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# LTL2TGBA

## Table of Contents

- NAME
- SYNOPSIS
- DESCRIPTION
- NOTE ON TGBA
- NOTE ON LBTT’S FORMAT
- NOTE ON GENERATING MONITORS
- BIBLIOGRAPHY
- REPORTING BUGS
- COPYRIGHT
- SEE ALSO

## NAME

ltl2tgba − translate LTL/PSL formulas into Büchi automata

## SYNOPSIS

ltl2tgba [OPTION...] [FORMULA...]

## DESCRIPTION

Translate linear−time formulas (LTL/PSL) into Büchi automata.

By default it will apply all available optimizations to output the smallest Transition−based Generalized Büchi Automata, in GraphViz’s format. If multiple formulas are supplied, several automata will be output.

### Input options:

−f, −−formula=STRING

process the formula STRING

−F, −−file=FILENAME[/COL]

process each line of FILENAME as a formula; if COL is a positive integer, assume a CSV file and read column COL; use a negative COL to drop the first line of the CSV file

−−lbt−input

read all formulas using LBT’s prefix syntax

−−lenient

parenthesized blocks that cannot be parsed as subformulas are considered as atomic properties

### Output automaton type:

−B, −−ba

Büchi Automaton (implies −S)

−C, −−complete

output a complete automaton

−G, −−generic

any acceptance condition is allowed

−M, −−monitor

Monitor (accepts all finite prefixes of the given property)

−S, −−state−based−acceptance, −−sbacc

define the acceptance using states

| −−tgba |  | Transition−based Generalized Büchi Automaton (default)

−U, −−unambiguous

output unambiguous automata

### Output format:

−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 any prefix of ’all’ (default), ’unambiguous’, ’stutter−invariant’, or ’strength’.

−d,
−−dot[=1|a|b|B|c|C(COLOR)|e|f(FONT)|h|k|n|N|o|r|R|s|t|v|y|+INT|<INT|#]

GraphViz’s format. Add letters for (1) force numbered states, (a) acceptance display, (b) acceptance sets as bullets, (B) bullets except for Büchi/co−Büchi automata, (c) force circular nodes, (C) color nodes with COLOR, (e) force elliptic nodes, (f(FONT)) use FONT, (h) horizontal layout, (k) use state labels when possible, (n) with name, (N) without 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, (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

output statistics about the automaton

### Any FORMAT string may use the following interpreted sequences:

| %< |  | 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, (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 edges
| %f |  | the formula, in Spot’s syntax
| %F |  | name of the input file
| %g |  | acceptance condition (in HOA syntax)
| %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 states |
| %t |  | number of transitions |
| %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

### 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 (default)

### Simplification level:

| −−high |  | all available optimizations (slow, default) |
| −−low |  | minimal optimizations (fast) |

−−medium

moderate optimizations

### Miscellaneous options:

−x, −−extra−options=OPTS

fine−tuning options (see spot−x (7))

| −−help |  | print this help |

−−version

print program version

Mandatory or optional arguments to long options are also mandatory or optional for any corresponding short options.

## NOTE ON TGBA

TGBA stands for Transition-based Generalized Büchi Automaton. The name was coined by Dimitra Giannakopoulou and Flavio Lerda in their FORTE’02 paper (From States to Transitions: Improving Translation of LTL Formulae to Büchi Automata), although similar automata have been used under different names long before that.

As its name implies a TGBA uses a generalized Büchi acceptance condition, meanings that a run of the automaton is accepted iff it visits ininitely often multiple acceptance sets, and it also uses transition-based acceptance, i.e., those acceptance sets are sets of transitions. TGBA are often more consise than traditional Büchi automata. For instance the LTL formula `GFa & GFb` can be translated into a single-state TGBA while a traditional Büchi automaton would need 3 states. Compare

    % ltl2tgba ’GFa & GFb’

with

    % ltl2tgba --ba ’GFa & GFb’

In the dot output produced by the above commands, the membership of the transitions to the various acceptance sets is denoted using names in braces. The actuall names do not really matter as they may be produced by the translation algorithm or altered by any latter postprocessing.

When the −−ba option is used to request a Büchi automaton, Spot builds a TGBA with a single acceptance set, and in which for any state either all outgoing transitions are accepting (this is equivalent to the state being accepting) or none of them are. Double circles are used to highlight accepting states in the output, but the braces denoting the accepting transitions are still shown because the underling structure really is a TGBA.

## NOTE ON LBTT’S FORMAT

LBTT’s format has support for both transition-based and state based generalized acceptance.

Because Spot uses transition-based generalized Büchi automata internally, it will normally use the transition-based flavor of that format, indicated with a ’t’ flag after the number of acceptance sets. For instance:

    % ltl2tgba --lbtt ’GFp0 & GFp1 & FGp2’
    2 2t                   // 2 states, 2 transition-based acceptance sets
    0 1                    // state 0: initial
    0 -1 t                 //   trans. to state 0, no acc., label: true
    1 -1 | & p0 p2 & p1 p2 //   trans. to state 1, no acc., label: (p0&p2)|(p1&p2)
    -1                     // end of state 0
    1 0                    // state 1: not initial
    1 0 1 -1 & & p0 p1 p2  //   trans. to state 1, acc. 0 and 1, label: p0&p1&p2
    1 0 -1 & & p1 p2 ! p0  //   trans. to state 1, acc. 0, label: !p0&p1&p2
    1 1 -1 & & p0 p2 ! p1  //   trans. to state 1, acc. 1, label: p0&!p1&p2
    1 -1 & & p2 ! p0 ! p1  //   trans. to state 1, no acc., label: !p0&!p1&p2
    -1                     // end if state 1

Here, the two acceptance sets are represented by the numbers 0 and 1, and they each contain two transitions (the first transition of state 1 belongs to both sets).

