sat-minimize: some documentation and associated fixes
* doc/org/satmin.org: Document the new DTωA-minimization procedure. * doc/org/tools.org: Fix link. * src/bin/autfilt.cc: Pass -S to sat_minimize(). * src/twa/twagraph.hh: (state_acc_sets) New method. * src/twaalgos/dotty.cc: Use it to correctly display co-Büchi automata. * src/twaalgos/dtbasat.cc: Set the deterministic property on the result. * src/twaalgos/dtgbasat.cc: Likewise, and preprocess the input automaton in sat_minimize(). * src/twaalgos/dtgbasat.hh: Fix documentation, and take the state-based information as an argument. * src/twaalgos/postproc.cc: Do not call simulation-based reduction on non-separated acceptances. * src/tests/satmin2.test: Use -S rather than 'state-based'. * NEWS: Update.
This commit is contained in:
Alexandre Duret-Lutz
parent
7b28f1ffb5
commit
96e2da8666
11 changed files with 467 additions and 60 deletions
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@ -1,26 +1,29 @@
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#+TITLE: SAT-based Minimization of Deterministic (Generalized) Büchi Automata
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#+TITLE: SAT-based Minimization of Deterministic ω-Automata
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#+SETUPFILE: setup.org
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#+HTML_LINK_UP: 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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This page explains how to use [[file:ltl2tgba.org][=ltl2tgba=]], [[file:dstar2tgba.org][=dstar2tgba=]], or [[file:autfilt.org][=autfilt=]]
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to minimize 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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2) Spot actually implements 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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(DTBA), and one that builds a deterministic transition-based
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ω-automaton with arbitrary acceptance condition (DTωA). In
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[[file:ltl2tgba.org][=ltl2tgba=]] and [[file:dstar2tgba.org][=dstar2tgba=]], the latter procedure is restricted to
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TGBA. In [[file:autfilt.org][=autfilt=]] it can use different and acceptance conditions
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for input and output, so you could for instance input a Rabin
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automaton, and produce a Streett automaton.
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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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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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@ -29,9 +32,12 @@ Let us first state a few facts about this minimization procedure.
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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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6) Details about the SAT encoding used in the two procedures can be
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found in our [[http://www.lrde.epita.fr/~adl/dl/adl/baarir.14.forte.pdf][FORTE'14 paper]].
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killed by a signal. [[file:autfilt.org][=autfilt --sat-minimize=]] will only output an
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automaton if the SAT-based minimization was successful.
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6) Our [[http://www.lrde.epita.fr/~adl/dl/adl/baarir.14.forte.pdf][FORTE'14 paper]] describes the SAT encoding for the minimization
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of deterministic BA and TGBA. Since then, the technique used in
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the SAT encoding for deterministic TGBA has been generalized to
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deal with any deterministic TωA.
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* How to change the SAT solver used
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@ -46,7 +52,7 @@ the input for the SAT solver and receiving its output. We assume that
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the SAT solver should follow the conventions of the [[http://www.satcompetition.org/][SAT competition]]
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for input and output.
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* Enabling SAT-based minimization for deterministic automata
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* Enabling SAT-based minimization in =ltl2tgba= or =dstar2tgba=
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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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@ -205,8 +211,8 @@ ltl2tgba -D -x sat-minimize "Ga R (F!b & (c U b))" --stats='states=%s, det=%d'
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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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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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@ -274,7 +280,9 @@ dstar2tgba -D -x sat-minimize,sat-acc=2 --stats='input(states=%S) output(states=
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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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Beware that the size of the SAT problem is exponential in the number
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of acceptance sets (adding one acceptance set, in the input automaton
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or in the output automaton, will double the size of the problem).
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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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@ -283,11 +291,11 @@ 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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The following figure (from our [[http://www.lrde.epita.fr/~adl/dl/adl/baarir.14.forte.pdf][FORTE'14 paper]]) gives an overview of
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the processing chains that can be used to turn an LTL formula into a
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minimal DBA/DTBA/DTGBA. The blue area at the top describes =ltl2tgba
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-D -x sat-minimize=, while the purple area at the bottom corresponds
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to =dstar2tgba -D -x stat-minimize=.
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[[file:satmin.png]]
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@ -343,8 +351,327 @@ The following options can be used to fine-tune this procedure:
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- =-x !wdba-minimize= :: disable WDBA minimization.
