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simplify: GF(f)=GF(dnf(f)) FG(f)=FG(cnf(f))

Browse source 
These rules come from Delag's paper, and help some cases
in issue #385.

* spot/tl/simplify.cc: Implement the simplification.
* doc/tl/tl.tex, NEWS: Document it.
* tests/core/385.test: New file.
* tests/Makefile.am: Add it.
* tests/core/reduccmp.test: More tests.
* tests/core/ltl2tgba2.test: Adjust one improved case.
* tests/python/automata.ipynb, tests/python/twagraph-internals.ipynb:
Adjust expected output, as the cnf/dnf reorder some subformulas.
This commit is contained in:
Alexandre Duret-Lutz 2019-06-17 00:19:02 +02:00
parent df326e032b
commit da5d23f0a2
9 changed files with 2517 additions and 2291 deletions
Download patch file Download diff file Expand all files Collapse all files

4
NEWS
View file

@ -105,9 +105,11 @@ New in spot 2.7.5.dev (not yet released)
- The LTL simplifier learned the following optional rules: - The LTL simplifier learned the following optional rules:
F(G(a | Fb)) = FGa | GFb (if option "favor_event_univ") F(G(a | Fb)) = FGa | GFb (if option "favor_event_univ")
G(F(a | Gb)) = GFa | FGb (if option "favor_event_univ") G(F(a | Gb)) = GFa | FGb (if option "favor_event_univ")
F(G(a & Fb) = FGa & GFb (unless option "reduce_size_strictly") F(G(a & Fb) = FGa & GFb (unless option "reduce_size_strictly")
G(F(a & Gb)) = GFa & FGb (unless option "reduce_size_strictly") G(F(a & Gb)) = GFa & FGb (unless option "reduce_size_strictly")
GF(f) = GF(dnf(f)) (unless option "reduce_size_strictly")
FG(f) = FG(cnf(f)) (unless option "reduce_size_strictly")
- cleanup_parity() and colorize_parity() were cleaned up a bit, - cleanup_parity() and colorize_parity() were cleaned up a bit,
resulting in fewer colors used in some cases. In particular, resulting in fewer colors used in some cases. In particular,

