org: document SAT-based minimization
* doc/org/satmin.org, doc/org/satmin.tex: New files. * doc/Makefile.am: Add them. * doc/org/tools.org: Point to satmin.org. * NEWS: Mention satmin.html.
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Alexandre Duret-Lutz
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doc/org/satmin.org
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#+TITLE: SAT-based Minimization of Deterministic (Generalized) Büchi Automata
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#+EMAIL spot@lrde.epita.fr
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#+OPTIONS: H:2 num:nil toc:t
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#+LINK_UP: file:tools.html
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This page explains how to use [[file:ltl2tgba.org][=ltl2tgba=]] or [[file:dstar2tgba.org][=dstar2tgba=]] to minimize
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deterministic automata using a SAT solver.
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Let us first state a few facts about this minimization procedure.
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1) The procedure works only on *deterministic* Büchi automata: any
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recurrence property can be converted into a deterministic Büchi
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automaton, and sometimes there are several ways of doing so.
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2) Spot actually implement two SAT-based minimization procedures: one
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that builds a deterministic transition-based Büchi automaton
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(DTBA), and one the builds a deterministic transition-based
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generalized Büchi automaton (DTGBA). For the latter, we can supply
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the number $m$ of acceptance sets to use.
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3) These two procedures can optionally constrain their output to
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use state-based acceptance. (They simply restrict all the outgoing
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transitions of a state to belong to the same acceptance sets.)
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4) A SAT solver should be installed for this to work. (Spot does not
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distribute any SAT solver.)
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5) [[file:ltl2tgba.org][=ltl2tgba=]] and [[file:dstar2tgba.org][=dstar2tgba=]] will always try to output an automaton
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If they fail to determinize the property, they will simply output a
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nondeterministic automaton, if they managed to obtain a
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deterministic automaton but failed to minimize it (e.g., the
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requested number of states in the final automaton is too low), they
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will return that "unminimized" deterministic automaton. There are
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only two cases where these tool will abort without returning an
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automaton: when the number of clauses output by Spot (and to be fed
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to the SAT solver) exceeds $2^{31}$, or when the SAT-solver was
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killed by a signal.
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* How change the SAT solver used
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The environment variable =SPOT_SATSOLVER= can be used to change the
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SAT solver used by Spot. The default is "=glucose %I >%O=", therefore
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if you have installed [[https://www.lri.fr/~simon/?page=glucose][=glucose=]] in your =$PATH=, it should work right
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away. Otherwise you may redefine this variable to point the correct
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location or to another SAT solver. The =%I= and =%O= sequences will be
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replaced by the names of temporary files containing the input for the
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SAT solver and receiving its output. We assume that the SAT solver
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should follow the conventions of the [[http://www.satcompetition.org/][SAT competition]] for input and
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output.
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* Enabling SAT-based minimization for deterministic automata
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Both tools follow the same interface, because they use the same
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post-processing steps internally (i.e., the =spot::postprocessor=
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class).
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First, option =-D= should be used to declare that you are looking for
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more determinism. This will tweak the translation algorithm used by
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=ltl2tgba= to improve determinism, and will also instruct the
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post-processing routine used by both tools to prefer a
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deterministic automaton over a smaller equivalent nondeterministic
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automaton.
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However =-D= is not a guarantee to obtain a deterministic automaton,
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even if one exists. For instance, =-D= fails to produce a
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deterministic automaton for =GF(a <-> XXb)=. Instead we get a 9-state
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non-deterministic automaton.
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#+BEGIN_SRC sh :results verbatim :exports both
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ltl2tgba -D "GF(a <-> XXb)" --stats='states=%s, det=%d'
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#+END_SRC
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#+RESULTS:
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: states=9, det=0
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Option =-x tba-det= enables an additional
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determinization procedure, that would otherwise not be used by =-D=
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alone. This procedure will work on any automaton that can be
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represented by a DTBA; if the automaton to process use multiple
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acceptance conditions, it will be degeneralized first.
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On our example, =-x tba-det= successfully produces a deterministic
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TBA, but a non-minimal one:
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#+BEGIN_SRC sh :results verbatim :exports both
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ltl2tgba -D -x tba-det "GF(a <-> XXb)" --stats='states=%s, det=%d'
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#+END_SRC
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#+RESULTS:
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: states=7, det=1
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Option =-x sat-minimize= will turn-on SAT-based minimization. It also
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implies =-x tba-det=, so there is no need to supply both options.