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When options =-B= and =-x sat-minimize= are both used, =-x
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state-based= and =-x sat-acc=1= are implied.
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state-based= and =-x sat-acc=1= are implied. Similarly, when option
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=-S= and =-x sat-minimize= are both used, then option =-x state-based=
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is implied.
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* Using =autfilt --sat-minimize= to minimize any deterministic ω-automaton
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This interface is new in Spot 1.99 and allows to minimize any
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deterministic ω-automaton, regardless of the acceptance condition
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used. By default, the procedure will try to use the same acceptance
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condition (or any inferior one) and produce transition-based
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acceptance.
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For our example, let us first generate an deterministic Rabin
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automaton with [[http://www.ltl2dstar.de/][=ltl2dstar=]].
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#+BEGIN_SRC sh :results verbatim :exports code
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ltlfilt -f "FGa | FGb" -l |
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ltl2dstar --ltl2nba=spin:../../src/bin/ltl2tgba@-Ds --output-format=hoa - - > output.hoa
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#+END_SRC
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#+RESULTS:
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Let's draw it:
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#+NAME: autfiltsm1
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#+BEGIN_SRC sh :results verbatim :exports code
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autfilt output.hoa --dot=.a
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#+END_SRC
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#+RESULTS: autfiltsm1
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#+begin_example
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digraph G {
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rankdir=LR
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label=<(Fin(<font color="#5DA5DA">⓿</font>) & Inf(<font color="#F17CB0">❶</font>)) | (Fin(<font color="#FAA43A">❷</font>) & Inf(<font color="#B276B2">❸</font>))>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label=<0<br/><font color="#5DA5DA">⓿</font><font color="#FAA43A">❷</font>>]
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0 -> 0 [label=<!a & !b>]
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0 -> 1 [label=<a & !b>]
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0 -> 2 [label=<!a & b>]
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0 -> 3 [label=<a & b>]
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1 [label=<1<br/><font color="#F17CB0">❶</font><font color="#FAA43A">❷</font>>]
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1 -> 0 [label=<!a & !b>]
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1 -> 1 [label=<a & !b>]
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1 -> 2 [label=<!a & b>]
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1 -> 3 [label=<a & b>]
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2 [label=<2<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
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2 -> 0 [label=<!a & !b>]
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2 -> 1 [label=<a & !b>]
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2 -> 2 [label=<!a & b>]
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2 -> 3 [label=<a & b>]
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3 [label=<3<br/><font color="#F17CB0">❶</font><font color="#B276B2">❸</font>>]
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3 -> 0 [label=<!a & !b>]
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3 -> 1 [label=<a & !b>]
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3 -> 2 [label=<!a & b>]
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3 -> 3 [label=<a & b>]
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}
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#+end_example
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#+BEGIN_SRC dot :file autfiltsm1.png :cmdline -Tpng :var txt=autfiltsm1 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:autfiltsm1.png]]
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So this is a state-based Rabin automaton with two pairs. If we call
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=autfilt= with the =--sat-minimize= option, we can get the following
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transition-based version (the output may change depending on the SAT
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solver used):
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#+NAME: autfiltsm2
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#+BEGIN_SRC sh :results verbatim :exports code
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autfilt --sat-minimize output.hoa --dot=.a
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#+END_SRC
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#+RESULTS: autfiltsm2
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#+begin_example
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digraph G {
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rankdir=LR
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label=<(Fin(<font color="#5DA5DA">⓿</font>) & Inf(<font color="#F17CB0">❶</font>)) | (Fin(<font color="#FAA43A">❷</font>) & Inf(<font color="#B276B2">❸</font>))>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 0 [label=<a & b<br/><font color="#F17CB0">❶</font>>]
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0 -> 0 [label=<!a & !b<br/><font color="#5DA5DA">⓿</font><font color="#FAA43A">❷</font>>]
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0 -> 0 [label=<a & !b<br/><font color="#F17CB0">❶</font><font color="#FAA43A">❷</font>>]
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0 -> 0 [label=<!a & b<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
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}
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#+end_example
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#+BEGIN_SRC dot :file autfiltsm2.png :cmdline -Tpng :var txt=autfiltsm2 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:autfiltsm2.png]]