22
doc/tl/tl.tex
View file

@ -116,7 +116,7 @@
\newcommand{\nsereMarked}[1]{\texttt{!+\{}#1\texttt{\}}} \newcommand{\nsereMarked}[1]{\texttt{!+\{}#1\texttt{\}}}
\newcommand{\seren}[1]{\texttt{\{}#1\texttt{\}!}} \newcommand{\seren}[1]{\texttt{\{}#1\texttt{\}!}}
% rewriting rules that enlarge the formula % rewriting rules that (may) enlarge the formula
\newcommand{\equiV}{\stackrel{\star}{\equiv}} \newcommand{\equiV}{\stackrel{\star}{\equiv}}
% rewriting rules that favors event/univ % rewriting rules that favors event/univ
\newcommand{\equivEU}{\stackrel{\blacktriangleup}{\equiv}} \newcommand{\equivEU}{\stackrel{\blacktriangleup}{\equiv}}
@ -1463,15 +1463,19 @@ instead of $\equiv$, and they can be disabled by setting the
\label{sec:basic-simp-ltl} \label{sec:basic-simp-ltl}
The following are simplification rules for unary operators (applied The following are simplification rules for unary operators (applied
from left to right, as usual): from left to right, as usual). The terms $\mathsf{dnf}(f)$ and
$\mathsf{cnf}(f)$ denote respectively the disjunctive and conjunctive
normal forms if $f$, handling non-Boolean sub-formulas as if they were
atomic propositions.
\begin{align*} \begin{align*}
\X\F\G f & \equiv \F\G f & \F(f\U g) & \equiv \F g & \G(f \R g) & \equiv \G g \\ \X\F\G f & \equiv \F\G f & \F(f\U g) & \equiv \F g & \G(f \R g) & \equiv \G g \\
\X\G\F f & \equiv \G\F f & \F(f\M g) & \equiv \F (f\AND g) & \G(f \W g) & \equiv \G(f\OR g) \\ \X\G\F f & \equiv \G\F f & \F(f\M g) & \equiv \F (f\AND g) & \G(f \W g) & \equiv \G(f\OR g) \\
\F\X f & \equiv \X\F f & \F\G(f\AND \X g) & \equiv \F\G(f\AND g) & \G\F(f\OR \X g) & \equiv \G\F(f\OR g) \\ \F\X f & \equiv \X\F f & \F\G(f\AND \X g) & \equiv \F\G(f\AND g) & \G\F(f\OR \X g) & \equiv \G\F(f\OR g) \\
\G\X f & \equiv \X\G f & \F\G(f\AND \G g) & \equiv \F\G(f\AND g) & \G\F(f\OR \F g) & \equiv \G\F(f\OR g) \\ \G\X f & \equiv \X\G f & \F\G(f\AND \G g) & \equiv \F\G(f\AND g) & \G\F(f\OR \F g) & \equiv \G\F(f\OR g) \\
& & \F\G(f\OR\G g) & \equiv \F(\G f\OR\G g) & \G\F(f\AND\F g) & \equiv \G(\F f\AND\F g) \\ & & \F\G(f\OR\G g) & \equiv \F(\G f\OR\G g) & \G\F(f\AND\F g) & \equiv \G(\F f\AND\F g) \\
& & \F\G(f\AND\F g) & \equiV \F\G f\AND\G\F g & \G\F(f\AND\G g) & \equiV \G\F f\AND\F\G g \\ \G\F f & \equiV \G\F(\mathsf{dnf}(f)) & \F\G(f\AND\F g) & \equiV \F\G f\AND\G\F g & \G\F(f\AND\G g) & \equiV \G\F f\AND\F\G g \\
& & \F\G(f\OR\F g) & \equivEU \F\G f\OR\G\F g & \G\F(f\OR\G g) & \equivEU \G\F f\OR\F\G g \F\G f & \equiV \F\G(\mathsf{cnf}(f)) & \F\G(f\OR\F g) & \equivEU \F\G f\OR\G\F g & \G\F(f\OR\G g) & \equivEU \G\F f\OR\F\G g
\end{align*} \end{align*}
\begin{align*} \begin{align*}
\G(f_1\OR\ldots\OR f_n \OR \G\F(g_1)\OR\ldots\OR \G\F(g_m)) & \equiv \G(f_1\OR\ldots\OR f_n)\OR \G\F(g_1\OR\ldots\OR g_m) \G(f_1\OR\ldots\OR f_n \OR \G\F(g_1)\OR\ldots\OR \G\F(g_m)) & \equiv \G(f_1\OR\ldots\OR f_n)\OR \G\F(g_1\OR\ldots\OR g_m)