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#+BEGIN_SRC sh :results verbatim :exports both
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ltl2tgba -D -x sat-minimize "GF(a <-> XXb)" --stats='states=%s, det=%d'
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#+END_SRC
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#+RESULTS:
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: states=4, det=1
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We can draw it:
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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba -D -x sat-minimize "GF(a <-> XXb)"
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#+END_SRC
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#+RESULTS:
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#+begin_example
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digraph G {
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0 [label="", style=invis, height=0]
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0 -> 1
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1 [label="1"]
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1 -> 1 [label="a & !b\n"]
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1 -> 2 [label="!b & !a\n"]
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1 -> 2 [label="b & !a\n{Acc[1]}"]
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1 -> 3 [label="a & b\n{Acc[1]}"]
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2 [label="2"]
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2 -> 4 [label="!b & !a\n"]
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2 -> 4 [label="b & !a\n{Acc[1]}"]
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2 -> 3 [label="a & !b\n"]
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2 -> 3 [label="a & b\n{Acc[1]}"]
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3 [label="4"]
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3 -> 1 [label="a & !b\n{Acc[1]}"]
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3 -> 1 [label="a & b\n"]
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3 -> 2 [label="!b & !a\n{Acc[1]}"]
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3 -> 2 [label="b & !a\n"]
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4 [label="3"]
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4 -> 2 [label="!b & !a\n{Acc[1]}"]
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4 -> 4 [label="b & !a\n"]
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4 -> 3 [label="a & !b\n{Acc[1]}"]
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4 -> 3 [label="a & b\n"]
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}
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#+end_example
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#+NAME: gfaexxb3
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#+BEGIN_SRC sh :results verbatim :exports none
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ltl2tgba -D -x sat-minimize "GF(a <-> XXb)" | sed 's/\\/\\\\/'
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#+END_SRC
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#+RESULTS: gfaexxb3
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#+begin_example
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digraph G {
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0 [label="", style=invis, height=0]
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0 -> 1
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1 [label="1"]
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1 -> 1 [label="a & !b\\n"]
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1 -> 1 [label="a & b\\n{Acc[1]}"]
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1 -> 2 [label="!b & !a\\n"]
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1 -> 2 [label="b & !a\\n{Acc[1]}"]
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2 [label="2"]
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2 -> 3 [label="!b & !a\\n"]
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2 -> 3 [label="b & !a\\n{Acc[1]}"]
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2 -> 4 [label="a & !b\\n"]
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2 -> 4 [label="a & b\\n{Acc[1]}"]
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3 [label="3"]
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3 -> 1 [label="a & !b\\n{Acc[1]}"]
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3 -> 3 [label="b & !a\\n"]
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3 -> 4 [label="!b & !a\\n{Acc[1]}"]
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3 -> 4 [label="a & b\\n"]
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4 [label="4"]
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4 -> 1 [label="a & !b\\n{Acc[1]}"]
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4 -> 1 [label="a & b\\n"]
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4 -> 2 [label="!b & !a\\n{Acc[1]}"]
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4 -> 2 [label="b & !a\\n"]
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}
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#+end_example
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#+BEGIN_SRC dot :file gfaexxb3.png :cmdline -Tpng :var txt=gfaexxb3 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:gfaexxb3.png]]
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Clearly this is automaton benefit from the transition-based
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acceptance. If we want a traditional Büchi automaton, with
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state-based acceptance, we only need to add the =-B= option. The
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result will of course be slightly bigger.
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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba -BD -x sat-minimize "GF(a <-> XXb)"
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#+END_SRC
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#+RESULTS:
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#+begin_example
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digraph G {
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0 [label="", style=invis, height=0]
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0 -> 1
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1 [label="1", peripheries=2]
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1 -> 2 [label="!a\n{Acc[1]}"]
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1 -> 3 [label="a & !b\n{Acc[1]}"]
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1 -> 4 [label="a & b\n{Acc[1]}"]
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2 [label="2", peripheries=2]
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2 -> 1 [label="!b & !a\n{Acc[1]}"]
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2 -> 4 [label="a\n{Acc[1]}"]
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2 -> 5 [label="b & !a\n{Acc[1]}"]
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3 [label="4"]
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3 -> 1 [label="a & b\n"]
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3 -> 2 [label="b & !a\n"]
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3 -> 3 [label="a & !b\n"]
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3 -> 6 [label="!b & !a\n"]
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4 [label="3"]
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4 -> 1 [label="!b\n"]
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4 -> 3 [label="a & b\n"]
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4 -> 6 [label="b & !a\n"]
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5 [label="6"]
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5 -> 1 [label="!b\n"]
