possible that i can be mapped to remap[j] even
-              // if j was last compatible states found.  Consider the
-              // following cases, taken from an actual test case:
-              // 18 is equal to 5, 35 is equal to 18, but 35 is not
-              // equal to 5.
-              //
-              // State: 5
-              // [0&1&2] 8 {3}
-              // [!0&1&2] 10 {1}
-              // [!0&!1&!2] 18 {1}
-              // [!0&!1&2] 19 {1}
-              // [!0&1&!2] 20 {1}
-              //
-              // State: 18
-              // [0&1&2] 8 {3}
-              // [!0&1&2] 10 {1}
-              // [!0&!1&!2] 18 {1} // self-loop
-              // [!0&!1&2] 19 {1}
-              // [!0&1&!2] 20 {1}
-              //
-              // State: 35
-              // [0&1&2] 8 {3}
-              // [!0&1&2] 10 {1}
-              // [!0&!1&!2] 35 {1} // self-loop
-              // [!0&!1&2] 19 {1}
-              // [!0&1&!2] 20 {1}
-              break;
-            }
+                // Because of the special self-loop tests we use above,
+                // it's possible that i can be mapped to remap[j] even
+                // if j was last compatible states found.  Consider the
+                // following cases, taken from an actual test case:
+                // 18 is equal to 5, 35 is equal to 18, but 35 is not
+                // equal to 5.
+                //
+                // State: 5
+                // [0&1&2] 8 {3}
+                // [!0&1&2] 10 {1}
+                // [!0&!1&!2] 18 {1}
+                // [!0&!1&2] 19 {1}
+                // [!0&1&!2] 20 {1}
+                //
+                // State: 18
+                // [0&1&2] 8 {3}
+                // [!0&1&2] 10 {1}
+                // [!0&!1&!2] 18 {1} // self-loop
+                // [!0&!1&2] 19 {1}
+                // [!0&1&!2] 20 {1}
+                //
+                // State: 35
+                // [0&1&2] 8 {3}
+                // [!0&1&2] 10 {1}
+                // [!0&!1&!2] 35 {1} // self-loop
+                // [!0&!1&2] 19 {1}
+                // [!0&1&!2] 20 {1}
+                break;
+              }
           }
       }
 
     for (auto& e: edges())
       if (remap[e.dst] != -1U)
-        e.dst = remap[e.dst];
+        {
+          assert((!spe || (sp->at(e.dst) == sp->at(remap[e.dst])))
+                 && "States do not have the same owner");
+          e.dst = remap[e.dst];
+        }
+
 
     if (remap[get_init_state_number()] != -1U)
       set_init_state(remap[get_init_state_number()]);
@@ -382,6 +677,10 @@ namespace spot
     unsigned merged = num_states() - st;
     if (merged)
       defrag_states(remap, st);
+    // Info hash coll 2
+    //std::cout << "Hash collission rate post merge: "
+    //          << ((env_map.size()+player_map.size())/((float)num_states()))
+    //          << '\n';
     return merged;
   }
 
@@ -942,8 +1241,36 @@ namespace spot
               s = newst[s];
           }
       }
+    // Reassign the state-players
+    if (auto sp = get_named_prop<std::vector<bool>>("state-player"))
+      {
+        const auto ns = (unsigned) used_states;
+        const auto sps = (unsigned) sp->size();
+        assert(ns <= sps);
+        assert(sps == newst.size());
+
+        for (unsigned i = 0; i < sps; ++i)
+          {
+            if (newst[i] == -1u)
+              continue;
+            (*sp)[newst[i]] = (*sp)[i];
+          }
+        sp->resize(ns);
+      }
     init_number_ = newst[init_number_];
     g_.defrag_states(newst, used_states);
+    // Make sure we did not mess up the structure
+    assert([&]()
+      {
+        if (auto sp = get_named_prop<std::vector<bool>>("state-player"))
+          {
+            for (const auto& e : edges())
+              if (sp->at(e.src) == sp->at(e.dst))
+                return false;
+            return true;
+          }
+        return true;
+      }() && "Game not alternating!");
   }
 
   void twa_graph::remove_unused_ap()
diff --git a/spot/twaalgos/game.cc b/spot/twaalgos/game.cc
index 6bb62500d..6d319eea8 100644
--- a/spot/twaalgos/game.cc
+++ b/spot/twaalgos/game.cc
@@ -896,10 +896,19 @@ namespace spot
                 arena->new_edge(sink_con, sink_env, bddtrue, um.second);
                 arena->new_edge(sink_env, sink_con, bddtrue, um.second);
