autfilt.org

# -*- coding: utf-8 -*-
#+TITLE: =autfilt=
#+DESCRIPTION: Spot command-line tool for filtering, tranforming, and converting ω-automata.
#+SETUPFILE: setup.org
#+HTML_LINK_UP: tools.html

The =autfilt= tool can filter, transform, and convert a stream of automata.

The tool operates a loop over 5 phases:
- input one automaton
- optionally preprocess the automaton
- optionally filter the automaton (i.e., decide whether to ignore the
  automaton or continue with it)
- optionally postprocess the automaton (to simply it or change its acceptance)
- output the automaton

The simplest way to use the tool is simply to use it for input and
output (i.e., format conversion) without any transformation and
filtering.

* Conversion between formats

=autfilt= can read automata written in the [[http://adl.github.io/hoaf/][Hanoi Omega Automata
Format]], as [[http://spinroot.com/spin/Man/never.html][Spin never claims]], using [[http://www.tcs.hut.fi/Software/lbtt/doc/html/Format-for-automata.html][LBTT's format]], or using
[[http://www.ltl2dstar.de/docs/ltl2dstar.html][=ltl2dstar='s format]].  Automata in those formats (even a mix of those
formats) can be concatenated in the same stream, =autfilt= will
process them in batch.  (The only restriction is that inside a file an
automaton in LBTT's format may not follow an automaton in
=ltl2dstar='s format.)

By default the output uses the HOA format.  This can be changed using
[[file:oaut.org][the common output options]] like =--spin=, =--lbtt=, =--dot=,
=--stats=...

#+BEGIN_SRC sh :results silent :exports both
cat >example.hoa <<EOF
HOA: v1
States: 1
Start: 0
AP: 1 "p0"
Acceptance: 1 Inf(0)
--BODY--
State: 0
[0] 0 {0}
[!0] 0
--END--
EOF
autfilt example.hoa --dot
#+END_SRC

#+BEGIN_SRC sh :results verbatim :exports results
SPOT_DOTEXTRA= autfilt example.hoa --dot
#+END_SRC

#+RESULTS:
: digraph G {
:   rankdir=LR
:   node [shape="circle"]
:   I [label="", style=invis, width=0]
:   I -> 0
:   0 [label="0"]
:   0 -> 0 [label="p0\n{0}"]
:   0 -> 0 [label="!p0"]
: }

The =--spin= option implicitly requires a degeneralization:

#+BEGIN_SRC sh :results verbatim :exports both
autfilt example.hoa --spin
#+END_SRC

#+RESULTS:
#+begin_example
never {
accept_init:
  if
  :: ((p0)) -> goto accept_init
  :: ((!(p0))) -> goto T0_S2
  fi;
T0_S2:
  if
  :: ((p0)) -> goto accept_init
  :: ((!(p0))) -> goto T0_S2
  fi;
}
#+end_example

Option =--lbtt= only works for Büchi or generalized Büchi acceptance.

#+BEGIN_SRC sh :results verbatim :exports both
autfilt example.hoa --lbtt
#+END_SRC

#+RESULTS:
: 1 1t
: 0 1
: 0 0 -1 p0
: 0 -1 ! p0
: -1

* Displaying statistics

One special output format of =autfilt= is the statistic output.  For
instance the following command calls [[file:randaut.org][=randaut=]] to generate 10 random
automata, and pipe the result into =autfilt= to display various
statistics.


#+BEGIN_SRC sh :results verbatim :exports both
randaut -n 10 -A0..2 -Q10..20 -e0.05 2 |
autfilt --stats='%s states, %e edges, %a acc-sets, %c SCCs, det=%d'
#+END_SRC

#+RESULTS:
#+begin_example
16 states, 30 edges, 1 acc-sets, 3 SCCs, det=0
20 states, 42 edges, 2 acc-sets, 1 SCCs, det=0
15 states, 27 edges, 2 acc-sets, 1 SCCs, det=0
10 states, 17 edges, 1 acc-sets, 1 SCCs, det=1
13 states, 25 edges, 1 acc-sets, 1 SCCs, det=0
11 states, 18 edges, 0 acc-sets, 1 SCCs, det=0
19 states, 41 edges, 2 acc-sets, 1 SCCs, det=0
11 states, 18 edges, 0 acc-sets, 1 SCCs, det=0
12 states, 21 edges, 1 acc-sets, 5 SCCs, det=0
18 states, 37 edges, 1 acc-sets, 5 SCCs, det=0
#+end_example