173
spot/tl/simplify.cc
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@ -33,6 +33,7 @@
#include <spot/tl/snf.hh> #include <spot/tl/snf.hh>
#include <spot/tl/length.hh> #include <spot/tl/length.hh>
#include <spot/twa/formula2bdd.hh> #include <spot/twa/formula2bdd.hh>
#include <spot/misc/minato.hh>
#include <cassert> #include <cassert>
#include <memory> #include <memory>
@ -45,6 +46,7 @@ namespace spot
{ {
typedef std::unordered_map<formula, formula> f2f_map; typedef std::unordered_map<formula, formula> f2f_map;
typedef std::unordered_map<formula, bdd> f2b_map; typedef std::unordered_map<formula, bdd> f2b_map;
typedef std::map<int, formula> b2f_map;
typedef std::pair<formula, formula> pairf; typedef std::pair<formula, formula> pairf;
typedef std::map<pairf, bool> syntimpl_cache_t; typedef std::map<pairf, bool> syntimpl_cache_t;
public: public:
@ -78,13 +80,18 @@ namespace spot
<< "syntactic implications: " << syntimpl_.size() << " entries\n" << "syntactic implications: " << syntimpl_.size() << " entries\n"
<< "boolean to bdd: " << as_bdd_.size() << " entries\n" << "boolean to bdd: " << as_bdd_.size() << " entries\n"
<< "star normal form: " << snf_cache_.size() << " entries\n" << "star normal form: " << snf_cache_.size() << " entries\n"
<< "boolean isop: " << bool_isop_.size() << " entries\n"; << "boolean isop: " << bool_isop_.size() << " entries\n"
<< "as dnf: " << as_dnf_.size() << " entries\n"
<< "as cnf: " << as_cnf_.size() << " entries\n";
} }
void void
clear_as_bdd_cache() clear_as_bdd_cache()
{ {
as_bdd_.clear(); as_bdd_.clear();
for (auto p: bdd_to_f_)
dict->unregister_variable(p.first, this);
bdd_to_f_.clear();
} }
// Convert a Boolean formula into a BDD for easier comparison. // Convert a Boolean formula into a BDD for easier comparison.
@ -136,7 +143,12 @@ namespace spot
break; break;
} }
default: default:
SPOT_UNIMPLEMENTED(); {
unsigned var = dict->register_anonymous_variables(1, this);
bdd_to_f_[var] = f;
result = bdd_ithvar(var);
break;
}
} }
// Cache the result before returning. // Cache the result before returning.
@ -144,6 +156,83 @@ namespace spot
return result; return result;
} }
formula as_xnf(formula f, bool cnf)
{
bdd in = as_bdd(f);
if (cnf)
in = !in;
minato_isop isop(in);
bdd cube;
vec clauses;
while ((cube = isop.next()) != bddfalse)
{
vec literals;
while (cube != bddtrue)
{
int var = bdd_var(cube);
const bdd_dict::bdd_info& i = dict->bdd_map[var];
formula res;
if (i.type == bdd_dict::var)
{
res = i.f;
}
else
{
res = bdd_to_f_[var];
assert(res);
}
bdd high = bdd_high(cube);
if (high == bddfalse)
{
if (!cnf)
res = formula::Not(res);
cube = bdd_low(cube);
}
else
{
if (cnf)
res = formula::Not(res);
// If bdd_low is not false, then cube was not a
// conjunction.
assert(bdd_low(cube) == bddfalse);
cube = high;
}
assert(cube != bddfalse);
literals.emplace_back(res);
}
if (cnf)
clauses.emplace_back(formula::Or(literals));
else
clauses.emplace_back(formula::And(literals));
}
if (cnf)
return formula::And(clauses);
else
return formula::Or(clauses);
}
formula as_dnf(formula f)
{
auto i = as_dnf_.find(f);
if (i != as_dnf_.end())
return i->second;
formula r = as_xnf(f, false);
as_dnf_[f] = r;
return r;
}
formula as_cnf(formula f)
{
auto i = as_cnf_.find(f);
if (i != as_cnf_.end())
return i->second;
formula r = as_xnf(f, true);
as_cnf_[f] = r;
return r;
}
formula formula
lookup_nenoform(formula f) lookup_nenoform(formula f)
{ {
@ -303,7 +392,10 @@ namespace spot
private: private:
f2b_map as_bdd_; f2b_map as_bdd_;
b2f_map bdd_to_f_;
f2f_map simplified_; f2f_map simplified_;
f2f_map as_dnf_;
f2f_map as_cnf_;
f2f_map nenoform_; f2f_map nenoform_;
syntimpl_cache_t syntimpl_; syntimpl_cache_t syntimpl_;
snf_cache snf_cache_; snf_cache snf_cache_;
@ -987,22 +1079,32 @@ namespace spot
return recurse(res); return recurse(res);
} }
// FG(a & Xb) = FG(a & b) // FG(a) = FG(dnf(a)) if a is not Boolean
// FG(a & Gb) = FG(a & b) // and contains some | above non-Boolean subformulas.
if (c.is({op::G, op::And})) if (c.is(op::G) && !c[0].is_boolean())
{ {
formula m = c[0]; formula m = c[0];
if (!m.is_boolean()) bool want_cnf = m.is(op::And);
{ if (!want_cnf && m.is(op::Or))
formula out = m.map([](formula f) for (auto cc : m)
{ if (cc.is(op::And))
if (f.is(op::X, op::G)) {
return f[0]; want_cnf = true;
return f; break;
}); }
if (out != m) if (want_cnf && !opt_.reduce_size_strictly)
return recurse(unop_unop(op::F, op::G, out)); m = c_->as_cnf(m);
} // FG(a & Xb) = FG(a & b)
// FG(a & Gb) = FG(a & b)
if (m.is(op::And))
m = m.map([](formula f)
{
if (f.is(op::X, op::G))
return f[0];
return f;
});
if (c[0] != m)
return recurse(unop_unop(op::F, op::G, m));
} }
} }
// if Fa => a, keep a. // if Fa => a, keep a.
@ -1248,22 +1350,33 @@ namespace spot
return recurse(res); return recurse(res);
} }
// GF(a | Xb) = GF(a | b)
// GF(a | Fb) = GF(a | b) // GF(a) = GF(dnf(a)) if a is not Boolean
if (c.is({op::F, op::Or})) // and contains some | above non-Boolean subformulas.
if (c.is(op::F) && !c[0].is_boolean())
{ {
formula m = c[0]; formula m = c[0];
if (!m.is_boolean()) bool want_dnf = m.is(op::Or);
{ if (!want_dnf && m.is(op::And))
formula out = m.map([](formula f) for (auto cc : m)
{ if (cc.is(op::Or))
if (f.is(op::X, op::F)) {
return f[0]; want_dnf = true;
return f; break;
}); }
if (out != m) if (want_dnf && !opt_.reduce_size_strictly)
return recurse(unop_unop(op::G, op::F, out)); m = c_->as_dnf(m);
} // GF(a | Xb) = GF(a | b)
// GF(a | Fb) = GF(a | b)
if (m.is(op::Or))
m = m.map([](formula f)
{
if (f.is(op::X, op::F))
return f[0];
return f;
});
if (c[0] != m)
return recurse(unop_unop(op::G, op::F, m));
} }
// GF(f1 & f2 & eu1 & eu2) = G(F(f1 & f2) & eu1 & eu2 // GF(f1 & f2 & eu1 & eu2) = G(F(f1 & f2) & eu1 & eu2
if (opt_.event_univ && c.is({op::F, op::And})) if (opt_.event_univ && c.is({op::F, op::And}))