               }
-            arena->new_edge(src, sink_con, missing, um.second);
+            arena->new_edge(src, sink_env, missing, um.second);
+            assert(owner->at(src) != owner->at(sink_env));
           }
       }
 
+    assert([&]()
+      {
+        for (const auto& e : arena->edges())
+          if (owner->at(e.src) == owner->at(e.dst))
+            return false;
+        return true;
+      }() && "Not alternating");
+
     arena->set_named_prop("state-player", owner);
   }
 
diff --git a/spot/twaalgos/synthesis.cc b/spot/twaalgos/synthesis.cc
index dc79f8cce..1708923c6 100644
--- a/spot/twaalgos/synthesis.cc
+++ b/spot/twaalgos/synthesis.cc
@@ -958,6 +958,10 @@ namespace spot
           *vs << "determinization done\nDPA has "
               << dpa->num_states() << " states, "
               << dpa->num_sets() << " colors\n";
+        // The named property "state-player" is set in split_2step
+        // but not propagated by ntgba2dpa
+        alternate_players(dpa);
+        // Merge states knows about players
         dpa->merge_states();
         if (bv)
           bv->paritize_time += sw.stop();
@@ -966,9 +970,6 @@ namespace spot
               << dpa->num_states() << " states\n"
               << "determinization and simplification took "
               << bv->paritize_time << " seconds\n";
-        // The named property "state-player" is set in split_2step
-        // but not propagated by ntgba2dpa
-        alternate_players(dpa);
         break;
       }
       case algo::ACD:
diff --git a/tests/python/_synthesis.ipynb b/tests/python/_synthesis.ipynb
index 5866057a1..b9e065c18 100644
--- a/tests/python/_synthesis.ipynb
+++ b/tests/python/_synthesis.ipynb
@@ -737,7 +737,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f7458055570> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc883a7720> >"
       ]
      },
      "execution_count": 8,
@@ -820,7 +820,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f7458055570> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc883a7720> >"
       ]
      },
      "execution_count": 9,
@@ -944,7 +944,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f74580553c0> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc8833aa80> >"
       ]
      },
      "execution_count": 10,
@@ -1042,7 +1042,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f743a5ca6c0> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c2ab0> >"
       ]
      },
      "execution_count": 11,
@@ -1210,7 +1210,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f7458059f90> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc8833ae70> >"
       ]
      },
      "execution_count": 12,
@@ -1419,7 +1419,7 @@
        "</svg>\n"
       ],
       "text/plain": [
-       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f7458055870> >"
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c2240> >"
       ]
      },
      "execution_count": 13,
@@ -1432,6 +1432,4535 @@
     "print(a_s.acc())\n",
     "a_s"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0ee90b2a",
+   "metadata": {},
+   "source": [
+    "## A problematic case for merge\n",
+    "\n",
+    "This is an example graph for which the self-loop optimisation in merge_states does not work"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "06b20a8c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/svg+xml": [
+       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
+       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
+       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
+       "<!-- Generated by graphviz version 2.43.0 (0)\n",
+       " -->\n",
+       "<!-- Pages: 1 -->\n",
+       "<svg width=\"243pt\" height=\"207pt\"\n",
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+       "</g>\n",
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+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880bdc30> >"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut = spot.make_twa_graph()\n",
+    "aut.set_buchi()\n",
+    "aut.new_states(5)\n",
+    "\n",
+    "a = buddy.bdd_ithvar(aut.register_ap(\"a\"))\n",
+    "b = buddy.bdd_ithvar(aut.register_ap(\"b\"))\n",
+    "c = buddy.bdd_ithvar(aut.register_ap(\"c\"))\n",
+    "x = buddy.bdd_ithvar(aut.register_ap(\"x\"))\n",
