dstar2tgba

Table of Contents

#+EMAIL spot@lrde.epita.fr

This tool converts deterministic Rabin and Streett automata, presented in the format output by ltl2dstar, into Büchi automata.

It's usage is almost similar to ltl2tgba except that instead of supplying a formula to translate, you should specify a filename containing the deterministic Rabin or Streett automaton to convert.

Two quick examples

Here are some brief examples before we discuss the behavior of dstar2tgba in more detail.

From Rabin to Büchi

The following command instructs ltl2dstar to:

  1. run ltl2tgba -sD to build a Büchi automaton for Fa & GFb, and then
  2. convert that Büchi automaton into a deterministic Rabin automaton (DRA) stored in fagfb.

Additionally we use ltlfilt to convert our formula to the prefix format used by ltl2dstar.

ltlfilt -f 'Fa & GFb' -l | ltl2dstar --ltl2nba=spin:../../src/bin/ltl2tgba@-sD - fagfb

By looking at the file fagfb you can see the ltl2dsar actually produced a 3-state DRA:

cat fagfb
DRA v2 explicit
Comment: "Safra[NBA=3]"
States: 3
Acceptance-Pairs: 1
Start: 1
AP: 2 "a" "b"
---
State: 0
Acc-Sig: +0
2
2
0
0
State: 1
Acc-Sig:
1
0
1
0
State: 2
Acc-Sig:
2
2
0
0

dstar2tgba can now be used to convert this DRA into a TGBA, a BA, or a Monitor, using the same options as ltl2tgba.

For instance here is the conversion to a Büchi automaton (-B) in GraphViz's format:

dstar2tgba -B fagfb
digraph G {
  0 [label="", style=invis, height=0]
  0 -> 1
  1 [label="0"]
  1 -> 2 [label="a\n"]
  1 -> 1 [label="!a\n"]
  2 [label="1", peripheries=2]
  2 -> 2 [label="b\n{Acc[1]}"]
  2 -> 3 [label="!b\n{Acc[1]}"]
  3 [label="2"]
  3 -> 2 [label="b\n"]
  3 -> 3 [label="!b\n"]
}

Which can be rendered as:

fagfb2ba.png

But we could as well require the output to be output as a never claim for Spin (option -s):

dstar2tgba -s fagfb
never {
T0_init:
  if
  :: ((a)) -> goto accept_S2
  :: ((!(a))) -> goto T0_init
  fi;
accept_S2:
  if
  :: ((b)) -> goto accept_S2
  :: ((!(b))) -> goto T0_S3
  fi;
T0_S3:
  if
  :: ((b)) -> goto accept_S2
  :: ((!(b))) -> goto T0_S3
  fi;
}

Streett to TGBA

Here is the translation of GFa & GFb to a 4-state Streett automaton:

ltlfilt -f 'GFa & GFb' -l | ltl2dstar --automata=streett --ltl2nba=spin:../../src/bin/ltl2tgba@-sD - gfagfb
cat gfagfb
DSA v2 explicit
Comment: "Streett{Union{Safra[NBA=2],Safra[NBA=2]}}"
States: 4
Acceptance-Pairs: 2
Start: 0
AP: 2 "a" "b"
---
State: 0
Acc-Sig: -0 -1
3
2
1
0
State: 1
Acc-Sig: +0 -1
3
2
1
0
State: 2
Acc-Sig: -0 +1
3
2
1
0
State: 3
Acc-Sig: +0 +1
3
2
1
0

And now its conversion by dstar2tgba to a 2-state Büchi automaton. We don't pass any option to dstar2tgba because converting to TGBA in GraphViz's format is the default:

dstar2tgba gfagfb

gfagfb2ba.png

(Obviously the resulting automaton could be simplified further, by starting with the second state right away.)

Details

General behavior

The dstar2tgba tool implement a 4-step process:

  1. read the DRA/DSA
  2. convert it into TGBA
  3. postprocess the resulting TGBA (simplifying the automaton, a degeneralizing it into a BA or Monitor if requested)
  4. output the resulting automaton

Controlling output

The last two steps are shared with ltl2tgba and use the same options.

The type of automaton to produce can be selected using the -B or -M switches:

-B, --ba                   Büchi Automaton
-M, --monitor              Monitor (accepts all finite prefixes of the given
                           formula)
    --tgba                 Transition-based Generalized Büchi Automaton
                           (default)

And these may be refined by a translation intent, should the post-processor routine had a choice to make:

-a, --any                  no preference
-C, --complete             output a complete automaton (combine with other
                           intents)
-D, --deterministic        prefer deterministic automata
    --small                prefer small automata (default)

The effort put into post-processing can be limited with the --low or --medium options:

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

For instance using -a --low will skip any optional post-processing, should you find dstar2tgba too slow.