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5 -> 4 [label="a & b\n"]
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5 -> 5 [label="b & !a\n"]
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6 [label="5"]
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6 -> 1 [label="a & b\n"]
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6 -> 2 [label="b & !a\n"]
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6 -> 4 [label="a & !b\n"]
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6 -> 5 [label="!b & !a\n"]
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}
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#+end_example
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#+NAME: gfaexxb4
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#+BEGIN_SRC sh :results verbatim :exports none
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ltl2tgba -BD -x sat-minimize "GF(a <-> XXb)" | sed 's/\\/\\\\/'
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#+END_SRC
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#+RESULTS: gfaexxb4
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#+begin_example
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digraph G {
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0 [label="", style=invis, height=0]
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0 -> 1
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1 [label="1", peripheries=2]
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1 -> 1 [label="!b & !a\\n{Acc[1]}"]
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1 -> 2 [label="b & !a\\n{Acc[1]}"]
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1 -> 3 [label="a\\n{Acc[1]}"]
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2 [label="2"]
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2 -> 1 [label="!b & !a\\n"]
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2 -> 2 [label="b & !a\\n"]
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2 -> 3 [label="a & !b\\n"]
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2 -> 4 [label="a & b\\n"]
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3 [label="3", peripheries=2]
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3 -> 5 [label="!a\\n{Acc[1]}"]
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3 -> 6 [label="a\\n{Acc[1]}"]
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4 [label="5"]
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4 -> 1 [label="!b & !a\\n"]
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4 -> 5 [label="b & !a\\n"]
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4 -> 3 [label="a & !b\\n"]
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4 -> 6 [label="a & b\\n"]
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5 [label="4"]
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5 -> 1 [label="b & !a\\n"]
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5 -> 2 [label="!b & !a\\n"]
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5 -> 3 [label="a & b\\n"]
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5 -> 4 [label="a & !b\\n"]
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6 [label="6"]
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6 -> 1 [label="b & !a\\n"]
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6 -> 5 [label="!b & !a\\n"]
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6 -> 3 [label="a & b\\n"]
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6 -> 6 [label="a & !b\\n"]
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}
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#+end_example
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#+BEGIN_SRC dot :file gfaexxb4.png :cmdline -Tpng :var txt=gfaexxb4 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:gfaexxb4.png]]
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There are cases where =ltl2tgba='s =tba-det= algorithm fails to produce a deterministic automaton.
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In that case, SAT-based minimization is simply skipped. For instance:
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#+BEGIN_SRC sh :results verbatim :exports both
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ltl2tgba -D -x sat-minimize "Ga R (F!b & (c U b))" --stats='states=%s, det=%d'
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#+END_SRC
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#+RESULTS:
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: states=4, det=0
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The question, of course, is whether there exist a deterministic
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automaton for this formula, in other words: is this a recurrence
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property? There are two ways to answer this question using Spot (and
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some help from [[http://www.ltl2dstar.de/][=ltl2dstar=]]).
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The first is purely syntactic. If a formula belongs to the class of
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"syntactic recurrence formulas", it expresses a syntactic property.
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(Of course there are formulas that expresses a syntactic properties
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without being syntactic recurrences.) [[file:ltlfilt.org][=ltlfilt=]] can be instructed to
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print only formulas that are syntactic recurrences:
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#+BEGIN_SRC sh :results verbatim :exports both
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ltlfilt --syntactic-recurrence -f "Ga R (F!b & (c U b))"
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#+END_SRC
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#+RESULTS:
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: Ga R (F!b & (c U b))
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Since our input formula was output, it expresses a recurrence property.
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The second way to check whether a formula is a recurrence is by
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converting a deterministic Rabin automaton using [[file:dstar2tgba.org][=dstar2tgba=]]. The
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output is guaranteed to be deterministic if and only if the input DRA
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expresses a recurrence property.
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#+BEGIN_SRC sh :results verbatim :exports both
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ltlfilt -f "Ga R (F!b & (c U b))" -l |
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ltl2dstar --ltl2nba=spin:../../src/bin/ltl2tgba@-Ds - - |
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dstar2tgba -D --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
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#+END_SRC
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#+RESULTS:
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: input(states=11) output(states=9, acc-sets=1, det=1)
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In the above command, =ltlfilt= is used to convert the LTL formula
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into =ltl2dstar='s syntax. Then =ltl2dstar= creates a deterministic
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Rabin automaton (using =ltl2tgba= as an LTL to BA translator), and the
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resulting 11-state DRA is converted into a 9-state DTBA by
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=dstar2tgba=. Since that result is deterministic, we can conclude
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that the formula was a recurrence.