+    "\n",
+    "\n",
+    "aut.new_edge(0, 1, buddy.bddtrue)\n",
+    "aut.new_edge(0, 4, buddy.bddtrue)\n",
+    "\n",
+    "# OK, edge conditions ensure \"correct\" ordering\n",
+    "aut.new_edge(1, 1, a)\n",
+    "aut.new_edge(1, 2, b)\n",
+    "aut.new_edge(1, 3, c)\n",
+    "\n",
+    "aut.new_edge(4, 4, a)\n",
+    "aut.new_edge(4, 2, b)\n",
+    "aut.new_edge(4, 3, c)\n",
+    "\n",
+    "aut.new_edge(2, 2, x)\n",
+    "aut.new_edge(3, 3, buddy.bdd_not(x))\n",
+    "\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "8a2f2e4d",
+   "metadata": {},
+   "outputs": [
+    {
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+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880bdc30> >"
+      ]
+     },
+     "execution_count": 15,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut.merge_states()\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "b40f8ce7",
+   "metadata": {},
+   "outputs": [
+    {
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+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c5210> >"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut = spot.make_twa_graph()\n",
+    "aut.set_buchi()\n",
+    "aut.new_states(5)\n",
+    "\n",
+    "a = buddy.bdd_ithvar(aut.register_ap(\"a\"))\n",
+    "b = buddy.bdd_ithvar(aut.register_ap(\"b\"))\n",
+    "c = buddy.bdd_ithvar(aut.register_ap(\"c\"))\n",
+    "x = buddy.bdd_ithvar(aut.register_ap(\"x\"))\n",
+    "\n",
+    "\n",
+    "aut.new_edge(0, 1, buddy.bddtrue)\n",
+    "aut.new_edge(0, 4, buddy.bddtrue)\n",
+    "\n",
+    "# Not OK, all edge equal -> sorted by destination\n",
+    "# Fails to merge\n",
+    "aut.new_edge(1, 1, a)\n",
+    "aut.new_edge(1, 2, a)\n",
+    "aut.new_edge(1, 3, a)\n",
+    "\n",
+    "aut.new_edge(4, 4, a)\n",
+    "aut.new_edge(4, 2, a)\n",
+    "aut.new_edge(4, 3, a)\n",
+    "\n",
+    "aut.new_edge(2, 2, x)\n",
+    "aut.new_edge(3, 3, buddy.bdd_not(x))\n",
+    "\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "1f596284",
+   "metadata": {},
+   "outputs": [
+    {
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+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c5210> >"
+      ]
+     },
+     "execution_count": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut.merge_states()\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "761b4c96",
+   "metadata": {},
+   "outputs": [
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+       "</g>\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c5d50> >"
+      ]
+     },
+     "execution_count": 18,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut = spot.make_twa_graph()\n",
+    "aut.set_buchi()\n",
+    "aut.new_states(8)\n",
+    "\n",
+    "a = buddy.bdd_ithvar(aut.register_ap(\"a\"))\n",
+    "b = buddy.bdd_ithvar(aut.register_ap(\"b\"))\n",
+    "c = buddy.bdd_ithvar(aut.register_ap(\"c\"))\n",
+    "x = buddy.bdd_ithvar(aut.register_ap(\"x\"))\n",
+    "\n",
+    "\n",
+    "aut.new_edge(0, 1, buddy.bddtrue)\n",
+    "aut.new_edge(0, 4, buddy.bddtrue)\n",
+    "\n",
+    "aut.new_edge(1, 1, a)\n",
+    "aut.new_edge(1, 2, b)\n",
+    "aut.new_edge(1, 3, c)\n",
+    "\n",
+    "aut.new_edge(4, 4, a)\n",
+    "aut.new_edge(4, 2, b)\n",
+    "aut.new_edge(4, 3, c)\n",
+    "\n",
+    "aut.new_edge(2, 2, x)\n",
+    "aut.new_edge(3, 3, buddy.bdd_not(x))\n",
+    "\n",
+    "aut.new_edge(5, 1, a)\n",
+    "aut.new_edge(5, 2, b)\n",
+    "aut.new_edge(5, 3, c)\n",
+    "\n",
+    "display(aut.show_storage())\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "d4e09261",
+   "metadata": {},
+   "outputs": [
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+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c5d50> >"
+      ]
+     },
+     "execution_count": 19,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut.merge_states()\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4a8ace82",
+   "metadata": {},
+   "source": [
+    "## Splitting can inhibit merging\n",
+    "\n",
+    "In split automata, no self-loops exist.\n",
+    "Therefore states that can be merged pre-split can not be merged in a split automaton"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "c9e38db9",
+   "metadata": {},
+   "outputs": [
+    {
