Omega-Automata: A Coalgebraic Perspective on Regular omega-Languages

Authors Vincenzo Ciancia, Yde Venema

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Vincenzo Ciancia
  • Istituto di Scienza e Tecnologie dell'Informazione "A. Faedo" - Consiglio Nazionale delle Ricerche, Pisa, Italy
Yde Venema
  • Institute for Logic, Language and Computation, Universiteit van Amsterdam, Amsterdam, The Netherlands


The authors wish to thank Oded Maler and Ludwig Staiger for useful discussions about right congruences and characterisations of omega-regularity.

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Vincenzo Ciancia and Yde Venema. Omega-Automata: A Coalgebraic Perspective on Regular omega-Languages. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In this work, we provide a simple coalgebraic characterisation of regular omega-languages based on languages of lassos, and prove a number of related mathematical results, framed into the theory of a new kind of automata called Omega-automata. In earlier work we introduced Omega-automata as two-sorted structures that naturally operate on lassos, pairs of words encoding ultimately periodic streams (infinite words). Here we extend the scope of these Omega-automata by proposing them as a new kind of acceptor for arbitrary streams. We prove that Omega-automata are expressively complete for the regular omega-languages. We show that, due to their coalgebraic nature, Omega-automata share some attractive properties with deterministic automata operating on finite words, properties that other types of stream automata lack. In particular, we provide a simple, coalgebraic definition of bisimilarity between Omega-automata that exactly captures language equivalence and allows for a simple minimization procedure. We also prove a coalgebraic Myhill-Nerode style theorem for lasso languages, and use this result, in combination with a closure property on stream languages called lasso determinacy, to give a characterization of regular omega-languages.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Automata over infinite objects
  • omega-automata
  • regular omega-languages
  • coalgebra
  • streams
  • bisimilarity


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