hierarchy: add a new way to check DBA-realizability via DPA
* spot/tl/hierarchy.cc, spot/tl/hierarchy.hh: Here. * tests/core/hierarchy.test: Test it. * bin/man/spot-x.x: Document SPOT_PR_CHECK. * doc/org/hierarchy.org, NEWS: Update.
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Alexandre Duret-Lutz
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6 changed files with 155 additions and 230 deletions
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@ -520,7 +520,7 @@ automata.
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For the subclass of /obligation/ properties, using =-D= is a sure way
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to obain a deterministic automaton (and even a minimal one), but for
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the /recurrence/ properties that are not /obligations/ the translator
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does not make any special effort to produce deterministic automata,
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does not make /too much/ effort to produce deterministic automata,
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even with =-D= (this might change in the future).
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All properties that are not in the /persistence/ class (this includes
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@ -585,106 +585,40 @@ $txt
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[[file:hier-recurrence-3.svg]]
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One way to obtain a deterministic Büchi automaton (it has to exist, since this is
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a /recurrence/ property), is to chain a few algorithms implemented in Spot:
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a /recurrence/ property), is to request a deterministic automaton with parity
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acceptance using =-P=. The number of color output with =-P= is always reduced
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to the minimal number possible, so for a /recurrence/ property the output
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automaton can only have one of three possible acceptance: =Inf(0)=, =t=, or =f=.
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1. Determinize the non-deterministic automaton to obtain a
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deterministic automaton with parity acceptance: this is done by
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using =ltl2tgba -P -D=, with option =-P= indicating that parity
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acceptance is desired.
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#+NAME: hier-recurrence-4
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#+BEGIN_SRC sh :exports code
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ltl2tgba -P -D 'G(Gb | Fa)' -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-recurrence-4.svg :var txt=hier-recurrence-4 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-recurrence-4.svg]]
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2. Transform the parity acceptance into Rabin acceptance: this is
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done with =autfilt --generalized-rabin=. Because of the type of
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parity acceptance used, the result will actually be Rabin and not
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generalized Rabin.
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#+NAME: hier-recurrence-5
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#+BEGIN_SRC sh :exports code
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ltl2tgba -P -D 'G(Gb | Fa)' |
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autfilt --generalized-rabin -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-recurrence-5.svg :var txt=hier-recurrence-5 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-recurrence-5.svg]]
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(The only change here is in the acceptance condition.)
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3. In step 4 we are going to convert the automaton to state-based
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Büchi, and this sometimes works better if the input Rabin automaton
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also uses state-based acceptance. So let us add =-S= to the
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previous command:
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#+NAME: hier-recurrence-6
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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba -P -D 'G(Gb | Fa)' |
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autfilt -S --generalized-rabin -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-recurrence-6.svg :var txt=hier-recurrence-6 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-recurrence-6.svg]]
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4. Finally, convert the resulting automaton to BA, using =autfilt
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-B=. Spot can convert automata with any acceptance condition to
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BA, but when the input is a deterministic Rabin automaton, it uses
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a dedicated algorithm that preserves determinism whenever possible
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(and we know it is possible, because we are working on a
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recurrence formula). Adding =-D= here to suggest that we are
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trying to obtain a deterministic automaton does not hurt, as it
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will enable simplifications as a side-effect (without =-D= we
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simply get a larger deterministic automaton).
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#+NAME: hier-recurrence-7
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#+BEGIN_SRC sh :exports code
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ltl2tgba -P -D 'G(Gb | Fa)' |
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autfilt -S --generalized-rabin |
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autfilt -B -D -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-recurrence-7.svg :var txt=hier-recurrence-7 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-recurrence-7.svg]]
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Here we are lucky that the deterministic Büchi automaton is even
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smaller than the original non-deterministic version. As said earlier,
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passing =-S= to the first =autfilt= was optional, but in this case it
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helps producing a smaller automaton. Here is what we get without it:
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#+NAME: hier-recurrence-8
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#+NAME: hier-recurrence-4
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#+BEGIN_SRC sh :exports code
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ltl2tgba -P -D 'G(Gb | Fa)' |
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autfilt --generalized-rabin |
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autfilt -B -D -d
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ltl2tgba -P -D 'G(Gb | Fa)' -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-recurrence-8.svg :var txt=hier-recurrence-8 :exports results
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#+BEGIN_SRC dot :file hier-recurrence-4.svg :var txt=hier-recurrence-4 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-recurrence-8.svg]]
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[[file:hier-recurrence-4.svg]]
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It is likely that =ltl2tgba -B -D= will implement all this processing
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chain in the future, but so originally =-D= was only expressing a
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preference not a requirement.