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+       "<text text-anchor=\"start\" x=\"198\" y=\"-85.8\" font-family=\"Times,serif\" font-size=\"14.00\">acceptance:</text>\n",
+       "<text text-anchor=\"start\" x=\"287\" y=\"-85.8\" font-family=\"Times,serif\" font-size=\"14.00\">Inf(0)</text>\n",
+       "<text text-anchor=\"start\" x=\"198\" y=\"-66.8\" font-family=\"Times,serif\" font-size=\"14.00\">ap_vars:</text>\n",
+       "<text text-anchor=\"start\" x=\"287\" y=\"-66.8\" font-family=\"Times,serif\" font-size=\"14.00\">a b c x</text>\n",
+       "</g>\n",
+       "<!-- meta&#45;&gt;states -->\n",
+       "<!-- props -->\n",
+       "<g id=\"node4\" class=\"node\">\n",
+       "<title>props</title>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-180.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_state_acc:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-180.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-161.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_inherently_weak:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-161.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-142.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_terminal:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-142.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-123.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_weak:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-123.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-104.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_very_weak:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-104.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-85.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_complete:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-85.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-66.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_universal:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-66.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-47.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_unambiguous:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-47.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-28.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_semi_deterministic:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-28.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "<text text-anchor=\"start\" x=\"1287\" y=\"-9.8\" font-family=\"Times,serif\" font-size=\"14.00\">prop_stutter_invariant:</text>\n",
+       "<text text-anchor=\"start\" x=\"1472\" y=\"-9.8\" font-family=\"Times,serif\" font-size=\"14.00\">maybe</text>\n",
+       "</g>\n",
+       "<!-- props&#45;&gt;edges -->\n",
+       "<!-- namedprops -->\n",
+       "<g id=\"node5\" class=\"node\">\n",
+       "<title>namedprops</title>\n",
+       "<text text-anchor=\"middle\" x=\"1622\" y=\"-110.3\" font-family=\"Times,serif\" font-size=\"14.00\">named properties:</text>\n",
+       "<text text-anchor=\"middle\" x=\"1622\" y=\"-95.3\" font-family=\"Times,serif\" font-size=\"14.00\">state&#45;player</text>\n",
+       "<text text-anchor=\"middle\" x=\"1622\" y=\"-80.3\" font-family=\"Times,serif\" font-size=\"14.00\">synthesis&#45;outputs</text>\n",
+       "</g>\n",
+       "<!-- namedprops&#45;&gt;edges -->\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<spot.jupyter.SVG object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
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+       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
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+       "<text text-anchor=\"start\" x=\"277.14\" y=\"-832.37\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
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+       "<text text-anchor=\"start\" x=\"314.14\" y=\"-832.37\" font-family=\"Lato\" font-size=\"14.00\">)</text>\n",
+       "<text text-anchor=\"start\" x=\"276.14\" y=\"-818.37\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
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+       "<g id=\"node2\" class=\"node\">\n",
+       "<title>0</title>\n",
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+       "<text text-anchor=\"middle\" x=\"56\" y=\"-394.87\" font-family=\"Lato\" font-size=\"14.00\">0</text>\n",
+       "</g>\n",
+       "<!-- I&#45;&gt;0 -->\n",
+       "<g id=\"edge1\" class=\"edge\">\n",
