On the Translation of Automata to Linear Temporal Logic

TL;DR

This paper establishes tight bounds for translating automata to LTL over unary alphabets and introduces a new translation method for counter-free deterministic automata, providing elementary bounds and preserving automaton acceptance conditions.

Contribution

It provides the first elementary bounds for translating deterministic automata to LTL and introduces a new translation method using Krohn-Rhodes decomposition.

Findings

Automata can be exponentially, quadratically, or linearly more succinct than LTL over unary alphabets.

The translation of counter-free deterministic automata to LTL has double exponential depth and triple exponential length.

The translation preserves acceptance conditions, enabling translation of safety automata to safety LTL fragments.

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, B\"uchi, coB\"uchi 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).