The following =%= sequences are available:
#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/^  %%/,/^$/p' | sed '$d'
#+END_SRC
#+RESULTS:
#+begin_example
  %%                         a single %
  %A, %a                     number of acceptance sets
  %C, %c                     number of SCCs
  %d                         1 if the output is deterministic, 0 otherwise
  %E, %e                     number of edges
  %F                         name of the input file
  %G, %g                     acceptance condition (in HOA syntax)
  %L                         location in the input file
  %M, %m                     name of the automaton
  %n                         number of nondeterministic states in output
  %p                         1 if the output is complete, 0 otherwise
  %r                         processing time (excluding parsing) in seconds
  %S, %s                     number of states
  %T, %t                     number of transitions
  %w                         one word accepted by the output automaton
#+end_example

When a letter is available both as uppercase and lowercase, the
uppercase version refer to the input automaton, while the lowercase
refer to the output automaton.  Of course this distinction makes sense
only if =autfilt= was instructed to perform an operation on the input
automaton.

* Filtering automata

=autfilt= offers multiple options to filter automata based on
different characteristics of the automaton.

#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/Filtering options.*:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
#+begin_example
      --acc-sccs=RANGE, --accepting-sccs=RANGE
                             keep automata whose number of non-trivial
                             accepting SCCs is in RANGE
      --acc-sets=RANGE       keep automata whose number of acceptance sets is
                             in RANGE
      --accept-word=WORD     keep automata that accept WORD
      --ap=RANGE             match automata with a number of (declared) atomic
                             propositions in RANGE
      --are-isomorphic=FILENAME   keep automata that are isomorphic to the
                             automaton in FILENAME
      --edges=RANGE          keep automata whose number of edges is in RANGE
      --equivalent-to=FILENAME   keep automata thare are equivalent
                             (language-wise) to the automaton in FILENAME
      --included-in=FILENAME keep automata whose languages are included in that
                             of the automaton from FILENAME
      --inherently-weak-sccs=RANGE
                             keep automata whose number of accepting
                             inherently-weak SCCs is in RANGE.  An accepting
                             SCC is inherently weak if it does not have a
                             rejecting cycle.
      --intersect=FILENAME   keep automata whose languages have an non-empty
                             intersection with the automaton from FILENAME
      --is-complete          keep complete automata
      --is-deterministic     keep deterministic automata
      --is-empty             keep automata with an empty language
      --is-inherently-weak   keep only inherently weak automata
      --is-stutter-invariant keep automata representing stutter-invariant
                             properties
      --is-terminal          keep only terminal automata
      --is-unambiguous       keep only unambiguous automata
      --is-weak              keep only weak automata
      --nondet-states=RANGE  keep automata whose number of nondeterministic
                             states is in RANGE
      --rej-sccs=RANGE, --rejecting-sccs=RANGE
                             keep automata whose number of non-trivial
                             rejecting SCCs is in RANGE
      --reject-word=WORD     keep automata that reject WORD
      --sccs=RANGE           keep automata whose number of SCCs is in RANGE
      --states=RANGE         keep automata whose number of states is in RANGE
      --terminal-sccs=RANGE  keep automata whose number of accepting terminal
                             SCCs is in RANGE.  Terminal SCCs are weak and
                             complete.
      --triv-sccs=RANGE, --trivial-sccs=RANGE
                             keep automata whose number of trivial SCCs is in
                             RANGE
      --unused-ap=RANGE      match automata with a number of declared, but
                             unused atomic propositions in RANGE
      --used-ap=RANGE        match automata with a number of used atomic
                             propositions in RANGE
  -u, --unique               do not output the same automaton twice (same in
                             the sense that they are isomorphic)
  -v, --invert-match         select non-matching automata
      --weak-sccs=RANGE      keep automata whose number of accepting weak SCCs
                             is in RANGE.  In a weak SCC, all transitions
                             belong to the same acceptance sets.
#+end_example

For instance =--states=2..5 --acc-sets=3= will /keep/ only automata that
use 3 acceptance sets, and that have between 2 and 5 states.

