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python: improve handling of formulas

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* src/misc/escape.hh, src/misc/escape.cc (escape_latex): New function.
* src/ltlvisit/tostring.cc: Escape atomic proposition in LaTeX output.
* wrap/python/spot.py: Make it easy to output formulas in different
syntaxes.  Also allow the AST to be shown.
* wrap/python/spot_impl.i: Catch std::runtime_error.
* wrap/python/tests/formulas.ipynb: New file.
* wrap/python/tests/Makefile.am: Add it.
This commit is contained in:
Alexandre Duret-Lutz 2015-03-11 19:41:24 +01:00
parent a6dbf5cf5e
commit 2362b9ab68
7 changed files with 667 additions and 35 deletions
Download patch file Download diff file Expand all files Collapse all files

10
src/ltlvisit/tostring.cc
View file

@ -30,7 +30,7 @@
#include "lunabbrev.hh" #include "lunabbrev.hh"
#include "wmunabbrev.hh" #include "wmunabbrev.hh"
#include "tostring.hh" #include "tostring.hh"
#include "misc/escape.hh"
namespace spot namespace spot
{ {
@ -416,7 +416,9 @@ namespace spot
{ {
// Spin 6 supports atomic propositions such as (a == 0) // Spin 6 supports atomic propositions such as (a == 0)
// as long as they are enclosed in parentheses. // as long as they are enclosed in parentheses.
if (kw_ != spin_kw) if (kw_ == sclatex_kw || kw_ == sclatex_kw)
escape_latex(os_ << "``\\mathit{", str) << "\\textrm{''}}";
else if (kw_ != spin_kw)
os_ << '"' << str << '"'; os_ << '"' << str << '"';
else if (!full_parent_) else if (!full_parent_)
os_ << '(' << str << ')'; os_ << '(' << str << ')';
@ -431,11 +433,11 @@ namespace spot
while (str[s - 1] >= '0' && str[s - 1] <= '9') while (str[s - 1] >= '0' && str[s - 1] <= '9')
{ {
--s; --s;
assert(s != 0); // bare words cannot start with letters assert(s != 0); // bare words cannot start with digits
} }
if (s > 1) if (s > 1)
os_ << "\\mathit{"; os_ << "\\mathit{";
os_ << str.substr(0, s); escape_latex(os_, str.substr(0, s));
if (s > 1) if (s > 1)
os_ << '}'; os_ << '}';
if (s != str.size()) if (s != str.size())

44
src/misc/escape.cc
View file

@ -1,5 +1,5 @@
// -*- coding: utf-8 -*- // -*- coding: utf-8 -*-
// Copyright (C) 2012, 2013 Laboratoire de Recherche et Developpement de // Copyright (C) 2012, 2013, 2015 Laboratoire de Recherche et Developpement de
// l'Epita (LRDE) // l'Epita (LRDE)
// Copyright (C) 2004 Laboratoire d'Informatique de Paris 6 (LIP6), // Copyright (C) 2004 Laboratoire d'Informatique de Paris 6 (LIP6),
// département Systèmes Répartis Coopératifs (SRC), Université Pierre // département Systèmes Répartis Coopératifs (SRC), Université Pierre
@ -34,14 +34,44 @@ namespace spot
std::ostream& std::ostream&
escape_rfc4180(std::ostream& os, const std::string& str) escape_rfc4180(std::ostream& os, const std::string& str)
{ {
for (std::string::const_iterator i = str.begin(); i != str.end(); ++i) for (auto i: str)
switch (*i) switch (i)
{ {
case '"': case '"':
os << "\"\""; os << "\"\"";
break; break;
default: default:
os << *i; os << i;
break;
}
return os;
}
std::ostream&
escape_latex(std::ostream& os, const std::string& str)
{
for (auto i: str)
switch (i)
{
case '~':
os << "\\textasciitilde";
break;
case '^':
os << "\\textasciicircum";
break;
case '\\':
os << "\\textbackslash";
break;
case '&':
case '%':
case '$':
case '#':
case '_':
case '{':
case '}':
os << '\\';
default:
os << i;
break; break;
} }
return os; return os;
@ -50,8 +80,8 @@ namespace spot
std::ostream& std::ostream&
escape_str(std::ostream& os, const std::string& str) escape_str(std::ostream& os, const std::string& str)
{ {
for (std::string::const_iterator i = str.begin(); i != str.end(); ++i) for (auto i: str)
switch (*i) switch (i)
{ {
case '\\': case '\\':
os << "\\\\"; os << "\\\\";
@ -63,7 +93,7 @@ namespace spot
os << "\\n"; os << "\\n";
break; break;
default: default:
os << *i; os << i;
break; break;
} }
return os; return os;