+       "<title>I&#45;&gt;0</title>\n",
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+       "</g>\n",
+       "<!-- 5 -->\n",
+       "<g id=\"node3\" class=\"node\">\n",
+       "<title>5</title>\n",
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+       "</g>\n",
+       "<!-- 0&#45;&gt;5 -->\n",
+       "<g id=\"edge2\" class=\"edge\">\n",
+       "<title>0&#45;&gt;5</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M74.4,-398.57C85.16,-398.57 99.25,-398.57 111.7,-398.57\"/>\n",
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+       "</g>\n",
+       "<!-- 1 -->\n",
+       "<g id=\"node4\" class=\"node\">\n",
+       "<title>1</title>\n",
+       "<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"227.96\" cy=\"-598.57\" rx=\"18\" ry=\"18\"/>\n",
+       "<text text-anchor=\"start\" x=\"223.46\" y=\"-594.87\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
+       "</g>\n",
+       "<!-- 5&#45;&gt;1 -->\n",
+       "<g id=\"edge19\" class=\"edge\">\n",
+       "<title>5&#45;&gt;1</title>\n",
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+       "</g>\n",
+       "<!-- 4 -->\n",
+       "<g id=\"node16\" class=\"node\">\n",
+       "<title>4</title>\n",
+       "<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"227.96\" cy=\"-218.57\" rx=\"18\" ry=\"18\"/>\n",
+       "<text text-anchor=\"middle\" x=\"227.96\" y=\"-214.87\" font-family=\"Lato\" font-size=\"14.00\">4</text>\n",
+       "</g>\n",
+       "<!-- 5&#45;&gt;4 -->\n",
+       "<g id=\"edge20\" class=\"edge\">\n",
+       "<title>5&#45;&gt;4</title>\n",
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+       "</g>\n",
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+       "<g id=\"node5\" class=\"node\">\n",
+       "<title>6</title>\n",
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+       "<text text-anchor=\"middle\" x=\"376.29\" y=\"-442.87\" font-family=\"Lato\" font-size=\"14.00\">6</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;6 -->\n",
+       "<g id=\"edge3\" class=\"edge\">\n",
+       "<title>1&#45;&gt;6</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M232.08,-580.8C236.3,-561.58 245.5,-531.08 263.96,-511.57 285.54,-488.74 300.97,-497.65 327.96,-481.57 338.66,-475.19 349.9,-466.95 358.74,-460.07\"/>\n",
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+       "<text text-anchor=\"start\" x=\"263.96\" y=\"-515.37\" font-family=\"Lato\" font-size=\"14.00\">!a &amp; !b &amp; c</text>\n",
+       "</g>\n",
+       "<!-- 7 -->\n",
+       "<g id=\"node6\" class=\"node\">\n",
+       "<title>7</title>\n",
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+       "<text text-anchor=\"middle\" x=\"376.29\" y=\"-328.87\" font-family=\"Lato\" font-size=\"14.00\">7</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;7 -->\n",
+       "<g id=\"edge4\" class=\"edge\">\n",
+       "<title>1&#45;&gt;7</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M230.27,-580.49C233.5,-540.72 243.17,-447.31 263.96,-424.57 283.97,-402.66 305.88,-425.38 327.96,-405.57 343.38,-391.73 334.51,-379.83 345.96,-362.57 349.66,-356.98 354.49,-351.53 359.16,-346.85\"/>\n",
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+       "<text text-anchor=\"start\" x=\"263.96\" y=\"-428.37\" font-family=\"Lato\" font-size=\"14.00\">!a &amp; b &amp; !c</text>\n",
+       "</g>\n",
+       "<!-- 8 -->\n",
+       "<g id=\"node7\" class=\"node\">\n",
+       "<title>8</title>\n",
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+       "</g>\n",
+       "<!-- 1&#45;&gt;8 -->\n",
+       "<g id=\"edge5\" class=\"edge\">\n",
+       "<title>1&#45;&gt;8</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M228.32,-580.48C228.32,-551.6 232.66,-495.04 263.96,-462.57 284.55,-441.2 303.6,-460.51 327.96,-443.57 338.9,-435.95 337.28,-429.69 345.96,-419.57 350.32,-414.48 355.34,-409.17 359.97,-404.46\"/>\n",
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+       "<text text-anchor=\"start\" x=\"265.96\" y=\"-466.37\" font-family=\"Lato\" font-size=\"14.00\">!a &amp; b &amp; c</text>\n",
+       "</g>\n",
+       "<!-- 9 -->\n",
+       "<g id=\"node8\" class=\"node\">\n",
+       "<title>9</title>\n",
+       "<polygon fill=\"#ffffaa\" stroke=\"black\" points=\"376.29,-641.57 353.33,-623.57 376.29,-605.57 399.25,-623.57 376.29,-641.57\"/>\n",