Except for =--unique=, all these filters can be inverted using option
=-v=.  Using =--states=2..5 --acc-sets=3 -v= will /drop/ all automata
that use 3 acceptance sets and that have between 2 and 5 states, and
keep the others.

* Simplifying automata

The standard set of automata simplification routines (these are often
referred to as the "post-processing" routines, because these are the
procedures performed by [[file:ltl2tgba.org][=ltl2tgba=]] after translating a formula into a
TGBA) are available through the following options.

This set of options controls the desired type of output automaton:

#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/Output automaton type:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
#+begin_example
  -B, --ba                   Büchi Automaton (with state-based acceptance)
  -C, --complete             output a complete automaton
  -G, --generic              any acceptance is allowed (default)
  -M, --monitor              Monitor (accepts all finite prefixes of the given
                             property)
  -p, --colored-parity[=any|min|max|odd|even|min odd|min even|max odd|max
      even]                  colored automaton with parity acceptance
  -P, --parity[=any|min|max|odd|even|min odd|min even|max odd|max even]
                             automaton with parity acceptance
  -S, --state-based-acceptance, --sbacc
                             define the acceptance using states
      --tgba                 Transition-based Generalized Büchi Automaton
#+end_example

These options specify any simplification goal:

#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/Simplification goal:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
:   -a, --any                  no preference, do not bother making it small or
:                              deterministic
:   -D, --deterministic        prefer deterministic automata
:       --small                prefer small automata

Finally, the following switches control the amount of effort applied
toward the desired goal:

#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/Simplification level:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
:       --high                 all available optimizations (slow)
:       --low                  minimal optimizations (fast)
:       --medium               moderate optimizations


By default, =--any --low= is used, which cause all simplifications to
be skipped.  However if any goal is given, than the simplification level
defaults to =--high= (unless specified otherwise).  If a simplification
level is given without specifying any goal, then the goal default to =--small=.

So if you want to reduce the size of the automaton, try =--small= and
if you want to try to make (or keep) it deterministic use
=--deterministic=.

Note that the =--deterministic= flag has two possible behaviors
depending on the constraints on the acceptance conditions:
- When =autfilt= is configured to work with generic acceptance (the
  =--generic= option, which is the default) or parity acceptance
  (using =--parity= or =--colored-parity=), then the =--deterministic=
  flag will do whatever it takes to output a deterministic automaton,
  and this includes changing the acceptance condition if needed (see
  below).
- If options =--tgba= or =--ba= are used, the =--deterministic= option
  is taken as a /preference/: =autfilt= will try to favor determinism
  in the output, but it may not always succeed and may output
  non-deterministic automata.  Note that if =autfilt --deterministic
  --tgba= fails to output a deterministic automaton, it does not
  necessarily implies that a deterministic TGBA does not exist: it
  just implies that =autfilt= could not find it.


** Determinization

Spot has basically two ways to determinize automata, and that it uses
when =--deterministic= is passed.

- Automata that express obligation properties (this can be decided),
  can be *determinized and minimized* into weak Büchi automata, as
  discussed by [[http://www.daxc.de/eth/paper/atva07.pdf][Dax at al. (ATVA'07)]].

- Büchi automata (preferably with transition-based acceptance) can be
  determinized into parity automata using a Safra-like procedure close
  to the one presented by [[http://www.romanredz.se/papers/FI2012.pdf][Redziejowski (Fund. Inform. 119)]], with a few
  additional tricks.  This procedure does not necessarily produce a
  minimal automaton.

When =--deterministic= is used, the first of these two procedures is
attempted on any supplied automaton.  (It's even attempted for
deterministic automata, because that might reduce them.)

If that first procedure failed, and the input automaton is not
deterministic and =--generic= (the default for =autfilt=), =--parity=
or =--colorized-parity= is used, then the second procedure is used.
In this case, automata will be first converted to transition-based
Büchi automata if their acceptance condition is more complex.

The difference between =--parity= and =--colored-parity= parity is
that the latter requests all transitions (or all states when
state-based acceptance is used) to belong to exactly one acceptance
set.