When both −−ba and −−lbtt options are used, the state-based flavor of the format is used instead. Note that the LBTT format supports generalized acceptance conditions on states, but Spot only use this format for Büchi automata, where there is always only one acceptance set. Unlike in the LBTT documentation, we do not use the optional ’s’ flag to indicate the state-based acceptance, this way our output is also compatible with that of LBT.

    % ltl2tgba --ba --lbtt FGp0
    2 1                 // 2 states, 1 (state-based) accepance set
    0 1 -1              // state 0: initial, non-accepting
    0 t                 //   trans. to state 0, label: true
    1 p0                //   trans. to state 1, label: p0
    -1                  // end of state 0
    1 0 0 -1            // state 1: not initial, in acceptance set 0
    1 p0                //   trans. to state 0, label: p0
    -1                  // end if state 1

You can force ltl2tgba to use the transition-based flavor of the format even for Büchi automaton using −−lbtt=t.

    % ltl2tgba --ba --lbtt=t FGp0
    2 1t                // 2 states, 1 transition-based accepance set.
    0 1                 // state 0: initial
    0 -1 t              //   trans. to state 0, no acc., label: true
    1 -1 p0             //   trans. to state 1, no acc., label: p0
    -1                  // end of state 0
    1 0                 // state 1: not initial
    1 0 -1 p0           //   trans. to state 1, acc. 0, label: p0
    -1                  // end if state 1

When representing a Büchi automaton using transition-based acceptance, all transitions leaving accepting states are put into the acceptance set.

A final note concerns the name of the atomic propositions. The original LBTT and LBT formats require these atomic propositions to have names such as ’p0’, ’p32’, ... We extend the format to accept atomic proposition with arbitrary names that do not conflict with LBT’s operators (e.g. ’i’ is the symbol of the implication operator so it may not be used as an atomic proposition), or as double-quoted strings. Spot will always output atomic-proposition that do not match p[0-9]+ as double-quoted strings.

    % ltl2tgba --lbtt ’GFa & GFb’
    1 2t
    0 1
    0 0 1 -1 & "a" "b"
    0 0 -1 & "b" ! "a"
    0 1 -1 & "a" ! "b"
    0 -1 & ! "b" ! "a"
    -1

## NOTE ON GENERATING MONITORS

The monitors generated with option −M are finite state automata used to reject finite words that cannot be extended to infinite words compatible with the supplied formula. The idea is that the monitor should progress alongside the system, and can only make decisions based on the finite prefix read so far.

Monitors can be seen as Büchi automata in which all recognized runs are accepting. As such, the only infinite words they can reject are those are not recognized, i.e., infinite words that start with a bad prefix.

Because of this limited expressiveness, a monitor for some given LTL or PSL formula may accept a larger language than the one specified by the formula. For instance a monitor for the LTL formula `a U b` will reject (for instance) any word starting with `!a&!b` as there is no way such a word can validate the formula, but it will not reject a finite prefix repeating only `a&!b` as such a prefix could be extented in a way that is comptible with `a U b`.

For more information about monitors, we refer the readers to the following two papers (the first paper describes the construction of the second paper in a more concise way):

| • | Deian Tabakov and Moshe Y. Vardi: Optimized Temporal Monitors for SystemC. Proceedings of RV’10. LNCS 6418.
| • | Marcelo d’Amorim and Grigoire Roşu: Efficient monitoring of ω-languages. Proceedings of CAV’05. LNCS 3576.

## BIBLIOGRAPHY

If you would like to give a reference to this tool in an article, we suggest you cite one of the following papers:

| • | Alexandre Duret-Lutz: LTL translation improvements in Spot 1.0. Int. J. on Critical Computer-Based Systems, 5(1/2):31--54, March 2014.
| • | Alexandre Duret-Lutz: Manipulating LTL formulas using Spot 1.0. Proceedings of ATVA’13. LNCS 8172.
| • | Tomáš Babiak, Thomas Badie, Alexandre Duret-Lutz, Mojmír Křetínský, and Jan Strejček: Compositional approach to suspension and other improvements to LTL translation. Proceedings of SPIN’13. LNCS 7976.
| • | Souheib Baarir and Alexandre Duret-Lutz: Mechanizing the minimization of deterministic generalized Büchi automata. Proceedings of FORTE’14. LNCS 8461.

## REPORTING BUGS

Report bugs to <spot@lrde.epita.fr>.

## COPYRIGHT

Copyright © 2017 Laboratoire de Recherche et Développement de l’Epita. License GPLv3+: GNU GPL version 3 or later.
This is free software: you are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by law.

## SEE ALSO

spot-x(7)