+       "<text text-anchor=\"middle\" x=\"376.29\" y=\"-619.87\" font-family=\"Lato\" font-size=\"14.00\">9</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;9 -->\n",
+       "<g id=\"edge6\" class=\"edge\">\n",
+       "<title>1&#45;&gt;9</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M238.84,-612.91C245.05,-620.61 253.8,-629.36 263.96,-633.57 293.2,-645.68 330.38,-638.29 353.51,-631.34\"/>\n",
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+       "<text text-anchor=\"start\" x=\"263.96\" y=\"-643.37\" font-family=\"Lato\" font-size=\"14.00\">a &amp; !b &amp; !c</text>\n",
+       "</g>\n",
+       "<!-- 10 -->\n",
+       "<g id=\"node9\" class=\"node\">\n",
+       "<title>10</title>\n",
+       "<polygon fill=\"#ffffaa\" stroke=\"black\" points=\"376.29,-787.57 346.12,-769.57 376.29,-751.57 406.46,-769.57 376.29,-787.57\"/>\n",
+       "<text text-anchor=\"middle\" x=\"376.29\" y=\"-765.87\" font-family=\"Lato\" font-size=\"14.00\">10</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;10 -->\n",
+       "<g id=\"edge7\" class=\"edge\">\n",
+       "<title>1&#45;&gt;10</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M229.52,-616.54C231.27,-659.11 238.16,-764.15 263.96,-785.57 289.23,-806.54 329.35,-793.05 353.9,-781.33\"/>\n",
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+       "<text text-anchor=\"start\" x=\"265.96\" y=\"-799.37\" font-family=\"Lato\" font-size=\"14.00\">a &amp; !b &amp; c</text>\n",
+       "</g>\n",
+       "<!-- 11 -->\n",
+       "<g id=\"node10\" class=\"node\">\n",
+       "<title>11</title>\n",
+       "<polygon fill=\"#ffffaa\" stroke=\"black\" points=\"376.29,-559.57 346.12,-541.57 376.29,-523.57 406.46,-541.57 376.29,-559.57\"/>\n",
+       "<text text-anchor=\"middle\" x=\"376.29\" y=\"-537.87\" font-family=\"Lato\" font-size=\"14.00\">11</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;11 -->\n",
+       "<g id=\"edge8\" class=\"edge\">\n",
+       "<title>1&#45;&gt;11</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M234.49,-581.5C239.77,-568.28 249.15,-550.86 263.96,-542.57 289.63,-528.18 324.15,-530.55 347.91,-534.84\"/>\n",
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+       "<text text-anchor=\"start\" x=\"265.96\" y=\"-546.37\" font-family=\"Lato\" font-size=\"14.00\">a &amp; b &amp; !c</text>\n",
+       "</g>\n",
+       "<!-- 12 -->\n",
+       "<g id=\"node11\" class=\"node\">\n",
+       "<title>12</title>\n",
+       "<polygon fill=\"#ffffaa\" stroke=\"black\" points=\"376.29,-700.57 346.12,-682.57 376.29,-664.57 406.46,-682.57 376.29,-700.57\"/>\n",
+       "<text text-anchor=\"middle\" x=\"376.29\" y=\"-678.87\" font-family=\"Lato\" font-size=\"14.00\">12</text>\n",
+       "</g>\n",
+       "<!-- 1&#45;&gt;12 -->\n",
+       "<g id=\"edge9\" class=\"edge\">\n",
+       "<title>1&#45;&gt;12</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M228.64,-616.85C229.3,-642.43 234.79,-688.18 263.96,-709.57 291.49,-729.75 332.16,-710.73 355.97,-696\"/>\n",
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+       "<text text-anchor=\"start\" x=\"267.96\" y=\"-721.37\" font-family=\"Lato\" font-size=\"14.00\">a &amp; b &amp; c</text>\n",
+       "</g>\n",
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+       "<g id=\"node14\" class=\"node\">\n",
+       "<title>3</title>\n",
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+       "<text text-anchor=\"middle\" x=\"469.62\" y=\"-413.87\" font-family=\"Lato\" font-size=\"14.00\">3</text>\n",
+       "</g>\n",
+       "<!-- 6&#45;&gt;3 -->\n",
+       "<g id=\"edge21\" class=\"edge\">\n",
+       "<title>6&#45;&gt;3</title>\n",
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+       "</g>\n",
+       "<!-- 2 -->\n",
+       "<g id=\"node12\" class=\"node\">\n",
+       "<title>2</title>\n",
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+       "<text text-anchor=\"middle\" x=\"469.62\" y=\"-332.87\" font-family=\"Lato\" font-size=\"14.00\">2</text>\n",
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+       "<g id=\"edge22\" class=\"edge\">\n",
+       "<title>7&#45;&gt;2</title>\n",
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+       "</g>\n",
+       "<!-- 8&#45;&gt;2 -->\n",