* Transformations

The following transformations are available:

#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/Transformations:/,/^$/p' | sed '1d;$d'
#+END_SRC

#+RESULTS:
#+begin_example
      --cleanup-acceptance   remove unused acceptance sets from the automaton
      --cnf-acceptance       put the acceptance condition in Conjunctive Normal
                             Form
      --complement           complement each automaton (currently support only
                             deterministic automata)
      --complement-acceptance   complement the acceptance condition (without
                             touching the automaton)
      --decompose-strength=t|w|s   extract the (t) terminal, (w) weak, or (s)
                             strong part of an automaton (letters may be
                             combined to combine more strengths in the output)
      --destut               allow less stuttering
      --dnf-acceptance       put the acceptance condition in Disjunctive Normal
                             Form
      --exclusive-ap=AP,AP,...   if any of those APs occur in the automaton,
                             restrict all edges to ensure two of them may not
                             be true at the same time.  Use this option
                             multiple times to declare independent groups of
                             exclusive propositions.
      --generalized-rabin[=unique-inf|share-inf], --gra[=unique-inf|share-inf]
                             rewrite the acceptance condition as generalized
                             Rabin; the default "unique-inf" option uses the
                             generalized Rabin definition from the HOA format;
                             the "share-inf" option allows clauses to share Inf
                             sets, therefore reducing the number of sets
      --generalized-streett[=unique-fin|share-fin], --gsa[=unique-fin|share-fin]                                                          rewrite the
                             acceptance condition as generalized Streett; the
                             "share-fin" option allows clauses to share Fin
                             sets, therefore reducing the number of sets; the
                             default "unique-fin" does not
      --instut[=1|2]         allow more stuttering (two possible algorithms)
      --keep-states=NUM[,NUM...]   only keep specified states.  The first state
                             will be the new initial state.  Implies
                             --remove-unreachable-states.
      --mask-acc=NUM[,NUM...]   remove all transitions in specified acceptance
                             sets
      --merge-transitions    merge transitions with same destination and
                             acceptance
      --product=FILENAME, --product-and=FILENAME
                             build the product with the automaton in FILENAME
                             to intersect languages
      --product-or=FILENAME  build the product with the automaton in FILENAME
                             to sum languages
      --randomize[=s|t]      randomize states and transitions (specify 's' or
                             't' to randomize only states or transitions)
      --remove-ap=AP[=0|=1][,AP...]
                             remove atomic propositions either by existential
                             quantification, or by assigning them 0 or 1
      --remove-dead-states   remove states that are unreachable, or that cannot
                             belong to an infinite path
      --remove-fin           rewrite the automaton without using Fin
                             acceptance
      --remove-unreachable-states
                             remove states that are unreachable from the
                             initial state
      --remove-unused-ap     remove declared atomic propositions that are not
                             used
      --sat-minimize[=options]   minimize the automaton using a SAT solver
                             (only works for deterministic automata)
      --separate-sets        if both Inf(x) and Fin(x) appear in the acceptance
                             condition, replace Fin(x) by a new Fin(y) and
                             adjust the automaton
      --simplify-exclusive-ap   if --exclusive-ap is used, assume those AP
                             groups are actually exclusive in the system to
                             simplify the expression of transition labels
                             (implies --merge-transitions)
      --strip-acceptance     remove the acceptance condition and all acceptance
                             sets
#+end_example

* Decorations

Decorations work by coloring some states or edges in the automaton.
They are only useful when the automaton is output in Dot format (with
=--dot= or =-d=) or HOA v1.1 format (with =-H1.1= or =--hoa=1.1=).

#+BEGIN_SRC sh :results verbatim :exports results
autfilt --help | sed -n '/^ *Decorations.*:/,/^$/p' | sed '1d;$d'
#+END_SRC

#+RESULTS:
:       --highlight-nondet[=NUM]   highlight nondeterministic states and edges
:                              with color NUM
:       --highlight-nondet-edges[=NUM]
:                              highlight nondeterministic edges with color NUM
:       --highlight-nondet-states[=NUM]
:                              highlight nondeterministic states with color NUM
:       --highlight-word=[NUM,]WORD
:                              highlight one run matching WORD using color NUM

Color numbers are indices in some hard-coded color palette.  It is the
same palette that is currently used to display colored acceptance
sets, but this might change in the future.