1
tests/Makefile.am
View file

@ -208,6 +208,7 @@ TESTS_misc = \
core/trival.test core/trival.test
TESTS_twa = \ TESTS_twa = \
core/385.test \
core/acc.test \ core/acc.test \
core/acc2.test \ core/acc2.test \
core/bdddict.test \ core/bdddict.test \

45
tests/core/385.test Executable file
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@ -0,0 +1,45 @@
#!/bin/sh
# -*- coding: utf-8 -*-
# Copyright (C) 2019 Laboratoire de Recherche et Développement de
# l'Epita (LRDE).
#
# This file is part of Spot, a model checking library.
#
# Spot is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 3 of the License, or
# (at your option) any later version.
#
# Spot is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
# License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
. ./defs
set -e
# Formulas from issue #385.
ltl2tgba -G --stats=%s,%f> out <<EOF
GFX((e & XXXa) -> (e & X!d))
G(Gc | Fa | F!G(c | Ge))
Ge | XGF(Ge & X(c & Fd))
G!GXXe -> GF(b & c & Gc)
G!(Gd & (c | Fb))
XF(F(Gc & Xb) -> a)
EOF
# Some of the following are still not optimal.
cat >expected <<EOF
1,GF(!a | !d | !e)
6,G(Gc | F(a | (!c & F!e)))
2,Ge | G(Fd & FGe & Fc)
1,F(G(Fb & FGc) | Ge)
4,G(F!d | (!c & G!b))
3,F(GF!c | G!b) | XFa
EOF
diff out expected

2
tests/core/ltl2tgba2.test
View file

@ -369,7 +369,7 @@ diff output expected
cat >formulas <<EOF cat >formulas <<EOF
GF((a & XXa) | (!a & XX!a)), 4,8, 4,8, 6,14, 7,14, 4,8 GF((a & XXa) | (!a & XX!a)), 4,8, 4,8, 6,14, 7,14, 4,8
GF((a & XXXa) | (!a & XXX!a)), 7,14, 8,16, 8,18, 15,30, 8,16 GF((a & XXXa) | (!a & XXX!a)), 7,14, 8,16, 8,18, 15,30, 8,16
GF(((a & Xb) | XXc) & Xd), 4,64, 4,64, 5,80, 5,80, 4,64 GF(((a & Xb) | XXc) & Xd), 3,58, 4,64, 3,58, 5,80, 4,64
GF((b | Fa) & (b R Xb)), 2,4, 2,4, 3,6, 3,12, 2,4 GF((b | Fa) & (b R Xb)), 2,4, 2,4, 3,6, 3,12, 2,4
G(F(a & Xa) & F(a & X!a)), 2,4, 2,4, 4,8, 4,8, 3,6 G(F(a & Xa) & F(a & X!a)), 2,4, 2,4, 4,8, 4,8, 3,6
G(!p0 & F(p1 & XG!p1)), 1,0, 1,0, 1,0, 1,0, 1,0 G(!p0 & F(p1 & XG!p1)), 1,0, 1,0, 1,0, 1,0, 1,0

