# ltl2tgba

This tool translates LTL or PSL formulas into different types of automata.

The inner algorithm produces Transition-based Generalized Büchi Automata, hence the name of the tool, but ltl2tgba has grown and now offers several options to adjust the type of automaton output. Those options will be covered in more detail below, but here is a quick summary:

• --tgba (the default) outputs Transition-based Generalized Büchi Automata,

= --buchi (or -b) outputs transition-based Büchi automata,

• --sba (or -B) outputs state-based Büchi automata (--ba is a historical alias for this option)
• --monitor (or -M) outputs monitors
• --generic --deterministic (or -DG) will do whatever it takes to produce a deterministic automaton, and may use any acceptance condition
• --parity --deterministic (or -DP) will produce a deterministic automaton with parity acceptance.

The latter two can also be used without --deterministic, but that is less frequent.

Except for -B which forces state-based acceptance, these options build transition-based automata by default. Internally, Spot only supports transition-based automata. However, while transition-based automata can be smaller then their state-based counterpart, there are many cases where transition-based acceptance does not bring any benefits.

In case where it is detected that the transition-based automaton looks like a state-based automaton, the different automaton printers will usually present the result as an automaton with state-based acceptance for simplicity.

## TGBA and BA

Formulas to translate may be specified using common input options for LTL/PSL formulas.

ltl2tgba -f 'Fa & GFb'

HOA: v1
name: "Fa & GFb"
States: 2
Start: 0
AP: 2 "a" "b"
acc-name: Buchi
Acceptance: 1 Inf(0)
properties: trans-labels explicit-labels trans-acc complete
properties: deterministic stutter-invariant
--BODY--
State: 0
[!0] 0
[0] 1
State: 1
[!1] 1
[1] 1 {0}
--END--


As ltl2tgba is often used with a single formula passed on the command line, the -f option can be omitted and any command-line parameter that is not the argument of some option will be assumed to be a formula to translate (this differs from ltlfilt, where such parameters are assumed to be filenames).

ltl2tgba honors the common options for selecting the output format. The default output format, as shown above, is the HOA format, as this can easily be piped to other tools.

To convert the automaton into a picture, or into vectorial format, use --dot or -d to request GraphViz output and process the result with dot or dotty. Typically, one can get a pdf of this TGBA using

ltl2tgba "Fa & GFb" -d | dot -Tpdf > tgba.pdf


The result would look like this (note that in this documentation we use some environment variables to produce a more colorful output by default)

Characters like ⓿, ❶, etc. denotes the acceptance sets a transition belongs to. In this case, there is only one acceptance set, called 0, containing a single transition. An acceptance set can contains multiple transitions, and a transition may also belong to multiple acceptance sets. An infinite path through this automaton is accepting iff it visits each acceptance set infinitely often. Therefore, in the above example, any accepted path will necessarily leave the initial state after a finite amount of steps, and then it will verify the property b infinitely often. It is also possible that an automaton do not use any acceptance set at all, in which any run is accepting.

Here is a TGBA with multiple acceptance sets (we omit the call to dot to render the output of ltl2tgba from now on):

ltl2tgba "GFa & GFb" -d


The above TGBA has two acceptance sets: ⓿ and ❶. The definition of these acceptance sets ensures that atomic propositions a and b must be true infinitely often.

A Büchi automaton for the previous formula can be obtained with the -B or -b options, depending on whether we insist on have state-based acceptance (-B) or if transition-based acceptance is OK (-b).

ltl2tgba -b 'GFa & GFb' -d


ltl2tgba -B 'GFa & GFb' -d


Although accepting states in state-based Büchi automaton are (traditionally) pictured with double-lines, internally this automaton is still handled as a TGBA with a single acceptance set such that the transitions leaving the state are either all accepting, or all non-accepting. The underlying TGBA can be seen by passing the --dot=t option (the t requests the use of transition-based acceptance, as it is done internally):

ltl2tgba -B 'GFa & GFb' --dot=t


Option -B is in fact a shorthand for -b -S, where -b requests Büchi acceptance, and -S requests state-based acceptance.