* Examples

** Acceptance transformations

Here is an automaton with transition-based acceptance:

#+BEGIN_SRC sh :results silent :exports both
cat >aut-ex1.hoa<<EOF
HOA: v1
States: 3
Start: 0
AP: 2 "a" "b"
Acceptance: 4 Inf(0)&Fin(1)&Fin(3) | Inf(2)&Inf(3) | Inf(1)
--BODY--
State: 0 {3}
[t] 0
[0] 1 {1}
[!0] 2 {0}
State: 1 {3}
[1] 0
[0&1] 1 {0}
[!0&1] 2 {2}
State: 2
[!1] 0
[0&!1] 1 {0}
[!0&!1] 2 {0}
--END--
EOF
#+END_SRC

(Note: that the =--dot= option used below uses some default options
discussed [[file:oaut.org::#default-dot][on another page]].)

#+NAME: autfilt-ex1
#+BEGIN_SRC sh :results verbatim :exports code
autfilt aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex1
#+begin_example
digraph G {
  rankdir=LR
  label=<(Fin(<font color="#F17CB0">❶</font>) &amp; Fin(<font color="#B276B2">❸</font>) &amp; Inf(<font color="#5DA5DA">⓿</font>)) | (Inf(<font color="#FAA43A">❷</font>)&amp;Inf(<font color="#B276B2">❸</font>)) | Inf(<font color="#F17CB0">❶</font>)>
  labelloc="t"
  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=<1<br/><font color="#B276B2">❸</font>>]
  0 -> 1 [label=<a<br/><font color="#F17CB0">❶</font><font color="#B276B2">❸</font>>]
  0 -> 2 [label=<!a<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 [label="1"]
  1 -> 0 [label=<b<br/><font color="#B276B2">❸</font>>]
  1 -> 1 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 -> 2 [label=<!a &amp; b<br/><font color="#FAA43A">❷</font><font color="#B276B2">❸</font>>]
  2 [label="2"]
  2 -> 0 [label=<!b>]
  2 -> 1 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 2 [label=<!a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex1.svg :var txt=autfilt-ex1 :exports results
  $txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex1.svgz]]

Using =-S= will "push" the acceptance membership of the transitions to the states:

#+NAME: autfilt-ex2
#+BEGIN_SRC sh :results verbatim :exports code
autfilt -S aut-ex1.hoa --dot=
#+END_SRC

#+RESULTS: autfilt-ex2
#+begin_example
digraph G {
  rankdir=LR
  label=<(Fin(<font color="#F17CB0">❶</font>) &amp; Fin(<font color="#B276B2">❸</font>) &amp; Inf(<font color="#5DA5DA">⓿</font>)) | (Inf(<font color="#FAA43A">❷</font>)&amp;Inf(<font color="#B276B2">❸</font>)) | Inf(<font color="#F17CB0">❶</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<br/><font color="#B276B2">❸</font>>]
  0 -> 0 [label=<1>]
  0 -> 1 [label=<a>]
  0 -> 2 [label=<!a>]
  1 [label=<1<br/><font color="#F17CB0">❶</font><font color="#B276B2">❸</font>>]
  1 -> 0 [label=<b>]
  1 -> 6 [label=<a &amp; b>]
  1 -> 7 [label=<!a &amp; b>]
  2 [label=<2<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  2 -> 3 [label=<!b>]
  2 -> 4 [label=<a &amp; !b>]
  2 -> 5 [label=<!a &amp; !b>]
  3 [label=<3>]
  3 -> 0 [label=<1>]
  3 -> 1 [label=<a>]
  3 -> 2 [label=<!a>]
  4 [label=<4<br/><font color="#5DA5DA">⓿</font>>]
  4 -> 0 [label=<b>]
  4 -> 6 [label=<a &amp; b>]
  4 -> 7 [label=<!a &amp; b>]
  5 [label=<5<br/><font color="#5DA5DA">⓿</font>>]
  5 -> 3 [label=<!b>]
  5 -> 4 [label=<a &amp; !b>]
  5 -> 5 [label=<!a &amp; !b>]
  6 [label=<6<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  6 -> 0 [label=<b>]
  6 -> 6 [label=<a &amp; b>]
  6 -> 7 [label=<!a &amp; b>]
  7 [label=<7<br/><font color="#FAA43A">❷</font><font color="#B276B2">❸</font>>]
  7 -> 3 [label=<!b>]
  7 -> 4 [label=<a &amp; !b>]
  7 -> 5 [label=<!a &amp; !b>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex2.svg :var txt=autfilt-ex2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex2.svg]]

Using =--cnf-acceptance= simply rewrites the acceptance condition in Conjunctive Normal Form:

#+NAME: autfilt-ex3
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --cnf-acceptance aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex3
#+begin_example
digraph G {
  rankdir=LR
  label=<(Inf(<font color="#5DA5DA">⓿</font>) | Inf(<font color="#F17CB0">❶</font>) | Inf(<font color="#B276B2">❸</font>)) &amp; (Fin(<font color="#B276B2">❸</font>) | Inf(<font color="#F17CB0">❶</font>) | Inf(<font color="#FAA43A">❷</font>))>
  labelloc="t"
  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=<1<br/><font color="#B276B2">❸</font>>]
  0 -> 1 [label=<a<br/><font color="#F17CB0">❶</font><font color="#B276B2">❸</font>>]
  0 -> 2 [label=<!a<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 [label="1"]
  1 -> 0 [label=<b<br/><font color="#B276B2">❸</font>>]
  1 -> 1 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 -> 2 [label=<!a &amp; b<br/><font color="#FAA43A">❷</font><font color="#B276B2">❸</font>>]
  2 [label="2"]
  2 -> 0 [label=<!b>]
  2 -> 1 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 2 [label=<!a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex3.svg :var txt=autfilt-ex3 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex3.svg]]


Using =--remove-fin= transforms the automaton to remove all traces
of Fin-acceptance: this usually requires adding non-deterministic jumps to
altered copies of strongly-connected components.

#+NAME: autfilt-ex4
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --remove-fin aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex4
#+begin_example
digraph G {
  rankdir=LR
  label=<Inf(<font color="#5DA5DA">⓿</font>) | Inf(<font color="#F17CB0">❶</font>) | (Inf(<font color="#FAA43A">❷</font>)&amp;Inf(<font color="#B276B2">❸</font>))>
  labelloc="t"
  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=<1<br/><font color="#B276B2">❸</font>>]
  0 -> 1 [label=<a<br/><font color="#F17CB0">❶</font><font color="#B276B2">❸</font>>]
  0 -> 2 [label=<!a<br/><font color="#B276B2">❸</font>>]
  1 [label="1"]
  1 -> 0 [label=<b<br/><font color="#B276B2">❸</font>>]
  1 -> 1 [label=<a &amp; b<br/><font color="#B276B2">❸</font>>]
  1 -> 2 [label=<!a &amp; b<br/><font color="#FAA43A">❷</font><font color="#B276B2">❸</font>>]
  2 [label="2"]
  2 -> 0 [label=<!b>]
  2 -> 1 [label=<a &amp; !b>]
  2 -> 2 [label=<!a &amp; !b>]
  2 -> 3 [label=<!a &amp; !b>]
  3 [label="3"]
  3 -> 3 [label=<!a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex4.svg :var txt=autfilt-ex4 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex4.svg]]

Use =--mask-acc=NUM= to remove some acceptances sets and all
transitions they contain.  The acceptance condition will be updated to
reflect the fact that these sets can never be visited.

#+NAME: autfilt-ex5
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --mask-acc=1,2 aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex5
#+begin_example
digraph G {
  rankdir=LR
  label=<Fin(<font color="#F17CB0">❶</font>) &amp; Inf(<font color="#5DA5DA">⓿</font>)>
  labelloc="t"
  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=<1<br/><font color="#F17CB0">❶</font>>]
  0 -> 1 [label=<!a<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
  1 [label="1"]
  1 -> 0 [label=<!b>]
  1 -> 2 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  1 -> 1 [label=<!a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  2 [label="2"]
  2 -> 0 [label=<b<br/><font color="#F17CB0">❶</font>>]
  2 -> 2 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex5.svg :var txt=autfilt-ex5 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex5.svg]]

** Atomic proposition removal

Atomic propositions can be removed from an automaton in three ways:
- use ~--remove-ap=a~ to remove =a= by existential quantification, i.e., both =a= and its negation will be replaced by true.
  This does not remove any transition.
- use ~--remove-ap=a=0~ to keep only transitions compatible with =!a= (i.e, transitions requiring =a= will be removed).
- use ~--remove-ap=a=1~ to keep only transitions compatible with =a= (i.e, transitions requiring =!a= will be removed).