+       "<g id=\"edge23\" class=\"edge\">\n",
+       "<title>8&#45;&gt;2</title>\n",
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+       "<title>14&#45;&gt;3</title>\n",
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+       "<title>4&#45;&gt;6</title>\n",
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+       "<title>4&#45;&gt;7</title>\n",
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+       "<title>4&#45;&gt;8</title>\n",
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+       "<title>4&#45;&gt;18</title>\n",
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+       "<title>15&#45;&gt;4</title>\n",
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+       "<title>17&#45;&gt;2</title>\n",
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+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880bdc00> >"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "aut = spot.make_twa_graph()\n",
+    "aut.set_buchi()\n",
+    "aut.new_states(5)\n",
+    "\n",
+    "a = buddy.bdd_ithvar(aut.register_ap(\"a\"))\n",
+    "b = buddy.bdd_ithvar(aut.register_ap(\"b\"))\n",
+    "c = buddy.bdd_ithvar(aut.register_ap(\"c\"))\n",
+    "x = buddy.bdd_ithvar(aut.register_ap(\"x\"))\n",
+    "\n",
+    "\n",
+    "aut.new_edge(0, 1, buddy.bddtrue)\n",
+    "aut.new_edge(0, 4, buddy.bddtrue)\n",
+    "\n",
+    "aut.new_edge(1, 1, a)\n",
+    "aut.new_edge(1, 2, b)\n",
+    "aut.new_edge(1, 3, c)\n",
+    "\n",
+    "aut.new_edge(4, 4, a)\n",
+    "aut.new_edge(4, 2, b)\n",
+    "aut.new_edge(4, 3, c)\n",
+    "\n",
+    "aut.new_edge(2, 2, x)\n",
+    "aut.new_edge(3, 3, buddy.bdd_not(x))\n",
+    "\n",
+    "display(aut)\n",
+    "\n",
+    "aut = spot.split_2step(aut, x, False)\n",
+    "\n",
+    "display(aut.show_storage())\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "2009f279",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0\n"
+     ]
+    },
+    {
+     "data": {
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+       "<title>2&#45;&gt;13</title>\n",
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+       "<!-- 13&#45;&gt;2 -->\n",
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+       "<title>13&#45;&gt;2</title>\n",
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+       "<title>3&#45;&gt;14</title>\n",
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+       "<!-- 14&#45;&gt;3 -->\n",
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+       "<title>14&#45;&gt;3</title>\n",
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+       "<title>4&#45;&gt;6</title>\n",
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+       "<title>4&#45;&gt;7</title>\n",
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+       "<title>4&#45;&gt;8</title>\n",
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+       "<title>4&#45;&gt;15</title>\n",
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+       "<title>4&#45;&gt;16</title>\n",
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+       "<title>4&#45;&gt;18</title>\n",
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+       "<title>15&#45;&gt;4</title>\n",
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+       "<title>16&#45;&gt;4</title>\n",
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+       "<title>17&#45;&gt;2</title>\n",
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+       "</g>\n",
+       "<!-- 18&#45;&gt;3 -->\n",
+       "<g id=\"edge41\" class=\"edge\">\n",
+       "<title>18&#45;&gt;3</title>\n",
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+       "</g>\n",
+       "<!-- 18&#45;&gt;4 -->\n",
+       "<g id=\"edge42\" class=\"edge\">\n",
+       "<title>18&#45;&gt;4</title>\n",
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+       "</g>\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880bdc00> >"
+      ]
+     },
+     "execution_count": 21,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "print(aut.merge_states())\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "17c8d6bc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+       "</g>\n",
+       "<!-- 3&#45;&gt;3 -->\n",
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+       "<title>3&#45;&gt;3</title>\n",
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+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880cd2a0> >"
