Conjunctive Grammars, Cellular Automata and Logic

Authors Théo Grente, Étienne Grandjean

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Théo Grente
  • GREYC, Université de Caen Normandie, France
Étienne Grandjean
  • GREYC, Université de Caen Normandie, France


This paper would not exist without the inspiration of Véronique Terrier. Her in-depth knowledge of cellular automata, their complexity classes and their closure properties in relation to formal language theory, the references and advice she generously gave us, as well as her careful reading, were essential in designing and finalizing the results and the presentation of the paper. E.g., the class diagram of Figure 8 is due to her. We also thank the three reviewers of this paper for their careful reading and their detailed constructive comments and suggestions which helped us improve some parts of this paper.

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Théo Grente and Étienne Grandjean. Conjunctive Grammars, Cellular Automata and Logic. In 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021). Open Access Series in Informatics (OASIcs), Volume 90, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The expressive power of the class Conj of conjunctive languages, i.e. languages generated by the conjunctive grammars of Okhotin, is largely unknown, while its restriction LinConj to linear conjunctive grammars equals the class of languages recognized by real-time one-dimensional one-way cellular automata. We prove two weakened versions of the open question Conj ⊆? RealTime1CA, where RealTime1CA is the class of languages recognized by real-time one-dimensional two-way cellular automata: 1) it is true for unary languages; 2) Conj ⊆ RealTime2OCA, i.e. any conjunctive language is recognized by a real-time two-dimensional one-way cellular automaton. Interestingly, we express the rules of a conjunctive grammar in two Horn logics, which exactly characterize the complexity classes RealTime1CA and RealTime2OCA.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Formal languages and automata theory
  • Computational complexity
  • Real-time
  • One-dimensional/two-dimensional cellular automaton
  • One-way/two-way communication
  • Grid-circuit
  • Unary language
  • Descriptive complexity
  • Existential second-order logic
  • Horn formula


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