Calling ltl2tgba -S will build an automaton with generalized Büchi acceptance (the default for ltltgba), but force state-based acceptance on the output.

Here is the same formula as above, for comparison.

ltl2tgba -S 'GFa & GFb' -d


Note that ltl2tgba is not very good at generating state-based generalized Büchi automata (GBA): all it does is generating a transition-based one internally, and then pushing acceptance sets onto states. On this example, the resulting GBA produced by -S is larger than the BA produced by -B.

As already discussed on the page about common output options, various options controls the output format of ltl2tgba:

-8, --utf8                 enable UTF-8 characters in output (ignored with
--lbtt or --spin)
--check[=PROP]         test for the additional property PROP and output
the result in the HOA format (implies -H).  PROP
may be some prefix of 'all' (default),
'unambiguous', 'stutter-invariant',
'stutter-sensitive-example', 'semi-determinism',
or 'strength'.
-d, --dot[=1|a|A|b|B|c|C(COLOR)|e|E|f(FONT)|h|i(ID)|k|K|n|N|o|r|R|s|t|u|v|y|+INT|<INT|#]
GraphViz's format.  Add letters for (1) force
numbered states, (a) show acceptance condition
(default), (A) hide acceptance condition, (b)
acceptance sets as bullets, (B) bullets except for
Büchi/co-Büchi automata, (c) force circular
nodes, (C) color nodes with COLOR, (d) show
origins when known, (e) force elliptic nodes, (E)
force rEctangular nodes, (f(FONT)) use FONT, (g)
hide edge labels, (h) horizontal layout, (i) or
(i(GRAPHID)) add IDs, (k) use state labels when
possible, (K) use transition labels (default), (n)
show name, (N) hide name, (o) ordered transitions,
(r) rainbow colors for acceptance sets, (R) color
acceptance sets by Inf/Fin, (s) with SCCs, (t)
force transition-based acceptance, (u) hide true
states, (v) vertical layout, (y) split universal
edges by color, (+INT) add INT to all set numbers,
(<INT) display at most INT states, (#) show
internal edge numbers
-H, --hoaf[=1.1|i|k|l|m|s|t|v]   Output the automaton in HOA format
(default).  Add letters to select (1.1) version
1.1 of the format, (i) use implicit labels for
complete deterministic automata, (s) prefer
state-based acceptance when possible [default],
(t) force transition-based acceptance, (m) mix
state and transition-based acceptance, (k) use
state labels when possible, (l) single-line
output, (v) verbose properties
--lbtt[=t]             LBTT's format (add =t to force transition-based
acceptance even on Büchi automata)
--name=FORMAT          set the name of the output automaton
-o, --output=FORMAT        send output to a file named FORMAT instead of
standard output.  The first automaton sent to a
file truncates it unless FORMAT starts with '>>'.
-q, --quiet                suppress all normal output
-s, --spin[=6|c]           Spin neverclaim (implies --ba).  Add letters to
select (6) Spin's 6.2.4 style, (c) comments on
states
--stats=FORMAT, --format=FORMAT


Option -8 can be used to improve the readability of the output on systems that can display UTF-8 correctly.

ltl2tgba -8B "GFa & GFb" -d


## Spin output

Using the --spin or -s option, ltl2tgba will produce a Büchi automaton (the -B option is implied) as a never claim that can be fed to Spin. ltl2tgba -s is therefore a drop-in replacement for spin -f.

ltl2tgba -s 'GFa & GFb'

never { /* G(Fa & Fb) */
accept_init:
if
:: ((a) && (b)) -> goto accept_init
:: (!(b)) -> goto T0_S1
:: ((!(a)) && (b)) -> goto T0_S2
fi;
T0_S1:
if
:: ((a) && (b)) -> goto accept_init
:: (!(b)) -> goto T0_S1
:: ((!(a)) && (b)) -> goto T0_S2
fi;
T0_S2:
if
:: (a) -> goto accept_init
:: (!(a)) -> goto T0_S2
fi;
}