Here are the results of these three options on our example:

#+NAME: autfilt-ex6a
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --remove-ap=a aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex6a
#+begin_example
digraph G {
  rankdir=LR
  label=<(Fin(<font color="#F17CB0">❶</font>) &amp; Fin(<font color="#B276B2">❸</font>) &amp; Inf(<font color="#5DA5DA">⓿</font>)) | (Inf(<font color="#FAA43A">❷</font>)&amp;Inf(<font color="#B276B2">❸</font>)) | Inf(<font color="#F17CB0">❶</font>)>
  labelloc="t"
  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=<1<br/><font color="#B276B2">❸</font>>]
  0 -> 1 [label=<1<br/><font color="#F17CB0">❶</font><font color="#B276B2">❸</font>>]
  0 -> 2 [label=<1<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 [label="1"]
  1 -> 0 [label=<b<br/><font color="#B276B2">❸</font>>]
  1 -> 1 [label=<b<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 -> 2 [label=<b<br/><font color="#FAA43A">❷</font><font color="#B276B2">❸</font>>]
  2 [label="2"]
  2 -> 0 [label=<!b>]
  2 -> 1 [label=<!b<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 2 [label=<!b<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex6a.svg :var txt=autfilt-ex6a :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex6a.svg]]

#+NAME: autfilt-ex6b
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --remove-ap=a=0 aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex6b
#+begin_example
digraph G {
  rankdir=LR
  label=<(Fin(<font color="#F17CB0">❶</font>) &amp; Fin(<font color="#B276B2">❸</font>) &amp; Inf(<font color="#5DA5DA">⓿</font>)) | (Inf(<font color="#FAA43A">❷</font>)&amp;Inf(<font color="#B276B2">❸</font>)) | Inf(<font color="#F17CB0">❶</font>)>
  labelloc="t"
  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=<1<br/><font color="#B276B2">❸</font>>]
  0 -> 1 [label=<1<br/><font color="#5DA5DA">⓿</font><font color="#B276B2">❸</font>>]
  1 [label="1"]
  1 -> 0 [label=<!b>]
  1 -> 1 [label=<!b<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-ex6b.svg :var txt=autfilt-ex6b :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex6b.svg]]

#+NAME: autfilt-ex6c
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --remove-ap=a=1 aut-ex1.hoa --dot
#+END_SRC

#+RESULTS: autfilt-ex6c

#+BEGIN_SRC dot :file autfilt-ex6c.svg :var txt=autfilt-ex6c :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex6c.svg]]

** Testing word acceptance

The following example checks whether the formula ~a U b U c~ accepts
the word ~a&!b&!c; cycle{!a&!b&c}~.

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba 'a U b U c' |
  autfilt --accept-word 'a&!b&!c; cycle{!a&!b&c}' -q  && echo "word accepted"
#+END_SRC
#+RESULTS:
: word accepted

Here is an example where we generate an infinite stream of random LTL
formulas using [[file:randltl.org][=randltl=]], convert them all to automata using
[[file:ltl2tgba.org][=ltl2tgba=]], filter out the first 10 automata that accept both the
words =a&!b;cycle{!a&!b}= and =!a&!b;cycle{a&b}= yet reject any word
of the form =cycle{b}=, and display the associated formula (which was
stored as the name of the automaton by =ltl2tgba=).

#+BEGIN_SRC sh :results verbatim :exports both
randltl -n -1 a b | ltlfilt --simplify --uniq | ltl2tgba |
  autfilt --accept-word='a&!b;cycle{!a&!b}' --accept-word='!a&!b;cycle{a&b}' \
          --reject-word='cycle{b}' --stats=%M -n 10
#+END_SRC

#+RESULTS:
#+begin_example
F!b
!b
F(!a & !b)
(!a & (XX!a | (!a W F!b))) R !b
F(Fb R !b)
Fa R F!b
Fa U !b
!b & X(!b W Ga)
Fb R F!b
XF!b U (!b & (!a | G!b))
#+end_example

Note that the above example could be simplified using the
=--accept-word= and =--reject-word= options of =ltlfilt= directly.
However this demonstrates that using =--stats=%M=, it is possible to
filter formulas based on some properties of automata that have been
generated by from them.  The translator needs not be =ltl2tgba=: other
tools can be wrapped with [[file:ltldo.org][=ltldo --name=%f=]] to ensure they work well
in a pipeline and preserve the formula name in the HOA output.  For
example Here is a list of 5 LTL formulas that =ltl2dstar= converts to
Rabin automata that have exactly 4 states:

