autfilt.org

# -*- coding: utf-8 -*-
#+TITLE: =autfilt=
#+DESCRIPTION: Spot command-line tool for filtering, transforming, and converting ω-automata.
#+INCLUDE: setup.org
#+HTML_LINK_UP: tools.html
#+PROPERTY: header-args:sh :results verbatim :exports both

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

The tool operates a loop over 5 phases:
- input one automaton
- optionally pre-process the automaton
- optionally filter the automaton (i.e., decide whether to ignore the
  automaton or continue with it)
- optionally post-process 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
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 :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
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
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
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 :exports results
ltl2tgba --help | sed -n '/ sequences:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
#+begin_example
  %<                         the part of the line before the formula if it
                             comes from a column extracted from a CSV file
  %>                         the part of the line after the formula if it comes
                             from a column extracted from a CSV file
  %%                         a single %
  %a                         number of acceptance sets
  %c, %[LETTERS]c            number of SCCs; you may filter the SCCs to count
                             using the following LETTERS, possibly
                             concatenated: (a) accepting, (r) rejecting, (c)
                             complete, (v) trivial, (t) terminal, (w) weak,
                             (iw) inherently weak. Use uppercase letters to
                             negate them.
  %d                         1 if the output is deterministic, 0 otherwise
  %e                         number of reachable edges
  %f                         the formula, in Spot's syntax
  %F                         name of the input file
  %g, %[LETTERS]g            acceptance condition (in HOA syntax); add brackets
                             to print an acceptance name instead and LETTERS to
                             tweak the format: (0) no parameters, (a)
                             accentuated, (b) abbreviated, (d) style used in
                             dot output, (g) no generalized parameter, (l)
                             recognize Street-like and Rabin-like, (m) no main
                             parameter, (p) no parity parameter, (o) name
                             unknown acceptance as 'other', (s) shorthand for
                             'lo0'.
  %h                         the automaton in HOA format on a single line (use
                             %[opt]h to specify additional options as in
                             --hoa=opt)
  %L                         location in the input file
  %m                         name of the automaton
  %n                         number of nondeterministic states in output
  %p                         1 if the output is complete, 0 otherwise
  %r                         wall-clock time elapsed in seconds (excluding
                             parsing)
  %R, %[LETTERS]R            CPU time (excluding parsing), in seconds; Add
                             LETTERS to restrict to(u) user time, (s) system
                             time, (p) parent process, or (c) children
                             processes.
  %s                         number of reachable states
  %t                         number of reachable transitions
  %u, %[e]u                  number of states (or [e]dges) with universal
                             branching
  %u, %[LETTER]u             1 if the automaton contains some universal
                             branching (or a number of [s]tates or [e]dges with
                             universal branching)
  %w                         one word accepted by the output automaton
  %x, %[LETTERS]x            number of atomic propositions declared in the
                             automaton;  add LETTERS to list atomic
                             propositions with (n) no quoting, (s) occasional
                             double-quotes with C-style escape, (d)
                             double-quotes with C-style escape, (c)
                             double-quotes with CSV-style escape, (p) between
                             parentheses, any extra non-alphanumeric character
                             will be used to separate propositions
#+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 :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
      --acceptance-is=NAME|FORMULA
                             match automata with given acceptance condition
      --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 that are equivalent
                             (language-wise) to the automaton in FILENAME
      --has-exist-branching  keep automata that use existential branching
                             (i.e., make non-deterministic choices)
      --has-univ-branching   keep alternating automata that use universal
                             branching
      --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-alternating       keep only automata using universal branching
      --is-colored           keep colored automata (i.e., exactly one
                             acceptance mark per transition or state)
      --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-semi-deterministic   keep semi-deterministic automata
      --is-stutter-invariant keep automata representing stutter-invariant
                             properties
      --is-terminal          keep only terminal automata
      --is-unambiguous       keep only unambiguous automata
      --is-very-weak         keep only very-weak automata
      --is-weak              keep only weak automata
      --nondet-states=RANGE  keep automata whose number of nondeterministic
                             states is in RANGE
  -N, --nth=RANGE            assuming input automata are numbered from 1, keep
                             only those 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 and changing acceptance conditions
   :PROPERTIES:
   :header-args:sh: :results verbatim :exports results
   :END:

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
autfilt --help | sed -n '/Output automaton type:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
#+begin_example
  -B, --ba                   Büchi Automaton (with state-based acceptance)
      --cobuchi, --coBuchi   automaton with co-Büchi acceptance (will
                             recognize a superset of the input language if not
                             co-Büchi realizable)
  -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
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 (combine with
:                              --generic to be sure to obtain a deterministic
:                              automaton)
:       --small                prefer small automata

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

#+BEGIN_SRC sh
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 the minimization 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 :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 (different strategies
                             are used)
      --complement-acceptance   complement the acceptance condition (without
                             touching the automaton)
      --decompose-scc=t|w|s|N|aN, --decompose-strength=t|w|s|N|aN
                             extract the (t) terminal, (w) weak, or (s) strong
                             part of an automaton or (N) the subautomaton
                             leading to the Nth SCC, or (aN) to the Nth
                             accepting SCC (option can be combined with commas
                             to extract multiple parts)
      --destut               allow less stuttering
      --dnf-acceptance       put the acceptance condition in Disjunctive Normal
                             Form
      --dualize              dualize each automaton
      --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). Supported
                             options are acc=STRING, states=N, max-states=N,
                             sat-incr=N, sat-incr-steps=N, sat-langmap,
                             sat-naive, colored, preproc=N. Spot uses by
                             default its PicoSAT distribution but an external
                             SATsolver can be set thanks to the SPOT_SATSOLVER
                             environment variable(see spot-x).
      --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-acceptance  simplify the acceptance condition by merging
                             identical acceptance sets and by simplifying some
                             terms containing complementary sets
      --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)
      --split-edges          split edges into transitions labeled by
                             conjunctions of all atomic propositions, so they
                             can be read as letters
      --streett-like         convert to an automaton with Streett-like
                             acceptance. Works only with acceptance condition
                             in DNF
      --strip-acceptance     remove the acceptance condition and all acceptance
                             sets
      --sum=FILENAME, --sum-or=FILENAME
                             build the sum with the automaton in FILENAME to
                             sum languages
      --sum-and=FILENAME     build the sum with the automaton in FILENAME to
                             intersect languages
#+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 :exports results
autfilt --help | sed -n '/^ *Decorations.*:/,/^$/p' | sed '1d;$d'
#+END_SRC

#+RESULTS:
#+begin_example
      --highlight-accepting-run[=NUM]
                             highlight one accepting run using color NUM
      --highlight-languages  highlight states that recognize identical
                             languages
      --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
#+end_example

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
   :PROPERTIES:
   :header-args:sh: :results verbatim :exports code
   :END:

Here is an automaton with transition-based acceptance:

#+BEGIN_SRC sh :results silent
cat >aut-ex1.hoa<<EOF
HOA: v1
States: 3
Start: 0
AP: 2 "a" "b"
Acceptance: 5 Inf(0)&Fin(1)&Fin(4) | Inf(2)&Inf(3) | Inf(1)
--BODY--
State: 0 {3}
[t] 0
[0] 1 {1}
[!0] 2 {0 4}
State: 1 {3}
[1] 0
[0&1] 1 {0}
[!0&1] 2 {2 4}
State: 2
[!1] 0
[0&!1] 1 {0}
[!0&!1] 2 {0 4}
--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
autfilt aut-ex1.hoa --dot
#+END_SRC

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

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

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

#+NAME: autfilt-ex2
#+BEGIN_SRC sh
autfilt -S aut-ex1.hoa --dot
#+END_SRC

#+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
autfilt --cnf-acceptance aut-ex1.hoa --dot
#+END_SRC

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

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


Using =--simplify-acc= applies several rules (like unit-propagation, detection
of identical acceptance sets, etc) to simplify the acceptance formula of an automaton.

#+NAME: autfilt-ex3b
#+BEGIN_SRC sh
autfilt --simplify-acc aut-ex1.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfilt-ex3b.svg :var txt=autfilt-ex3b :exports results
  $txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex3b.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.  Fin removal does
not simplify the automaton constructed, so additionally passing
=--small= will help reduce the automaton.

#+NAME: autfilt-ex4
#+BEGIN_SRC sh
autfilt --remove-fin --small aut-ex1.hoa --dot
#+END_SRC

#+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
autfilt --mask-acc=1,2 aut-ex1.hoa --dot
#+END_SRC

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

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


The =--colored-parity= request a transformation to parity acceptance.
The "colored" part of the option mean that each edge should be
colored by one acceptance sets.  (Using =--parity= would allow edges
without any color.)

#+NAME: autfilt-ex6
#+BEGIN_SRC sh
autfilt --colored-parity aut-ex1.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfilt-ex6.svg :var txt=autfilt-ex6 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfilt-ex6.svg]]

A specific type of parity acceptance can be forced by passing it as an
argument of the =--parity= or =--colored-parity= option.

#+NAME: autfilt-ex6b
#+BEGIN_SRC sh
autfilt --parity='min odd' aut-ex1.hoa --dot
#+END_SRC

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

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



** Atomic proposition removal
   :PROPERTIES:
   :header-args:sh: :results verbatim :exports code
   :END:

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
autfilt --remove-ap=a aut-ex1.hoa --dot
#+END_SRC

#+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
autfilt --remove-ap=a=0 aut-ex1.hoa --dot
#+END_SRC

#+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
autfilt --remove-ap=a=1 aut-ex1.hoa --dot
#+END_SRC

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

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

** Testing word acceptance
   :PROPERTIES:
   :header-args:sh: :results verbatim :exports both
   :END:


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
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
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
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:
   :header-args:sh: :results verbatim :exports code
   :END:

We know from a previous example 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
ltl2tgba 'a U b U c' | autfilt --highlight-word='a&!b&!c; cycle{!a&!b&c}' -d
#+END_SRC

#+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
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

#+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
ltl2tgba 'F((b R a) W Gb)' |
    autfilt --highlight-nondet-states=5 --highlight-nondet-edges=1 -d
#+END_SRC

#+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

#  LocalWords:  utf html args pre LBTT's dstar lbtt SRC EOF DOTEXTRA
#  LocalWords:  rankdir invis init fi randaut acc SCCs det tgba sed
#  LocalWords:  CSV iw dges tates sccs ap SCC nondet rej triv ba Dax
#  LocalWords:  cobuchi coBuchi superset sbacc determinized al ATVA
#  LocalWords:  Safra Redziejowski cnf scc aN subautomaton destut dnf
#  LocalWords:  dualize APs rabin gra streett gsa instut NUM incr aut
#  LocalWords:  langmap preproc PicoSAT SATsolver SATSOLVER svg txt
#  LocalWords:  randltl ltlfilt uniq Fb XF ltldo wm Gb GFa FG hlword
#  LocalWords:  hlnondet