Since Spin 6 extended its syntax to support arbitrary atomic propositions, you may also need put the parser in --lenient mode to support these:

ltl2tgba -s --lenient '(a < b) U (process[2]@ok)'

never { /* "a < b" U "process[2]@ok" */
T0_init:
if
:: (process[2]@ok) -> goto accept_all
:: ((a < b) && (!(process[2]@ok))) -> goto T0_init
fi;
accept_all:
skip
}


## Do you favor deterministic or small automata?

The translation procedure can be controled by a few switches. A first set of options specifies the goal of the simplification routines: whenever possible, would you prefer a small automaton (--small) or a deterministic (--deterministic) automaton?

-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 (default)


The --any option tells the translator that it should attempt to reduce or produce a deterministic result result: any automaton denoting the given formula is OK. This effectively disables post-processings and speeds up the translation.

With the -D or --deterministic option, the translator will attempt to produce a deterministic automaton, even if this requires a lot of states. ltl2tgba knows how to produce the minimal deterministic Büchi automaton for any obligation property (this includes safety properties).

With the --small option (the default), the translator will not produce a deterministic automaton when it knows how to build smaller automaton.

Note that options --deterministic and --small express preferences. They certainly do not guarantee that the output will be deterministic, or will be the smallest automaton possible.

In particular, for properties more complex than obligations, it is possible that no deterministic TGBA exist, and even if it exists, ltl2tgba might not find it: so a non-deterministic automaton can be returned in this case. If you absolutely want a deterministic automaton, see the --generic option or the --parity option.

An example formula where the difference between -D and --small is flagrant is Ga|Gb|Gc:

ltl2tgba 'Ga|Gb|Gc' -d


ltl2tgba -D 'Ga|Gb|Gc' -d


You can augment the number of terms in the disjunction to magnify the difference. For N terms, the --small automaton has N+1 states, while the --deterministic automaton needs 2^N-1 states.

Add the -C or --complete option if you want to obtain a complete automaton, with a sink state capturing that rejected words that would not otherwise have a run in the output automaton.

Add the -U or --unambiguous option if you want unambiguous automata to be produced. An automaton is unambiguous if any word is recognized by at most one accepting run of the automaton (however a word can be rejected by multiple runs, so unambiguous automata can be non-deterministic).

The following example is an ambiguous Büchi automaton, because the are two ways to accept a run that repeats continuously the configuration $$\bar ab$$.

ltl2tgba -B 'GFa -> GFb' -d


Here is an unambiguous automaton for the same formula, in which there is only one run that recognizes this example word:

ltl2tgba -B -U 'GFa -> GFb' -d


Unlike --small and --deterministic that express preferences, options --complete and --unambiguous do guarantee that the output will be complete and unambiguous.

A last parameter that can be used to tune the translation is the amount of pre- and post-processing performed. These two steps can be adjusted via a common set of switches:

--high                 all available optimizations (slow, default)
--low                  minimal optimizations (fast)
--medium               moderate optimizations


Pre-processings are rewritings done on the LTL formulas, usually to reduce its size, but mainly to put it in a form that will help the translator (for instance F(a|b) is easier to translate than F(a)|F(b)). At --low level, only simple syntactic rewritings are performed. At --medium level, additional simplifications based on syntactic implications are performed. At --high level, language containment is used instead of syntactic implications.

Post-processings are cleanups and simplifications of the automaton produced by the core translator. The algorithms used during post-processing are

• SCC filtering: removing useless strongly connected components, and useless acceptance sets.
• direct simulation: merge states based on suffix inclusion.
• iterated simulations: merge states based on suffix inclusion, or prefix inclusion, in a loop.
• WDBA minimization: determinize and minimize automata representing obligation properties.
• degeneralization: convert a TGBA into a BA
• BA simulation (again direct or iterated)

The chaining of these various algorithms depends on the selected combination of optimization level (--low, --medium, --high), translation intent (--small, --deterministic) and type of automaton desired (--tgba, --ba).

