First-Order vs. Second-Order Encodings for \(\textsc {ltl}_f\)-to-Automata Translation

  • Shufang Zhu
  • Geguang PuEmail author
  • Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)


Translating formulas of Linear Temporal Logic (ltl) over finite traces, or \({\textsc {ltl}}_f\), to symbolic Deterministic Finite Automata (DFA) plays an important role not only in \({\textsc {ltl}}_f\) synthesis, but also in synthesis for Safety ltl formulas. The translation is enabled by using \(\mathsf {MONA}\), a powerful tool for symbolic, BDD-based, DFA construction from logic specifications. Recent works used a first-order encoding of \({\textsc {ltl}}_f\) formulas to translate \({\textsc {ltl}}_f\) to First Order Logic (fol), which is then fed to \(\mathsf {MONA}\) to get the symbolic DFA. This encoding was shown to perform well, but other encodings have not been studied. Specifically, the natural question of whether second-order encoding, which has significantly simpler quantificational structure, can outperform first-order encoding remained open.

In this paper we address this challenge and study second-order encodings for \({\textsc {ltl}}_f\) formulas. We first introduce a specific mso encoding that captures the semantics of \({\textsc {ltl}}_f\) in a natural way and prove its correctness. We then explore is a Compact mso encoding, which benefits from automata-theoretic minimization, thus suggesting a possible practical advantage. To that end, we propose a formalization of symbolic DFA in second-order logic, thus developing a novel connection between BDDs and mso. We then show by empirical evaluations that the first-order encoding does perform better than both second-order encodings. The conclusion is that first-order encoding is a better choice than second-order encoding in \({\textsc {ltl}}_f\)-to-Automata translation.



Work supported in part by China HGJ Project No. 2017ZX01038102-002, NSFC Projects No. 61572197, No. 61632005 and No. 61532019, NSF grants IIS-1527668, IIS-1830549, and by NSF Expeditions in Computing project “ExCAPE: Expeditions in Computer Augmented Program Engineering”. Special thanks to Jeffrey M. Dudek and Dror Fried for useful discussions.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.East China Normal UniversityShanghaiChina
  2. 2.Rice UniversityHoustonUSA

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