9
src/misc/escape.hh
View file

@ -1,5 +1,5 @@
// -*- coding: utf-8 -*- // -*- coding: utf-8 -*-
// Copyright (C) 2011, 2012, 2013 Laboratoire de Recherche et // Copyright (C) 2011, 2012, 2013, 2015 Laboratoire de Recherche et
// Developpement de l'Epita (LRDE). // Developpement de l'Epita (LRDE).
// Copyright (C) 2004 Laboratoire d'Informatique de Paris 6 (LIP6), // Copyright (C) 2004 Laboratoire d'Informatique de Paris 6 (LIP6),
// département Systèmes Répartis Coopératifs (SRC), Université Pierre // département Systèmes Répartis Coopératifs (SRC), Université Pierre
@ -39,6 +39,13 @@ namespace spot
SPOT_API std::ostream& SPOT_API std::ostream&
escape_rfc4180(std::ostream& os, const std::string& str); escape_rfc4180(std::ostream& os, const std::string& str);
/// \brief Escape special LaTeX characters.
///
/// The following characters are rewritten:
/// <code>& % $ # _ { } ~ ^ \\</code>
SPOT_API std::ostream&
escape_latex(std::ostream& os, const std::string& str);
/// \brief Escape characters <code>"</code>, <code>\\</code>, and /// \brief Escape characters <code>"</code>, <code>\\</code>, and
/// <code>\\n</code> in \a str. /// <code>\\n</code> in \a str.
SPOT_API std::ostream& SPOT_API std::ostream&

69
wrap/python/spot.py
View file

@ -1,5 +1,5 @@
# -*- coding: utf-8 -*- # -*- coding: utf-8 -*-
# Copyright (C) 2014 Laboratoire de # Copyright (C) 2014, 2015 Laboratoire de
# Recherche et Développement de l'Epita (LRDE). # Recherche et Développement de l'Epita (LRDE).
# #
# This file is part of Spot, a model checking library. # This file is part of Spot, a model checking library.
@ -23,27 +23,59 @@ import sys
_bdd_dict = make_bdd_dict() _bdd_dict = make_bdd_dict()
def render_automaton_as_svg(a): def _ostream_to_svg(ostr):
dotsrc = ostringstream()
dotty_reachable(dotsrc, a)
dotty = subprocess.Popen(['dot', '-Tsvg'], dotty = subprocess.Popen(['dot', '-Tsvg'],
stdin=subprocess.PIPE, stdin=subprocess.PIPE,
stdout=subprocess.PIPE) stdout=subprocess.PIPE)
dotty.stdin.write(dotsrc.str().encode('utf-8')) dotty.stdin.write(ostr.str().encode('utf-8'))
res = dotty.communicate() res = dotty.communicate()
return res[0].decode('utf-8') return res[0].decode('utf-8')
tgba._repr_svg_ = render_automaton_as_svg def _render_automaton_as_svg(a):
ostr = ostringstream()
dotty_reachable(ostr, a)
return _ostream_to_svg(ostr)
def formula_str_ctor(self, str): tgba._repr_svg_ = _render_automaton_as_svg
def _render_formula_as_svg(a):
# Load the SVG function only if we need it. This way the bindings
# can still be used outside of IPython if IPython is not
# installed.
from IPython.display import SVG
ostr = ostringstream()
dotty(ostr, a)
return SVG(_ostream_to_svg(ostr))
def _formula_str_ctor(self, str):
self.this = parse_formula(str) self.this = parse_formula(str)
formula.__init__ = formula_str_ctor def _formula_to_str(self, format = 'spot'):
if format == 'spot':
return to_string(self)
elif format == 'spin':
return to_spin_string(self)
elif format == 'utf8':
return to_utf8_string(self)
elif format == 'lbt':
return to_lbt_string(self)
elif format == 'wring':
return to_wring_string(self)
elif format == 'latex':
return to_latex_string(self)
elif format == 'sclatex':
return to_sclatex_string(self)
else:
raise ValueError("unknown string format: " + format)
def tgba_str_ctor(self, str): formula.__init__ = _formula_str_ctor
formula.to_str = _formula_to_str
formula.show_ast = _render_formula_as_svg
def _tgba_str_ctor(self, str):
self.this = ltl_to_tgba_fm(parse_formula(str), _bdd_dict) self.this = ltl_to_tgba_fm(parse_formula(str), _bdd_dict)
tgba.__init__ = tgba_str_ctor tgba.__init__ = _tgba_str_ctor
# Wrapper around a formula iterator to which we add some methods of formula # Wrapper around a formula iterator to which we add some methods of formula
# (using _addfilter and _addmap), so that we can write things like # (using _addfilter and _addmap), so that we can write things like
@ -211,15 +243,20 @@ def simplify(f, **kwargs):
boolean_to_isop = kwargs.get('boolean_to_isop', False) boolean_to_isop = kwargs.get('boolean_to_isop', False)
favor_event_univ = kwargs.get('favor_event_univ', False) favor_event_univ = kwargs.get('favor_event_univ', False)
simp_opts = ltl_simplifier_options(basics, synt_impl, event_univ, simp_opts = ltl_simplifier_options(basics,
containment_checks, containment_checks_stronger, nenoform_stop_on_boolean, synt_impl,
reduce_size_strictly, boolean_to_isop, favor_event_univ) event_univ,
containment_checks,
containment_checks_stronger,
nenoform_stop_on_boolean,
reduce_size_strictly,
boolean_to_isop,
favor_event_univ)
return ltl_simplifier(simp_opts).simplify(f) return ltl_simplifier(simp_opts).simplify(f)
for fun in dir(formula): for fun in dir(formula):
if callable(getattr(formula, fun)) and fun[:3] == 'is_' \ if (callable(getattr(formula, fun)) and
or fun[:4] == 'has_': (fun[:3] == 'is_' or fun[:4] == 'has_')):
_addfilter(fun) _addfilter(fun)
for fun in ['remove_x', 'get_literal', 'relabel', 'relabel_bse', for fun in ['remove_x', 'get_literal', 'relabel', 'relabel_bse',

