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From Linear Temporal Logics to Büchi Automata: The Early and Simple Principle

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Model Checking, Synthesis, and Learning

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13030))


The automata-theoretic approach advocates reducing problems in an application domain to those in automata theory. When there are multiple paths for the reduction, leaving the realm of application and entering that of automata as early as possible should be preferred, to take full advantages of the abundant algorithmic techniques from the latter. This makes the entire reduction simpler for intuitive understanding and easier for correctness proofs. Indeed, for linear-time temporal logic model checking, there are quite a few ways for translating a temporal formula into an equivalent Büchi automaton. They all go through one or more types of automata as intermediaries, with various interspersing formula manipulation and automaton generation along the way. Among them, translations via alternating automata apparently better adhere to the aforementioned “early and simple” principle. When it comes to translating temporal formulae with past operators, algorithms following the principle generalize more easily by using a two-way alternating automaton as the first intermediary.

In this paper, we give a tutorial presentation of two translation algorithms adhering to the early and simple principle, one for formulae with only future operators and the other for formulae with both future and past operators. They are adaptations of existing works, with a substantially different exposition, further improving simplicity for understanding and easiness for proofs. In particular, we have tried wherever possible to avoid using types of automata or notations that are less common. The relevant notion of a very weak automaton is introduced with two equivalent defining conditions, each offering its unique advantage in a suitable context. Finally, we discuss the role of minimization in such an approach to translation of temporal formulae.

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  1. 1.

    The equivalence between two temporal formulae can be conveniently checked with, e.g., the GOAL tool [29].

  2. 2.

    The name LTL follows the naming convention in model checking [3]. In other contexts, LTL may refer to the temporal logic of Manna and Pnueli [21, 22] with past operators, variables of infinite domains, and quantification. On the other hand, the name PTL was used instead of LTL in some early works on propositional linear temporal logics without past operators, e.g., [34].


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This paper has been written on the occasion of Bengt Jonsson’s 60th birthday. The first author Yih-Kuen Tsay worked as a postdoctoral researcher with Bengt at Uppsala University for near two years, from 1993 to 1995. One of their collaborative works [16] shows how the LTL of Manna and Pnueli [22], which includes past temporal operators, can be conveniently used to write and reason about modular specifications in the “assume-guarantee” style.

We thank the anonymous reviewers and our colleagues Ming-Hsien Tsai, Wayne Zeng, Dror Fried, Suguman Bansal, and Kevin Smith for many helpful comments and suggestions on earlier drafts of this work.

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Tsay, YK., Vardi, M.Y. (2021). From Linear Temporal Logics to Büchi Automata: The Early and Simple Principle. In: Olderog, ER., Steffen, B., Yi, W. (eds) Model Checking, Synthesis, and Learning. Lecture Notes in Computer Science(), vol 13030. Springer, Cham.

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