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As far as SAT-based minimization goes, =dstar2tgba= will take the same
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options as =ltl2tgba=, so we for instance check that the smallest DTBA
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has 6 states:
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#+BEGIN_SRC sh :results verbatim :exports both
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ltlfilt -f "Ga R (F!b & (c U b))" -l |
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ltl2dstar --ltl2nba=spin:../../src/bin/ltl2tgba@-Ds - - |
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dstar2tgba -D -x sat-minimize --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
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#+END_SRC
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#+RESULTS:
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: input(states=11) output(states=6, acc-sets=1, det=1)
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* More acceptance sets
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The formula "=Ga R (F!b & (c U b))=" can in fact be minimized into an
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even smaller automaton if we use multiple acceptance sets.
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Unfortunately because =dstar2tgba= does not know the formula being
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translated, and it always convert a DRA into a DBA (with a single
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acceptance set) before further processing, it does not know if using
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more acceptance sets could be useful to further minimize it. This
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number of acceptance sets can however be specified on the command-line
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with option =-x sat-acc=M=. For instance:
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#+BEGIN_SRC sh :results verbatim :exports both
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ltlfilt -f "Ga R (F!b & (c U b))" -l |
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ltl2dstar --ltl2nba=spin:../../src/bin/ltl2tgba@-Ds - - |
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dstar2tgba -D -x sat-minimize,sat-acc=2 --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
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#+END_SRC
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#+RESULTS:
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: input(states=11) output(states=5, acc-sets=2, det=1)
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Beware that the size of the SAT problem is exponential in the number of acceptance sets.
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The case of =ltl2tgba= is slightly different because it can remember
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the number of acceptance sets used by the translation algorithm, and
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reuse that for SAT-minimization even if the automaton had to be
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degeneralized in the meantime for the purpose of determinization.
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* Low-level details
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The following figure gives an overview of the processing chains that
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can be used to turn an LTL formula into a minimal DBA/DTBA/DTGBA. The
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blue area at the top describes =ltl2tgba -D -x sat-minimize=, while
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the purple area at the bottom corresponds to =dstar2tgba -D -x
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stat-minimize=.
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[[file:satmin.png]]
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The picture is slightly inaccurate in the sense that both =ltl2tgba=
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and =dstar2tgba= are actually using the same post-processing chain:
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only the initial translation to TGBA or conversion to DBA differs, the
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rest is the same. However in the case of =dstar2tgba=, no
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degeneration or determinization are needed.
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Also the picture does not show what happens when =-B= is used: any
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DTBA is degeneralized into a DBA, before being sent to "DTBA SAT
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minimization", with a special option to request state-based output.
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The WDBA-minimization boxes are able to produce minimal Weak DBA from
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any TGBA representing an obligation property. In that case using
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transition-based or generalized acceptance will not allow further
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reduction. This minimal WDBA is always used when =-D= is given
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(otherwise, for the default =--small= option, the minimal WDBA is only
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used if it is smaller than the nondeterministic automaton it has been
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built from).
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The "simplify" boxes are actually simulation-based reductions, and
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SCC-based simplifications.
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The red boxes "not in TCONG" or "not a recurrence" correspond to
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situations where the tools will produce non-deterministic automata.
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The following options can be used to fine-tune this procedure:
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- =-x tba-det= :: attempt a powerset construction and check if
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there exists a acceptance set such that the
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resulting DTBA is equivalent to the input
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- =-x sat-minimize= :: enable SAT-based minimization. By default it
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tries to reduce the size of the automaton one state at a time.
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This option implies =-x tba-det=.
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- =-x sat-minimize=2= :: enabled SAT-based minimization, but perform a
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dichotomy to locate the correct automaton size. Use this only if
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you suspect that the optimal size is far away from the input
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size. This option implies =-x tba-det=.
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- =-x sat-acc=$m$= :: attempt to build a minimal DTGBA with $m$ acceptance sets.
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This options implies =-x sat-minimize=.
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- =-x sat-states=$n$= :: attempt to build an equivalent DTGBA with $n$
|
||||
states. This also implies =-x sat-minimize= but won't perform
|
||||
any loop to lower the number of states. Note that $n$ should be
|
||||
the number of states in a complete automaton, while =ltl2tgba=
|
||||
and =dstar2tgba= both remove sink states in their output by
|
||||
default (use option =--complete=) to output a complete automaton.
|
||||
Also note that even with the =--complete= option, the output
|
||||
automaton may have appear to have less states because the other
|
||||
are unreachable.
|
||||
- =-x state-based= :: for all outgoing transition of each state
|
||||
to belong to the same acceptance sets.
|
||||
- =-x !wdba-minimize= :: disable WDBA minimization.
|
||||
|
||||
When options =-B= and =-x sat-minimize= both used, =-x state-based= and
|
||||
=-x sat-acc=1= are implied.
|
||||
Loading…
Add table
Add a link
Reference in a new issue