8
wrap/python/spot_impl.i
View file

@ -190,9 +190,11 @@ using namespace spot;
} }
catch (const std::invalid_argument& e) catch (const std::invalid_argument& e)
{ {
std::string er("\n"); SWIG_exception(SWIG_ValueError, e.what());
er += e.what(); }
SWIG_exception(SWIG_ValueError, er.c_str()); catch (const std::runtime_error& e)
{
SWIG_exception(SWIG_RuntimeError, e.what());
} }
} }

8
wrap/python/tests/Makefile.am
View file

@ -1,7 +1,8 @@
## Copyright (C) 2010, 2012, 2013, 2014 Labortatoire de Recherche et Développement de ## -*- coding: utf-8 -*-
## l'EPITA. ## Copyright (C) 2010, 2012, 2013, 2014, 2015 Labortatoire de
## Recherche et Développement de l'EPITA.
## Copyright (C) 2003, 2004, 2005 Laboratoire d'Informatique de Paris 6 (LIP6), ## Copyright (C) 2003, 2004, 2005 Laboratoire d'Informatique de Paris 6 (LIP6),
## département Systèmes Répartis Coopératifs (SRC), Université Pierre ## département Systèmes Répartis Coopératifs (SRC), Université Pierre
## et Marie Curie. ## et Marie Curie.
## ##
## This file is part of Spot, a model checking library. ## This file is part of Spot, a model checking library.
@ -32,6 +33,7 @@ check_SCRIPTS = run
TESTS = \ TESTS = \
alarm.py \ alarm.py \
bddnqueen.py \ bddnqueen.py \
formulas.ipynb \
implies.py \ implies.py \
interdep.py \ interdep.py \
ltl2tgba.test \ ltl2tgba.test \

