Church Synthesis on Register Automata over Linearly Ordered Data Domains

Authors Léo Exibard, Emmanuel Filiot, Ayrat Khalimov

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Léo Exibard
  • Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France
  • Université libre de Bruxelles, Brussels, Belgium
Emmanuel Filiot
  • Université libre de Bruxelles, Brussels, Belgium
Ayrat Khalimov
  • Université libre de Bruxelles, Brussels, Belgium

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Léo Exibard, Emmanuel Filiot, and Ayrat Khalimov. Church Synthesis on Register Automata over Linearly Ordered Data Domains. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Register automata are finite automata equipped with a finite set of registers in which they can store data, i.e. elements from an unbounded or infinite alphabet. They provide a simple formalism to specify the behaviour of reactive systems operating over data ω-words. We study the synthesis problem for specifications given as register automata over a linearly ordered data domain (e.g. (ℕ, ≤) or (ℚ, ≤)), which allow for comparison of data with regards to the linear order. To that end, we extend the classical Church synthesis game to infinite alphabets: two players, Adam and Eve, alternately play some data, and Eve wins whenever their interaction complies with the specification, which is a language of ω-words over ordered data. Such games are however undecidable, even when the specification is recognised by a deterministic register automaton. This is in contrast with the equality case, where the problem is only undecidable for nondeterministic and universal specifications. Thus, we study one-sided Church games, where Eve instead operates over a finite alphabet, while Adam still manipulates data. We show they are determined, and deciding the existence of a winning strategy is in ExpTime, both for ℚ and ℕ. This follows from a study of constraint sequences, which abstract the behaviour of register automata, and allow us to reduce Church games to ω-regular games. Lastly, we apply these results to the transducer synthesis problem for input-driven register automata, where each output data is restricted to be the content of some register, and show that if there exists an implementation, then there exists one which is a register transducer.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Transducers
  • Synthesis
  • Church Game
  • Register Automata
  • Transducers
  • Ordered Data Words


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