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Note that if the acceptance is =t=, the property is a monitor, and if
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its =f=, the property is =false=. In any way, if you would like to
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obtain a DBA for any recurrent property, a sure way to avoid these
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difference is to pipe the result through =autfilt -B=
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#+NAME: hier-recurrence-5
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#+BEGIN_SRC sh :exports code
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ltl2tgba -P -D 'G(Gb | Fa)' | autfilt -B -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-recurrence-5.svg :var txt=hier-recurrence-5 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-recurrence-5.svg]]
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It is likely that =ltl2tgba -B -D= will implement these steps in the
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future, but so originally =-D= was only expressing a preference not a
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requirement.
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** Persistence
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@ -725,8 +659,9 @@ $txt
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Note that in this example, we know that =GFa= is trivial enough that
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=ltl2tgba -D GFa= will generate a deterministic automaton. In the
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general case we might have to determinize the automaton as we did in
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the previous section (we will do it again below).
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general case we might have to determinize the automaton using =-P -D= as
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we did in the previous section. For persistence properties, =-P -D= should
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return an automaton whose acceptance is one of =Fin(0)=, =t=, or =f=.
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/Persistence/ properties can be represented by weak Büchi automata.
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The translator is aware of that, so when it detects that the input
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@ -739,10 +674,8 @@ optimization is simply not applied.)
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If the input is a weak property that is not syntactically weak, the
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output will not necessarily be weak. One costly way to obtain a weak
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automaton for a formula $\varphi$ would be to first compute a
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deterministic Büchi automaton of the recurrence $\lnot\varphi$ then
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complement the acceptance of the resulting automaton, yielding a
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deterministic co-Büchi automaton, and then transform that into a Büchi
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automaton.
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deterministic co-Büchi automaton $\varphi$ then transform that into a
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Büchi automaton.
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Let's do that on the persistence formula =F(G!a | G(b U a))=, just for
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the fun of it.
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@ -766,15 +699,11 @@ $txt
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#+RESULTS:
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[[file:hier-persistence-3.svg]]
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Furthermore it appears that =ltl2tgba -D= does generate a deterministic
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Büchi automaton for the complement, instead we get a non-deterministic
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generalized Büchi automaton:
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So let's determinize using parity acceptance:
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#+NAME: hier-persistence-4
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#+BEGIN_SRC sh :exports code
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ltlfilt --negate -f 'F(G!a | G(b U a))' |
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ltl2tgba -D |
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autfilt --highlight-nondet=5 -d
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ltl2tgba -P -D 'F(G!a | G(b U a))' -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-persistence-4.svg :var txt=hier-persistence-4 :exports results
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$txt
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@ -783,58 +712,16 @@ $txt
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#+RESULTS:
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[[file:hier-persistence-4.svg]]
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So let us use the same tricks as in the previous section,
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determinizing this automaton into a Rabin automaton, and then back to
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deterministic Büchi:
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#+NAME: hier-persistence-5
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#+BEGIN_SRC sh :exports code
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ltlfilt --negate -f 'F(G!a | G(b U a))' |
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ltl2tgba -P -D |
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autfilt --generalized-rabin |
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autfilt --tgba -D -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-persistence-5.svg :var txt=hier-persistence-5 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-persistence-5.svg]]
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This is a deterministic Büchi automaton for the negation of our formula.
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Now we can complement it to obtain a deterministic co-Büchi automaton for =F(G!a | G(b U a))=:
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#+NAME: hier-persistence-6
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#+BEGIN_SRC sh :exports code
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ltlfilt --negate -f 'F(G!a | G(b U a))' |
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ltl2tgba -P -D |
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autfilt --generalized-rabin |
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autfilt --tgba -D |
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autfilt --complement -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-persistence-6.svg :var txt=hier-persistence-6 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-persistence-6.svg]]
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And finally we convert the result back to Büchi:
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And finally we convert the result back to Büchi with =autfilt -B=.
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#+NAME: hier-persistence-7
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#+BEGIN_SRC sh :exports code
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ltlfilt --negate -f 'F(G!a | G(b U a))' |
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ltl2tgba -P -D |
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autfilt --generalized-rabin |
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autfilt --tgba -D |
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autfilt --complement -B -d
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ltl2tgba -P -D 'F(G!a | G(b U a))' | autfilt -B --highlight-nondet --small -d
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#+END_SRC
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#+BEGIN_SRC dot :file hier-persistence-7.svg :var txt=hier-persistence-7 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:hier-persistence-7.svg]]
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That is indeed, a weak automaton.
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That is indeed, a weak non-deterministic automaton.
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