Rice-Like Theorems for Automata Networks

Authors Guilhem Gamard, Pierre Guillon, Kevin Perrot, Guillaume Theyssier

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Guilhem Gamard
  • Aix-Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France
Pierre Guillon
  • Aix-Marseille Université, CNRS, I2M, Marseille, France
Kevin Perrot
  • Aix-Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France
  • Université Côte d'Azur, CNRS, I3S, Sophia Antipolis, France
Guillaume Theyssier
  • Aix-Marseille Université, CNRS, I2M, Marseille, France


We would like to thank the colleagues that were involved in some discussions on the topic at the early stages of this work and somehow convinced us to pursue and finally settle the main result.

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Guilhem Gamard, Pierre Guillon, Kevin Perrot, and Guillaume Theyssier. Rice-Like Theorems for Automata Networks. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We prove general complexity lower bounds on automata networks, in the style of Rice’s theorem, but in the computable world. Our main result is that testing any fixed first-order property on the dynamics of an automata network is either trivial, or NP-hard, or coNP-hard. Moreover, there exist such properties that are arbitrarily high in the polynomial-time hierarchy. We also prove that testing a first-order property given as input on an automata network (also part of the input) is PSPACE-hard. Besides, we show that, under a natural effectiveness condition, any nontrivial property of the limit set of a nondeterministic network is PSPACE-hard. We also show that it is PSPACE-hard to separate deterministic networks with a very high and a very low number of limit configurations; however, the problem of deciding whether the number of limit configurations is maximal up to a polynomial quantity belongs to the polynomial-time hierarchy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Complexity classes
  • Automata networks
  • Rice theorem
  • complexity classes
  • polynomial hierarchy
  • hardness


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