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On the Translation of Automata to Linear Temporal Logic

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13242)

Abstract

While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, the size increase involved in turning automata back to LTL is not. In particular, there is no known elementary bound on the complexity of translating deterministic \(\omega \)-regular automata to LTL.

Our first contribution consists of tight bounds for LTL over a unary alphabet: alternating, nondeterministic and deterministic automata can be exactly exponentially, quadratically and linearly more succinct, respectively, than any equivalent LTL formula. Our main contribution consists of a translation of general counter-free deterministic \(\omega \)-regular automata into LTL formulas of double exponential temporal-nesting depth and triple exponential length, using an intermediate Krohn-Rhodes cascade decomposition of the automaton. To our knowledge, this is the first elementary bound on this translation. Furthermore, our translation preserves the acceptance condition of the automaton in the sense that it turns a looping, weak, Büchi, coBüchi or Muller automaton into a formula that belongs to the matching class of the syntactic future hierarchy. In particular, it can be used to translate an LTL formula recognising a safety language to a formula belonging to the safety fragment of LTL (over both finite and infinite words).

Keywords

  • Linear temporal logic
  • Automata
  • Cascade decomposition

The omitted proofs of this chapter can be found in the full version [5].

Salomon Sickert is supported by the Deutsche Forschungsgemeinschaft (DFG) under project number 436811179.

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Boker, U., Lehtinen, K., Sickert, S. (2022). On the Translation of Automata to Linear Temporal Logic. In: Bouyer, P., Schröder, L. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2022. Lecture Notes in Computer Science, vol 13242. Springer, Cham. https://doi.org/10.1007/978-3-030-99253-8_8

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