552
wrap/python/tests/formulas.ipynb Normal file
View file

@ -0,0 +1,552 @@
{
"metadata": {
"name": "",
"signature": "sha256:c5ce337080296dc7af1d530e86502099bdfb3f039b96f749f38390b8d5ade44b"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Handling LTL and PSL formulas\n",
"============================="
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import spot"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For interactive use, formulas can be entered as text strings and passed to the `spot.formula` constructor."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"f = spot.formula('p1 U p2 R (p3 & !p4)')\n",
"f"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{R}} (p_{3} \\land \\lnot p_{4}))$"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 2,
"text": [
"p1 U (p2 R (p3 & !p4))"
]
}
],
"prompt_number": 2
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"g = spot.formula('{a;b*;c[+]}<>->GFb'); g"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$\\{a \\mathbin{\\mathsf{;}} b^{\\star} \\mathbin{\\mathsf{;}} c^+\\}\\mathrel{\\Diamond\\kern-1.7pt\\raise.4pt\\hbox{$\\mathord{\\rightarrow}$}} \\mathsf{G} \\mathsf{F} b$"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 3,
"text": [
"{a;b[*];c[+]}<>-> GFb"
]
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"By default the parser recognizes an infix syntax, but when this fails, it tries to read the formula with the [LBT](http://www.tcs.hut.fi/Software/maria/tools/lbt/) syntax:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"h = spot.formula('& | a b c'); h"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$c \\land (a \\lor b)$"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 4,
"text": [
"c & (a | b)"
]
}
],
"prompt_number": 4
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"By default, a formula object is presented using mathjax as above.\n",
"When a formula is converted to string you get Spot's syntax by default:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"str(f)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 5,
"text": [
"'p1 U (p2 R (p3 & !p4))'"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If you prefer to print the string in another syntax, you may use the `to_str()` method, with an argument that indicates the output format to use. The `latex` format assumes that you will the define macros such as `\\U`, `\\R` to render all operators as you wish. On the otherhand, the `sclatex` (with `sc` for self-contained) format hard-codes the rendering of each of those operators: this is typically the output that is used to render formulas using MathJax in a notebook. "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"for i in ['spot', 'spin', 'lbt', 'wring', 'utf8', 'latex', 'sclatex']:\n",
" print(\"%-10s%s\" % (i, f.to_str(i)))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"spot p1 U (p2 R (p3 & !p4))\n",
"spin p1 U (p2 V (p3 && !p4))\n",
"lbt U p1 V p2 & p3 ! p4\n",
"wring (p1=1) U ((p2=1) R ((p3=1) * (p4=0)))\n",
"utf8 p1 U (p2 R (p3\u2227\u00acp4))\n",
"latex p_{1} \\U (p_{2} \\R (p_{3} \\land \\lnot p_{4}))\n",
"sclatex p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{R}} (p_{3} \\land \\lnot p_{4}))\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A `spot.formula` object has a number of built-in predicates whose value have been computed when the formula was constructed. For instance you can check whether a formula is in negative normal form using `is_in_nenoform()`, and you can make sure it is an LTL formula (i.e. not a PSL formula) using `is_ltl_formula()`:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"f.is_in_nenoform() and f.is_ltl_formula()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 7,
"text": [
"True"
]
}
],
"prompt_number": 7
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"g.is_ltl_formula()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 8,
"text": [
"False"
]
}
],
"prompt_number": 8
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Similarly, `is_syntactic_stutter_invariant()` tells wether the structure of the formula guarranties it to be stutter invariant. For LTL formula, this means the `X` operator should not be used. For PSL formula, this function capture all formulas built using the [siPSL grammar](http://www.daxc.de/eth/paper/09atva.pdf)."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"f.is_syntactic_stutter_invariant()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 9,
"text": [
"True"
]
}
],
"prompt_number": 9
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spot.formula('{a[*];b}<>->c').is_syntactic_stutter_invariant()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 10,
"text": [
"False"
]
}
],
"prompt_number": 10
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spot.formula('{a[+];b[*]}<>->d').is_syntactic_stutter_invariant()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 11,
"text": [
"True"
]
}
],
"prompt_number": 11