+      ]
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+     "metadata": {},
+     "output_type": "display_data"
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+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c55a0> >"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Merging possible even in split case\n",
+    "aut = spot.make_twa_graph()\n",
+    "aut.set_buchi()\n",
+    "aut.new_states(5)\n",
+    "\n",
+    "a = buddy.bdd_ithvar(aut.register_ap(\"a\"))\n",
+    "b = buddy.bdd_ithvar(aut.register_ap(\"b\"))\n",
+    "c = buddy.bdd_ithvar(aut.register_ap(\"c\"))\n",
+    "x = buddy.bdd_ithvar(aut.register_ap(\"x\"))\n",
+    "\n",
+    "\n",
+    "aut.new_edge(0, 1, buddy.bddtrue)\n",
+    "aut.new_edge(0, 4, buddy.bddtrue)\n",
+    "\n",
+    "aut.new_edge(1, 2, b)\n",
+    "aut.new_edge(1, 3, c)\n",
+    "\n",
+    "aut.new_edge(4, 2, b)\n",
+    "aut.new_edge(4, 3, c)\n",
+    "\n",
+    "aut.new_edge(2, 2, x)\n",
+    "aut.new_edge(3, 3, buddy.bdd_not(x))\n",
+    "\n",
+    "display(aut)\n",
+    "\n",
+    "aut = spot.split_2step(aut, x, False)\n",
+    "\n",
+    "display(aut.show_storage())\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "b3e90235",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "1\n"
+     ]
+    },
+    {
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+       "<!-- 7&#45;&gt;3 -->\n",
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+       "</g>\n",
+       "<!-- 2&#45;&gt;8 -->\n",
+       "<g id=\"edge6\" class=\"edge\">\n",
+       "<title>2&#45;&gt;8</title>\n",
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+       "<!-- 8&#45;&gt;2 -->\n",
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+       "<title>8&#45;&gt;2</title>\n",
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+       "<!-- 9 -->\n",
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+       "</g>\n",
+       "<!-- 3&#45;&gt;9 -->\n",
+       "<g id=\"edge7\" class=\"edge\">\n",
+       "<title>3&#45;&gt;9</title>\n",
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+       "</g>\n",
+       "<!-- 9&#45;&gt;3 -->\n",
+       "<g id=\"edge15\" class=\"edge\">\n",
+       "<title>9&#45;&gt;3</title>\n",
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+       "<text text-anchor=\"start\" x=\"479\" y=\"-105.8\" font-family=\"Lato\" font-size=\"14.00\">!x</text>\n",
+       "</g>\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c55a0> >"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "print(aut.merge_states())\n",
+    "aut"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "05785bb1",
+   "metadata": {},
+   "source": [
+    "Fail case for alternate_players"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "df4aa681",
+   "metadata": {},
+   "outputs": [
+    {
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+       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
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+       "<text text-anchor=\"start\" x=\"74\" y=\"-86.64\" font-family=\"Lato\" font-size=\"14.00\">[Büchi]</text>\n",
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+       "</svg>\n"
+      ],
+      "text/plain": [
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+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
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+       " -->\n",
+       "<!-- Pages: 1 -->\n",
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+       " viewBox=\"0.00 0.00 269.00 154.46\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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+       "</g>\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fcc880c5a50> >"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "aut = spot.make_twa_graph()\n",
+    "aut.set_buchi()\n",
+    "i = buddy.bdd_ithvar(aut.register_ap(\"i\"))\n",
+    "o = buddy.bdd_ithvar(aut.register_ap(\"o\"))\n",
+    "\n",
+    "spot.set_synthesis_outputs(aut, o)\n",
+    "\n",
+    "aut.new_states(2)\n",
+    "aut.new_edge(0,1,i)\n",
+    "aut.new_edge(1,0,o,spot.mark_t([0]))\n",
+    "display(aut)\n",
+    "spot.alternate_players(aut)\n",
+    "display(aut)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f3b2d981",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {