`autfilt`

## Table of Contents

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 Hanoi Omega Automata
Format, as Spin never claims, using LBTT's format, or using
`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
the common output options like `--spin`

, `--lbtt`

, `--dot`

,
`--stats`

…

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

digraph "" { rankdir=LR label=<Inf(<font color="#1F78B4">⓿</font>)<br/>[Büchi]> labelloc="t" node [shape="circle"] node [style="filled", fillcolor="#ffffa0"] fontname="Lato" node [fontname="Lato"] edge [fontname="Lato"] I [label="", style=invis, width=0] I -> 0 0 [label=<0>] 0 -> 0 [label=<p0<br/><font color="#1F78B4">⓿</font>>] 0 -> 0 [label=<!p0>] }

The `--spin`

option implicitly requires a degeneralization:

autfilt example.hoa --spin

never { T0_init: if :: (!(p0)) -> goto T0_init :: (p0) -> goto accept_S1 fi; accept_S1: if :: (!(p0)) -> goto T0_init :: (p0) -> goto accept_S1 fi; }

Option `--lbtt`

only works for Büchi or generalized Büchi acceptance.

autfilt example.hoa --lbtt

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 `randaut`

to generate 10 random
automata, and pipe the result into `autfilt`

to display various
statistics.

```
randaut -n 10 -A0..2 -Q10..20 -e0.05 2 |
autfilt --stats='%s states, %e edges, %a acc-sets, %c SCCs, det=%d'
```

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

The following `%`

sequences are available:

%< 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, %[LETTER]e number of edges (add one LETTER to select (r) reachable [default], (u) unreachable, (a) all). %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, %[LETTER]s number of states (add one LETTER to select (r) reachable [default], (u) unreachable, (a) all). %t, %[LETTER]t number of transitions (add one LETTER to select (r) reachable [default], (u) unreachable, (a) all). %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

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.

--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.

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

The standard set of automata simplification routines (these are often
referred to as the "post-processing" routines, because these are the
procedures performed by `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:

-b, --buchi, --Buchi automaton with Büchi acceptance -B, --sba, --ba state-based Büchi Automaton (same as -S -b) --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, --gba automaton with Generalized Büchi acceptance

These options specify any simplification goal:

-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:

--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 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 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:

--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. --kill-states=NUM[,NUM...] mark the specified states as dead (no successor), and remove them. Implies --remove-dead-states. --mask-acc=NUM[,NUM...] remove all transitions in specified acceptance sets --merge-transitions merge transitions with same destination and acceptance --partial-degeneralize[=NUM1,NUM2,...] Degeneralize automata according to sets NUM1,NUM2,... If no sets are given, partial degeneralization is performed for all conjunctions of Inf and disjunctions of Fin. --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 --to-finite[=alive] Convert an automaton with "alive" and "!alive" propositions into a Büchi automaton interpretable as a finite automaton. States with a outgoing "!alive" edge are marked as accepting.

## 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`

).

--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

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:

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

(Note: that the `--dot`

option used below uses some default options
discussed on another page.)

autfilt aut-ex1.hoa --dot

Using `-S`

will "push" the acceptance membership of the transitions to the states:

autfilt -S aut-ex1.hoa --dot

Using `--cnf-acceptance`

simply rewrites the acceptance condition in Conjunctive Normal Form:

autfilt --cnf-acceptance aut-ex1.hoa --dot

Using `--simplify-acc`

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

autfilt --simplify-acc aut-ex1.hoa --dot

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.

autfilt --remove-fin --small aut-ex1.hoa --dot

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.

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

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.)

autfilt --colored-parity aut-ex1.hoa --dot

A specific type of parity acceptance can be forced by passing it as an
argument of the `--parity`

or `--colored-parity`

option.

```
autfilt --parity='min odd' aut-ex1.hoa --dot
```

### 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:

autfilt --remove-ap=a aut-ex1.hoa --dot

```
autfilt --remove-ap=a=0 aut-ex1.hoa --dot
```

```
autfilt --remove-ap=a=1 aut-ex1.hoa --dot
```

### 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}`

.

ltl2tgba 'a U b U c' | autfilt --accept-word 'a&!b&!c; cycle{!a&!b&c}' -q && echo "word accepted"

word accepted

Here is an example where we generate an infinite stream of random LTL
formulas using `randltl`

, convert them all to automata using
`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`

).

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

F!b !b F(!a & !b) !a 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))

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 `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:

randltl -n -1 a b | ltlfilt --simplify --remove-wm | ltldo ltl2dstar --name=%f | autfilt --states=4 --stats=%M -n 5

Gb | G!b b R (a | b) (a & !b & (b | F(!b & F!a))) | (!a & (b | (!b & G(b | Ga)))) (a & (a U !b)) | (!a & (!a R b)) a | G((a & GFa) | (!a & FG!a))

### Decorations

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:

ltl2tgba 'a U b U c' | autfilt --highlight-word='a&!b&!c; cycle{!a&!b&c}' -d

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.

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

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

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