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We can also attempt to build a state-based version with
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#+NAME: autfiltsm3
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#+BEGIN_SRC sh :results verbatim :exports code
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autfilt -S --sat-minimize output.hoa --dot=.a
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#+END_SRC
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#+RESULTS: autfiltsm3
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#+begin_example
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digraph G {
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rankdir=LR
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label=<(Fin(<font color="#5DA5DA">⓿</font>) & Inf(<font color="#F17CB0">❶</font>)) | (Fin(<font color="#FAA43A">❷</font>) & Inf(<font color="#B276B2">❸</font>))>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label=<0<br/><font color="#F17CB0">❶</font><font color="#FAA43A">❷</font>>]
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0 -> 0 [label=<b>]
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0 -> 1 [label=<!b>]
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1 [label=<1<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
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1 -> 0 [label=<!a>]
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1 -> 1 [label=<a>]
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}
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#+end_example
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#+BEGIN_SRC dot :file autfiltsm3.png :cmdline -Tpng :var txt=autfiltsm3 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:autfiltsm3.png]]
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This is clearly smaller than the input automaton. In this example the
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acceptance condition did not change. The SAT-based minimization only
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tries to minimize the number of states, but sometime the
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simplifications algorithms that are run before we attempt SAT-solving
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will simplify the acceptance, because even removing a single
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acceptance set can halve the run time.
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You can however force a specific acceptance to be used as output.
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Let's try with generalized co-Büchi for instance:
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#+NAME: autfiltsm4
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#+BEGIN_SRC sh :results verbatim :exports code
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autfilt -S --sat-minimize='acc="generalized-co-Buchi 2"' output.hoa --dot=.a
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#+END_SRC
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#+RESULTS: autfiltsm4
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#+begin_example
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digraph G {
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rankdir=LR
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label=<Fin(<font color="#5DA5DA">⓿</font>)|Fin(<font color="#F17CB0">❶</font>)>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label=<0<br/><font color="#5DA5DA">⓿</font>>]
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0 -> 0 [label=<a>]
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0 -> 1 [label=<!a>]
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1 [label=<1<br/><font color="#F17CB0">❶</font>>]
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1 -> 0 [label=<!b>]
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1 -> 1 [label=<b>]
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}
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#+end_example
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#+BEGIN_SRC dot :file autfiltsm4.png :cmdline -Tpng :var txt=autfiltsm4 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:autfiltsm4.png]]
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Note that instead of naming the acceptance condition, you can actually
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give an acceptance formula in the [[http://adl.github.io/hoaf/#acceptance][HOA syntax]]. For example we can
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attempt to create a co-Büchi automaton with
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#+NAME: autfiltsm5
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#+BEGIN_SRC sh :results verbatim :exports code
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autfilt -S --sat-minimize='acc="Fin(0)"' output.hoa --dot=.a
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#+END_SRC
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#+RESULTS: autfiltsm5
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#+BEGIN_SRC dot :file autfiltsm5.png :cmdline -Tpng :var txt=autfiltsm5 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:autfiltsm5.png]]
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When forcing an acceptance condition, you should keep in mind that the
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SAT-based minimization algorithm will look for automata that have
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fewer states than the original automaton (after preliminary
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simplifications). This is not always reasonable. For instance
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constructing a Streett automaton from a Rabin automaton might require
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more states. An upper bound on the number of state be passed
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using a =max-states=123= argument to =--sat-minimize=.
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If the input automaton is transition-based, but option =-S= is used to
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produce a state-based automaton, then the original automaton is
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temporarily converted into an automaton with state-based acceptance to
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obtain an upper bound on the number of states if you haven't specified
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=max-state=. This upper bound might be larger than the one you would
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specify by hand.