A notable configuration is --any --low, which will produce a TGBA as fast as possible. In this case, post-processing is disabled, and only syntactic rewritings are performed. This can be used for satisfiability checking, although in this context even building an automaton is overkill (you only need an accepted run).

Finally, it should be noted that the default optimization options (--small --high) are usually overkill. --low will produce good automata most of the time. Most of pattern formulas of genltl will be efficiently translated in this configuration (meaning that --small --high will not produce a better automaton). If you are planning to generate automata for large family of pattern formulas, it makes sense to experiment with the different settings on a small version of the pattern, and select the lowest setting that satisfies your expectations.

## Deterministic automata with --generic --deterministic

The --generic (or -G) option allows ltl2tgba to use more complex acceptance conditions. This is done by splitting the LTL formulas on Boolean connectives to recognize some subformulas that are either to translate with different types of acceptance conditions, and then combining everything back together.

Combined with --deterministic (or -D) this allows the use of a determinization algorithm that produces automata with parity acceptance. This is only used for subformulas for which we do not know a better way to get a deterministic automaton.

For instance FGa is the typical formula for which not deterministic TGBA exists.

ltl2tgba "FGa" -D -d


But with --generic, ltl2tgba will output the following co-Büchi automaton:

ltl2tgba "FGa" -G -D -d


If we translate Fb|Gc as a deterministic automaton with any acceptance condition, we get a weak and deterministic Büchi automaton:

ltl2tgba "Fb|Gc" -G -D -d


Finally if we translate the conjunction of these two subformulas, a product of these two automata will be made, producing:

ltl2tgba "(Fb|Gc)&FGa" -G -D -d


Disabling the splitting of the original formula LTL formulas can be done using option -x ltl-split=0. In that case the formula (Fb|Gc)&FGa will be translated into a single TGBA, and because this TGBA is non-deterministic, it will then be determinized into an automaton with parity acceptance:

ltl2tgba "(Fb|Gc)&FGa" -G -D -xltl-split=0 -d


The spot-x(7) man page lists a few -x options (det-scc, det-simul, det-stutter) of the determinization algorithm that are enabled by default, but that you may want to disable for experimental purpose.

For instance the following deterministic automaton

ltl2tgba "F(a W FGb)" -G -D -d


would be larger if SCC-based optimizations were disabled:

ltl2tgba "F(a W FGb)" -xdet-scc=0 -G -D -d


## Deterministic automata with --parity --deterministic

Using the --parity (or upper-case -P) option will force the acceptance condition to be of a parity type. This has to be understood in the sense of the HOA format, where:

• multiple parity types are defined (min odd n, min even n, max odd n, and max even n where n is the number of acceptance sets), and
• the parity acceptance is only a type of acceptance condition, i.e., a formula expressed in terms of acceptance sets, and does not have additional constraints on these sets. In particular it is not necessary the case that each transition or state belongs to exactly one acceptance set (this is the "colored" property, see below).

Under these assumptions, Büchi acceptance is just one kind of parity (in HOA Buchi is equivalent to parity max even 1 or parity min even 1), Rabin with one pair is also a parity acceptance (Rabin 1 is equivalent to parity min odd 2), and Streett with one pair as well (Streett 1 is equivalent to parity max odd 2).

In the current implementation, using ltl2tgba --parity (without --deterministic) will always produce a Büchi automaton, and when --deterministic (or -D) is added, it will produce an deterministic automaton with Büchi acceptance (parity min even 1) or with parity min odd n acceptance, because the latter is the type of parity acceptance that our determinization procedure outputs.

