An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata

Abstract

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form n i =1 GFφ i ∨FGψ i , where φ i andψ i contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.CCS Concepts: • Theory of computation → Modal and temporal logics; Automata over infinite objects.where a ∈ Ap is an atomic proposition and X, U, W, R, and M are the next, (strong) until, weak until, (weak) release, and strong release operators, respectively.

A Unified Translation of Linear Temporal Logic to ω-Automata

“…Hence, the result presented here successfully finishes the journey embarked upon in Reference [45]: A single theorem provides an arguably elegant (unified, symmetric, syntaxindependent, not overly complex) and efficient (asymptotically optimal, practically relevant, direct) translation of LTL to ω-automata of your choice. Moreover, based on the Master Theorem one can derive an efficient, syntactic, and exponential normalisation procedure for LTL [71] that limits the nesting of greatest-fixed-point and least-fixed-pointer operators yielding a normal form described in Reference [12].…”

From linear temporal logic and limit-deterministic Büchi automata to deterministic parity automata

“…U, M) and greatest-fixed-point operators (G, W, R) symmetrically. LD p ltl2dpa (portfolio) 7 : Here, we combine the previous translations with a portfolio of translations for fragments that directly yield DPAs [9,34,35]. This portfolio approach is important in comparison to the configuration N 1 where similar steps are taken.…”

The Reactive Synthesis Competition (SYNTCOMP): 2018-2021

“…• LTL formulas are now translated to transitionbased deterministic Emerson-Lei automata (tDELA) by combining constructions (∆ 2normalisation, direct translation to deterministic automata) from (Sickert and Esparza, 2020) with a product construction adapted from (Müller and Sickert, 2017). Then either a tDELA-to-tDPW construction based on Zielonka-trees or the Alternating Cycle Decomposition is applied (Casares et al (2021(Casares et al ( , 2022).…”

“…It proceeds as follows: the LTL formula is decomposed into a Boolean combination of simpler formulas, each of these formulas are separately translated to BDD-encoded deterministic automata, and then recomposed by computing the (deterministic) union and intersection on the BDD-representation. Concretely, Otus makes use of the ∆ 2 -normalisation and the translation to deterministic (co-)Büchi automata found in (Sickert and Esparza, 2020) implemented by Owl (Kretínský et al, 2018), then computes the symbolic representation of a deterministic Rabin automaton by union and intersection, and then applies a symbolic implementation (Boker et al, 2010) to obtain a parity automaton. This symbolic automaton is reinterpreted as a parity game and a symbolic implementation of the distraction fix-point iteration (van Dijk and Rubbens, 2019; Lijzenga and van Dijk, 2020a) is applied.…”