2
tests/core/reduccmp.test
View file

@ -346,6 +346,8 @@ F(G(a & Fb)), G(Fb & FGa)
G(F(a & Gb)), G(Fa & FGb) G(F(a & Gb)), G(Fa & FGb)
Ge | XGF(Ge & X(c & Fd)), Ge | G(Fd & FGe & Fc) Ge | XGF(Ge & X(c & Fd)), Ge | G(Fd & FGe & Fc)
G!GXXe -> GF(b & c & Gc), F(G(Fb & FGc) | Ge) G!GXXe -> GF(b & c & Gc), F(G(Fb & FGc) | Ge)
GFX((e & XXXa) -> (e & X!d)), GF(!a | !d | !e)
!GFX((e & XXXa) -> (e & X!d)), FG(a & d & e)
# Because reduccmp will translate the formula, # Because reduccmp will translate the formula,
# this also check for an old bug in ltl2tgba_fm. # this also check for an old bug in ltl2tgba_fm.

415
tests/python/automata.ipynb
View file

@ -178,7 +178,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c0b2de0> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6034a960> >"
] ]
}, },
"execution_count": 2, "execution_count": 2,
@ -643,7 +643,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c02c930> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60301210> >"
] ]
}, },
"execution_count": 6, "execution_count": 6,
@ -719,7 +719,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c02c960> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60301bd0> >"
] ]
}, },
"execution_count": 7, "execution_count": 7,
@ -802,7 +802,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c02ca80> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60301690> >"
] ]
}, },
"execution_count": 8, "execution_count": 8,
@ -1321,7 +1321,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c03ec90> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a603010c0> >"
] ]
}, },
"execution_count": 12, "execution_count": 12,
@ -1435,7 +1435,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c03edb0> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60301090> >"
] ]
}, },
"execution_count": 13, "execution_count": 13,
@ -1566,7 +1566,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c03ecf0> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60301c90> >"
] ]
}, },
"execution_count": 14, "execution_count": 14,
@ -1785,7 +1785,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1540> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6030f4b0> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -1935,7 +1935,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c02c630> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60392630> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -2107,7 +2107,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1780> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6030f930> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -2282,7 +2282,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1660> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6030f870> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -2468,7 +2468,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
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"<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=\"edge14\">\n",
"<title>3-&gt;3</title>\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",
"</g>\n", "</g>\n",
"</svg></div>" "</svg></div>"
@ -3062,7 +3121,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1e40> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6034ad20> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -3162,7 +3221,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c02c750> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a60301ba0> >"
] ]
}, },
"execution_count": 24, "execution_count": 24,
@ -3235,7 +3294,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c02c780> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6030f9c0> >"
] ]
}, },
"execution_count": 25, "execution_count": 25,
@ -3399,7 +3458,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d1c03ed80> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6030f750> >"
] ]
}, },
"execution_count": 27, "execution_count": 27,
@ -3482,7 +3541,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1e40> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6034ad20> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -3547,7 +3606,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1e40> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6034ad20> >"
] ]
}, },
"metadata": {}, "metadata": {},
@ -3634,7 +3693,7 @@
"</svg>\n" "</svg>\n"
], ],
"text/plain": [ "text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f0d077c1e40> >" "<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f5a6034ad20> >"
] ]
}, },
"execution_count": 29, "execution_count": 29,
@ -3667,7 +3726,7 @@
"name": "python", "name": "python",
"nbconvert_exporter": "python", "nbconvert_exporter": "python",
"pygments_lexer": "ipython3", "pygments_lexer": "ipython3",
"version": "3.6.7" "version": "3.7.3"
} }
}, },
"nbformat": 4, "nbformat": 4,

4144
tests/python/twagraph-internals.ipynb
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