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Here is an example demonstrating the case where the input automaton is
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smaller than the output. Let's take this small TGBA as input:
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#+NAME: autfiltsm6
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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba 'GFa & GFb' -H > output2.hoa
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autfilt output2.hoa --dot=.a
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#+END_SRC
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#+RESULTS: autfiltsm6
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#+begin_example
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digraph G {
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rankdir=LR
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label=<Inf(<font color="#5DA5DA">⓿</font>)&Inf(<font color="#F17CB0">❶</font>)>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 0 [label=<a & b<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
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0 -> 0 [label=<!a & !b>]
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0 -> 0 [label=<!a & b<br/><font color="#F17CB0">❶</font>>]
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0 -> 0 [label=<a & !b<br/><font color="#5DA5DA">⓿</font>>]
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}
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#+end_example
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#+BEGIN_SRC dot :file autfiltsm6.png :cmdline -Tpng :var txt=autfiltsm6 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:autfiltsm6.png]]
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If we attempt to minimize it into a transition-based Büchi automaton,
|
||||
with fewer states, it will fail, output no result, and return with a
|
||||
non-zero exit code (because no automata where output).
|
||||
|
||||
#+NAME: autfiltsm7
|
||||
#+BEGIN_SRC sh :results verbatim :exports both
|
||||
autfilt --sat-minimize='acc="Buchi"' output2.hoa
|
||||
echo $?
|
||||
#+END_SRC
|
||||
#+RESULTS: autfiltsm7
|
||||
: 1
|
||||
|
||||
However if we allow more states, it will work:
|
||||
|
||||
#+NAME: autfiltsm8
|
||||
#+BEGIN_SRC sh :results verbatim :exports code
|
||||
autfilt --sat-minimize='acc="Buchi",max-states=3' output2.hoa --dot=.a
|
||||
#+END_SRC
|
||||
|
||||
#+RESULTS: autfiltsm8
|
||||
#+begin_example
|
||||
digraph G {
|
||||
rankdir=LR
|
||||
label=<Inf(<font color="#5DA5DA">⓿</font>)>
|
||||
labelloc="t"
|
||||
node [shape="circle"]
|
||||
fontname="Lato"
|
||||
node [fontname="Lato"]
|
||||
edge [fontname="Lato"]
|
||||
node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
|
||||
I [label="", style=invis, width=0]
|
||||
I -> 0
|
||||
0 [label="0"]
|
||||
0 -> 0 [label=<!b>]
|
||||
0 -> 0 [label=<a & b<br/><font color="#5DA5DA">⓿</font>>]
|
||||
0 -> 1 [label=<!a & b<br/><font color="#5DA5DA">⓿</font>>]
|
||||
1 [label="1"]
|
||||
1 -> 0 [label=<a>]
|
||||
1 -> 1 [label=<!a>]
|
||||
}
|
||||
#+end_example
|
||||
|
||||
#+BEGIN_SRC dot :file autfiltsm8.png :cmdline -Tpng :var txt=autfiltsm8 :exports results
|
||||
$txt
|
||||
#+END_SRC
|
||||
|
||||
#+RESULTS:
|
||||
[[file:autfiltsm8.png]]
|
||||
|
||||
|
||||
The =--sat-minimize= option takes a comma separated list of arguments
|
||||
that can be any of the following:
|
||||
|
||||
- =acc=DOUBLEQUOTEDSTRING= :: where the =DOUBLEQUOTEDSTRING= is an
|
||||
acceptance formula in the [[http://adl.github.io/hoaf/#acceptance][HOA syntax]], or a parametrized acceptance
|
||||
name (the different [[http://adl.github.io/hoaf/#acceptance-specifications][=acc-name:= options from HOA]]).
|
||||
- =max-states=N= :: where =N= is an upper-bound on the maximum
|
||||
number of states of the constructed automaton.
|
||||
- =states=M= :: where =M= is a fixed number of states to use in the
|
||||
result (all the states needs not be accessible in the result,
|
||||
so the output might be smaller nonetheless). If this option is
|
||||
used the SAT-based procedure is just used once to synthesize
|
||||
one automaton, and no further minimization is attempted.
|
||||
- =dichotomy= :: instead of looking for a smaller automaton starting
|
||||
from =N=, and then checking =N-1=, =N-2=, etc., do a binary
|
||||
search starting from =N/2=.
|
||||
|
||||
* Logging statistics
|
||||
|
||||
|
|
@ -397,3 +724,8 @@ failed. The final output is reported with 5 states, because by
|
|||
default we output trim automata. If the =--complete= option had been
|
||||
given, the useless sink state would have been kept and the output
|
||||
automaton would have 6 states.
|
||||
|
||||
|
||||
#+BEGIN_SRC sh :results verbatim :exports none
|
||||
rm -f output.hoa output2.hoa
|
||||
#+END_SRC
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue