On the Relationship between LTL Normal Forms and Buechi Automata

TL;DR

This paper presents a novel method for translating LTL formulas into Buechi automata using a disjunctive-normal form, with optimized constructions for specific subclasses of formulas, improving efficiency and state bounds.

Contribution

It introduces a direct translation approach from LTL to Buechi automata via DNF, with on-the-fly construction and improved bounds for certain formula classes.

Findings

Automata size is 2^{2n+1} in the worst case.

For Until- or Release-free formulas, size reduces to 2^{n+1}.

The method simplifies automata construction for specific LTL subclasses.

Abstract

In this paper, we consider the problem of translating LTL formulas to Buechi automata. We first translate the given LTL formula into a special disjuctive-normal form (DNF). The formula will be part of the state, and its DNF normal form specifies the atomic properties that should hold immediately (labels of the transitions) and the formula that should hold afterwards (the corresponding successor state). Surprisingly, if the given formula is Until-free or Release-free, the Buechi automaton can be obtained directly in this manner. For a general formula, the construction is slightly involved: an additional component will be needed for each formula that helps us to identify the set of accepting states. Notably, our construction is an on-the-fly construction, and the resulting Buechi automaton has in worst case 2^{2n+1} states where n denotes the number of subformulas. Moreover, it has a better bound 2^{n+1} when the formula is Until- (or Release-) free.