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`spot.relabel` renames the atomic propositions that occur in a formula, using either letters, or numbered propositions:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"gf = spot.formula('(GF_foo_) && \"a > b\" && \"proc[2]@init\"'); gf"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$``\\mathit{a > b\\textrm{''}} \\land ``\\mathit{proc[2]@init\\textrm{''}} \\land \\mathsf{G} \\mathsf{F} \\mathit{\\_foo\\_}$"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 12,
"text": [
"\"a > b\" & \"proc[2]@init\" & GF_foo_"
]
}
],
"prompt_number": 12
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spot.relabel(gf, spot.Abc)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$a \\land b \\land \\mathsf{G} \\mathsf{F} c$"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 13,
"text": [
"a & b & GFc"
]
}
],
"prompt_number": 13
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spot.relabel(gf, spot.Pnn)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$p_{0} \\land p_{1} \\land \\mathsf{G} \\mathsf{F} p_{2}$"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 14,
"text": [
"p0 & p1 & GFp2"
]
}
],
"prompt_number": 14
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The AST of any formula can be displayed with `show_ast()`. Despite the name, this is not a tree but a DAG, because identical subtrees are merged. Binary operators have their left and right operands denoted with `L` and `R`, while non-commutative n-ary operators have their operands numbered."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"print(g); g.show_ast()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"{a;b[*];c[+]}<>-> GFb\n"
]
},
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 15,
"svg": [
"<svg height=\"260pt\" viewBox=\"0.00 0.00 269.00 260.00\" width=\"269pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g class=\"graph\" id=\"graph0\" transform=\"scale(1 1) rotate(0) translate(4 256)\">\n",
"<title>G</title>\n",
"<polygon fill=\"white\" points=\"-4,4 -4,-256 265,-256 265,4 -4,4\" stroke=\"none\"/>\n",
"<!-- 0 -->\n",
"<g class=\"node\" id=\"node1\"><title>0</title>\n",
"<ellipse cx=\"106\" cy=\"-234\" fill=\"none\" rx=\"40.8928\" ry=\"18\" stroke=\"black\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"106\" y=\"-230.3\">EConcat</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g class=\"node\" id=\"node2\"><title>1</title>\n",
"<ellipse cx=\"155\" cy=\"-162\" fill=\"none\" rx=\"35.9954\" ry=\"18\" stroke=\"black\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"155\" y=\"-158.3\">Concat</text>\n",
"</g>\n",
"<!-- 0&#45;&gt;1 -->\n",
"<g class=\"edge\" id=\"edge1\"><title>0-&gt;1</title>\n",
"<path d=\"M117.612,-216.411C123.593,-207.868 131.005,-197.278 137.649,-187.787\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"140.604,-189.669 143.471,-179.47 134.869,-185.655 140.604,-189.669\" stroke=\"black\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"113.112\" y=\"-205.211\">L</text>\n",
"</g>\n",
"<!-- 7 -->\n",
"<g class=\"node\" id=\"node8\"><title>7</title>\n",
"<ellipse cx=\"58\" cy=\"-162\" fill=\"none\" rx=\"27\" ry=\"18\" stroke=\"black\"/>\n",
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"input": [
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],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$\\mathsf{F} (a \\land \\mathsf{X} (\\lnot a \\land b))$"
],
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"output_type": "pyout",
"prompt_number": 16,
"text": [
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]
}
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},
{
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"input": [
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],
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{
"latex": [
"$\\mathsf{F} (a \\land ((a \\land (a \\mathbin{\\mathsf{U}} (\\lnot a \\land b)) \\land ((\\lnot b \\mathbin{\\mathsf{U}} \\lnot a) \\lor (b \\mathbin{\\mathsf{U}} \\lnot a))) \\lor (\\lnot a \\land (\\lnot a \\mathbin{\\mathsf{U}} (a \\land \\lnot a \\land b)) \\land ((\\lnot b \\mathbin{\\mathsf{U}} a) \\lor (b \\mathbin{\\mathsf{U}} a))) \\lor (b \\land (b \\mathbin{\\mathsf{U}} (\\lnot a \\land b \\land \\lnot b)) \\land ((\\lnot a \\mathbin{\\mathsf{U}} \\lnot b) \\lor (a \\mathbin{\\mathsf{U}} \\lnot b))) \\lor (\\lnot b \\land (\\lnot b \\mathbin{\\mathsf{U}} (\\lnot a \\land b)) \\land ((\\lnot a \\mathbin{\\mathsf{U}} b) \\lor (a \\mathbin{\\mathsf{U}} b))) \\lor (\\lnot a \\land b \\land (\\mathsf{G} \\lnot a \\lor \\mathsf{G} a) \\land (\\mathsf{G} \\lnot b \\lor \\mathsf{G} b))))$"
],
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"F(a & ((a & (a U (!a & b)) & ((!b U !a) | (b U !a))) | (!a & (!a U (a & !a & b)) & ((!b U a) | (b U a))) | (b & (b U (!a & b & !b)) & ((!a U !b) | (a U !b))) | (!b & (!b U (!a & b)) & ((!a U b) | (a U b))) | (!a & b & (G!a | Ga) & (G!b | Gb))))"
]
}
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