#+BEGIN_SRC sh :results verbatim :exports both
randltl -n -1 a b | ltlfilt --simplify --remove-wm |
    ltldo ltl2dstar --name=%f | autfilt --states=4 --stats=%M -n 5
#+END_SRC

#+RESULTS:
: Gb | G!b
: b R (a | b)
: (a & !b & (b | (F!a U (!b & F!a)))) | (!a & (b | (!b & (Ga R (b | Ga)))))
: (a & (a U !b)) | (!a & (!a R b))
: a | G((a & GFa) | (!a & FG!a))

** Decorations
   :PROPERTIES:
   :CUSTOM_ID: decorations
   :END:

We know from a previous exemple that formula =a U b U c= accepts the
word =b; cycle{c}=.  We can actually highlight the corresponding
run in the automaton:

#+NAME: highlight-word
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba 'a U b U c' | autfilt --highlight-word='a&!b&!c; cycle{!a&!b&c}' -d
#+END_SRC

#+RESULTS: highlight-word
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  node [style="filled", fillcolor="#ffffa0"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 2
  0 [label="0", peripheries=2]
  0 -> 0 [label=<1>, style=bold, color="#F17CB0"]
  1 [label="1"]
  1 -> 0 [label=<c>]
  1 -> 1 [label=<b &amp; !c>]
  2 [label="2"]
  2 -> 0 [label=<c>, style=bold, color="#F17CB0"]
  2 -> 1 [label=<!a &amp; b &amp; !c>]
  2 -> 2 [label=<a &amp; !c>, style=bold, color="#F17CB0"]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-hlword.svg :var txt=highlight-word :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-hlword.svg]]


We can change the color by prefixing the word with a number and a
comma.  Also it is possible to highlight multiple words, but a
transition may only have one color so late highlights will overwrite
previous ones.

#+NAME: highlight-word2
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba 'a U b U c' |
  autfilt --highlight-word=5,'a&!b&!c; cycle{!a&!b&c}' \
          --highlight-word=4,'!a&b&!c; cycle{!a&!b&c}' -d
#+END_SRC

#+RESULTS: highlight-word2
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  node [style="filled", fillcolor="#ffffa0"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 2
  0 [label="0", peripheries=2]
  0 -> 0 [label=<1>, style=bold, color="#60BD68"]
  1 [label="1"]
  1 -> 0 [label=<c>, style=bold, color="#60BD68"]
  1 -> 1 [label=<b &amp; !c>]
  2 [label="2"]
  2 -> 0 [label=<c>, style=bold, color="#F15854"]
  2 -> 1 [label=<!a &amp; b &amp; !c>, style=bold, color="#60BD68"]
  2 -> 2 [label=<a &amp; !c>, style=bold, color="#F15854"]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-hlword2.svg :var txt=highlight-word2 :exports results
  $txt
#+END_SRC

#+RESULTS:
[[file:autfilt-hlword2.svg]]



Another useful thing to highlight is nondeterminism.  One can
highlight states or edges where nondeterministic choices need to be
made.

#+NAME: highlight-nondet
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba 'F((b R a) W Gb)' |
    autfilt --highlight-nondet-states=5 --highlight-nondet-edges=1 -d
#+END_SRC

#+RESULTS: highlight-nondet
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  node [style="filled", fillcolor="#ffffa0"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0"]
  0 -> 0 [label=<b<br/><font color="#5DA5DA">⓿</font>>]
  1 [label="1", style="bold,filled", color="#F15854"]
  1 -> 0 [label=<!a &amp; b>, style=bold, color="#5DA5DA"]
  1 -> 1 [label=<1>, style=bold, color="#5DA5DA"]
  1 -> 2 [label=<a &amp; b>]
  1 -> 3 [label=<a &amp; !b>]
  2 [label="2"]
  2 -> 0 [label=<!a &amp; b>]
  2 -> 2 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 3 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  3 [label="3"]
  3 -> 2 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  3 -> 3 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file autfilt-hlnondet.svg :var txt=highlight-nondet :exports results
  $txt
#+END_SRC

#+RESULTS:
[[file:autfilt-hlnondet.svg]]


#+BEGIN_SRC sh :results silent :exports results
rm -f example.hoa aut-ex1.hoa
#+END_SRC