Finally, the output format can be changed with the following options:

-8, --utf8                 enable UTF-8 characters in output (ignored with
                           --lbtt or --spin)
    --dot                  GraphViz's format (default)
-H, --hoaf[=s|t|m|l]       Output the automaton in HOA format.  Add letters
                           to select (s) state-based acceptance, (t)
                           transition-based acceptance, (m) mixed acceptance,
                           (l) single-line output
    --lbtt[=t]             LBTT's format (add =t to force transition-based
                           acceptance even on Büchi automata)
-s, --spin                 Spin neverclaim (implies --ba)
    --spot                 SPOT's format
    --stats=FORMAT         output statistics about the automaton

The --stats options can output statistics about the input and the output automaton, so it can be useful to search for particular pattern.

For instance here is a complex command that will

  1. generate an infinite stream of random LTL formulas with randltl,
  2. use ltlfilt to rewrite the W and M operators away (--remove-wm), simplify the formulas (-r), remove duplicates (u) as well as formulas that have a size less then 3 (--size-min=3),
  3. use head to keep only 10 of such formula
  4. loop to process each of these formula:
    • print it
    • then convert the formula into ltl2dstar's input format, process it with ltl2dstar (using ltl2tgba as the actual LTL->BA transltor), and process the result with dstar2tgba to build a Büchi automaton (-B), favoring determinism if we can (-D), and finally displaying some statistics about this conversion.

The statistics displayed in this case are: %S, the number of states of the input (Rabin) automaton, %s, the number of states of the output (Büchi) automaton, %d, whether the output automaton is deterministic, and %p whether the automaton is complete.

randltl -n -1 --tree-size=10..15 a b c |
ltlfilt --remove-wm -r -u --size-min=3 |
head -n 10 |
while read f; do
  echo "$f"
  ltlfilt -l -f "$f" |
  ltl2dstar --ltl2nba=spin:../../src/bin/ltl2tgba@-sD - - |
  dstar2tgba -B --stats='  DRA: %Sst.; BA: %sst.; det.? %d; complete? %p'
done
G((!a & c & F!c) | (a & (!c | Gc)))
  DRA: 6st.; BA: 3st.; det.? 1; complete? 0
G!c R (G!c | (!a & XGa) | (a & XF!a))
  DRA: 7st.; BA: 4st.; det.? 1; complete? 0
G!c | G(a | c)
  DRA: 5st.; BA: 3st.; det.? 1; complete? 0
XGa
  DRA: 3st.; BA: 2st.; det.? 1; complete? 0
!b | ((Xb R (!a | Fb | Xb)) U (!c & (Xb R (!a | Fb | Xb))))
  DRA: 8st.; BA: 6st.; det.? 1; complete? 1
G((b & X!c) | (!b & Xc))
  DRA: 4st.; BA: 3st.; det.? 1; complete? 0
XX!b
  DRA: 5st.; BA: 4st.; det.? 1; complete? 0
GFb
  DRA: 3st.; BA: 3st.; det.? 1; complete? 1
F(G!a R (!b | G!a))
  DRA: 10st.; BA: 10st.; det.? 0; complete? 0
F!c R F!a
  DRA: 8st.; BA: 5st.; det.? 1; complete? 1

An important point you should be aware of when comparing these numbers of states is that the deterministic automata produced by ltl2dstar are complete, while the automata produced by dstar2tgba (deterministic or not) are not complete by default. This can explain a difference of one state (the so called "sink" state).

You can instruct dstar2tgba to output a complete automaton using the --complete option (or -C for short).

Conversion from Rabin and Streett to TGBA

The algorithms used to convert Rabin and Streett into TGBA/BA are different.

  • Rabin to BA

    The conversion implemented is a variation of Krishnan et al.'s "Deterministic ω-Automata vis-a-vis Deterministic Büchi Automata" (ISAAC'94) paper. They explain how to convert a deterministic Rabin automaton (DRA) into a deterministic Büchi automaton (DBA) when such an automaton exist. The surprising result is that when a DRA is DBA-realizable, a DBA can be obtained from the DRA without changing its transition structure.

    Spot implements a slight refinement to the above technique: any DRA will be converted into a BA, and the determinism will be conserved only in strongly connected components where determinism can be conserved.

  • Streett to TGBA

    Streett automata are converted into non-deterministic TGBA. When a Streett automaton uses multiple acceptance pairs, we use generalized acceptance conditions in the TGBA to limit the combinatorial explosion.

    A straightforward translation from Streett to BA, as described for instance by Löding's diploma thesis, will create a BA with \(|Q|\cdot(4^n-3^n+2)\) states if the input Streett automaton has \(|Q|\) states and \(n\) acceptance pairs. Our translation to TGBA limits this to \(|Q|\cdot(2^n+1)\) states.

    Sometimes, as in the example for GFa & GFb the output of this conversion will happen to be deterministic. Let's say that this is luck: Spot does not implement any algorithm to preserve the determinism of Streett automata.

Author: Alexandre Duret-Lutz

Created: 2014-12-06 Sat 12:29

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