For instance, FGa gets translated into an automaton with Rabin 1 acceptance (another name for parity min odd 2):

ltl2tgba "FGa" -D -P -d


And GFa & GFb gets translated into a Büchi automaton (another name for parity min even 1):

ltl2tgba "GFa & GFb" -D -P -d


If we really want to use the same style of parity acceptance for all outputs, we can specify it as an argument to the --parity option. For instance

ltl2tgba "GFa & GFb" -D -P'min odd' -d


The --colored-parity (or lower-case -p) option works similarly to --parity, but additionally requests that the automaton be colored. I.e., each transition (or state if state-based acceptance is requested) should belong to exactly one acceptance set.

ltl2tgba "GFa & GFb" -D -p -d


ltl2tgba "GFa & GFb" -D -p'min odd' -d


Note that all these options can be combined with state-based acceptance if needed:

ltl2tgba "GFa & GFb" -D -S -p'max even' -d


## Translating multiple formulas for statistics

If multiple formulas are given to ltl2tgba, the corresponding automata will be output one after the other. The default output format HOA is designed to allow streaming automata this way to build processing pipelines, but Spot's automaton parser can also read a stream of automata in other formats.

Another situation where passing many formulas to ltl2tgba is useful is in combination with the --stats=FORMAT option. This option will output statistics about the translated automata instead of the automata themselves. The FORMAT string should indicate which statistics should be output, and how they should be output using the following sequence of characters (other characters are output as-is):

%<                         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


For instance we can study the size of the automata generated for the right-nested U formulas as follows:

genltl --u-right=1..8 | ltl2tgba --stats '%s states and %e edges for "%f"'

2 states and 2 edges for "p1"
2 states and 3 edges for "p1 U p2"
3 states and 6 edges for "p1 U (p2 U p3)"
4 states and 10 edges for "p1 U (p2 U (p3 U p4))"
5 states and 15 edges for "p1 U (p2 U (p3 U (p4 U p5)))"
6 states and 21 edges for "p1 U (p2 U (p3 U (p4 U (p5 U p6))))"
7 states and 28 edges for "p1 U (p2 U (p3 U (p4 U (p5 U (p6 U p7)))))"
8 states and 36 edges for "p1 U (p2 U (p3 U (p4 U (p5 U (p6 U (p7 U p8))))))"


Note that because no formula have been passed as argument to ltl2tgba, it defaulted to reading them from standard input. Such a behavior can be requested explicitly with -F - if needed (e.g., to read from standard input in addition to processing other formula supplied with -f).

When computing the size of an automaton, we distinguish transitions and edges. An edge between two states is labeled by a Boolean formula and may in fact represent several transitions labeled by compatible Boolean assignment.

For instance if the atomic propositions are x and y, an edge labeled by the formula !x actually represents two transitions labeled respectively with !x&y and !x&!y.

Two automata with the same structures (states and edges) but differing labels, may have a different count of transitions, e.g., if one has more restricted labels.

## Building Monitors

In addition to TGBA and BA, ltl2tgba can output monitor using the -M option. These are finite automata that accept all prefixes of a formula. The idea is that you can use these automata to monitor a system as it is running, and report a violation as soon as no compatible outgoing transition exist.

ltl2tgba -M may output non-deterministic monitors while ltl2tgba -MD (short for --monitor --deterministic) will output the minimal deterministic monitor for the given formula.

ltl2tgba -M '(Xa & Fb) | Gc' -d


ltl2tgba -MD '(Xa & Fb) | Gc' -d


Because they accept all finite executions that could be extended to match the formula, monitor cannot be used to check for eventualities such as F(a): indeed, any finite execution can be extended to match F(a).

For more discussion and examples about monitor, see also our separate page showing how to build them in Python and C++.

Because Monitors accept every recognized run (in other words, they only reject words that are not recognized), it makes little sense to use option -C to request complete monitors. If you combine -C with -M, the result will output as a Büchi automaton if (and only if) a sink state had to be added. For instance, here is the "complete" version of the previous monitor.

ltl2tgba -C -M -D '(Xa & Fb) | Gc' -d