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ltl2tgta

This tool generates various form of Testing Automata, i.e., automata that observe the changes of atomic propositions, not their values.

Three types of automata can be output. The only output format supported currently is GraphViz's format, with option -8 to enable UTF-8 characters as in other tools.

The --ta option will translate a formula into Testing Automaton, as described by Geldenhuys and Hansen (Spin'06).

Here is the output on a U Gb (we omit the call to dot, as shown while discussing ltl2tgba).

ltl2tgta --ta --multiple-init 'a U Gb'

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As always, the labels of the states have no influence on the language recognized by the automaton. This automaton has three possible initial states. The initial state should be chosen depending on the initial valuation of a and b in the system to be synchronized with this testing automaton. For instance if a is true and b false in the initial state, the testing automaton should start in the state pointed to by the initial arrow labeled a & !b. In the rest of the testing automaton, the transitions are labeled by the sets of atomic propositions that should change in the system for the testing automaton to progress. States with a double enclosure are Büchi accepting, meaning that any execution that visits one of these states is accepting. All states have an implicit self-loop labeled by {}: if the system progress without changing the value of a and b, the testing automaton remains in the same state. Rectangle states are livelock-accepting: any execution of the system that get stuck into one of these state is accepting.

Without the --multiple-init option, a fake initial state is added. This is the default since it often makes the result more readable.

ltl2tgta --ta 'a U Gb'
digraph G {
  0 [label="", style=invis, height=0]
  0 -> 1
  1 [label=init]
  1 -> 2 [label="!a & b\n"]
  1 -> 3 [label="a & b\n"]
  1 -> 4 [label="a & !b\n"]
  2 [label="2",shape=box]
  2 -> 5 [label="{a}\n"]
  3 [label="3",shape=box]
  3 -> 5 [label="{a}\n"]
  3 -> 2 [label="{a}\n"]
  3 -> 4 [label="{b}\n"]
  4 [label="1"]
  4 -> 3 [label="{b}\n"]
  4 -> 2 [label="{a, b}\n"]
  5 [label="4",peripheries=2,shape=box]
  5 -> 5 [label="{a}\n{0}"]
}

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The --gba option can be used to request a Generalized Testing Automaton, i.e., a Testing Automaton with Generalized Büchi acceptance. In that case double-enclosures are not used anymore, and Büchi accepting transitions are marked with the same {0,1} notation used in TGBA.

ltl2tgta --gta 'GFa & GFb'
digraph G {
  0 [label="", style=invis, height=0]
  0 -> 1
  1 [label=init]
  1 -> 2 [label="a & b\n"]
  1 -> 3 [label="!a & b\n"]
  1 -> 4 [label="a & !b\n"]
  1 -> 5 [label="!a & !b\n"]
  2 [label="1",shape=box]
  2 -> 3 [label="{a}\n{0,1}"]
  2 -> 4 [label="{b}\n{0,1}"]
  2 -> 5 [label="{a, b}\n{0,1}"]
  3 [label="3"]
  3 -> 2 [label="{a}\n{1}"]
  3 -> 4 [label="{a, b}\n{1}"]
  3 -> 5 [label="{b}\n{1}"]
  4 [label="2"]
  4 -> 2 [label="{b}\n{0}"]
  4 -> 3 [label="{a, b}\n{0}"]
  4 -> 5 [label="{a}\n{0}"]
  5 [label="4"]
  5 -> 2 [label="{a, b}\n"]
  5 -> 3 [label="{b}\n"]
  5 -> 4 [label="{a}\n"]
}

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The interpretation is similar to that of the TA. Execution that stutter in a livelock-accepting (square) state are accepting as well as execution that visit the 0 and 1 acceptance sets infinitely often. Those acceptance sets are carried by transitions, as in TGBAs.

Finally, the default is to output a Transition-based Generalized Testing Automaton 1. In TGTAs, the stuttering states are made explicit with {} self-loops. Since these self-loop can be in acceptance sets, livelock acceptance states are no longer needed.

ltl2tgta 'GFa & GFb'
digraph G {
  0 [label="", style=invis, height=0]
  0 -> 1
  1 [label=init]
  1 -> 2 [label="a & b\n"]
  1 -> 3 [label="!a & b\n"]
  1 -> 4 [label="a & !b\n"]
  1 -> 5 [label="!a & !b\n"]
  2 [label="3"]
  2 -> 3 [label="{a}\n{0,1}"]
  2 -> 4 [label="{b}\n{0,1}"]
  2 -> 5 [label="{a, b}\n{0,1}"]
  2 -> 2 [label="{}\n{0,1}"]
  3 [label="2"]
  3 -> 2 [label="{a}\n{1}"]
  3 -> 4 [label="{a, b}\n{1}"]
  3 -> 5 [label="{b}\n{1}"]
  3 -> 3 [label="{}\n"]
  4 [label="4"]
  4 -> 2 [label="{b}\n{0}"]
  4 -> 3 [label="{a, b}\n{0}"]
  4 -> 5 [label="{a}\n{0}"]
  4 -> 4 [label="{}\n"]
  5 [label="1"]
  5 -> 2 [label="{a, b}\n"]
  5 -> 3 [label="{b}\n"]
  5 -> 4 [label="{a}\n"]
  5 -> 5 [label="{}\n"]
}

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Footnotes:

1

: This new class of automaton, as well as the implementation of the previous testing automata classes, is part of Ala Eddine BEN SALEM's PhD work, and should appear in a future edition of ToPNoC (LNCS 7400).