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Languages of Words of Low Automatic Complexity Are Hard to Compute

Authors Joey Chen, Bjørn Kjos-Hanssen , Ivan Koswara , Linus Richter , Frank Stephan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Regular languages
Keywords
  • Automatic complexity
  • automata theory
  • formal languages
  • Boolean circuits
  • Shannon effect

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Abstract

The automatic complexity of a finite word (string) is an analogue for finite automata of Sipser’s distinguishing complexity (1983) and was introduced by Shallit and Wang (2001). For a finite alphabet Σ of at least two elements, we consider the non-deterministic automatic complexity given by exactly - yet not necessarily uniquely - accepting automata: a word x ∈ Σ^* has exact non-deterministic automatic complexity k ∈ ℕ if there exists a non-deterministic automaton of k states which accepts x while rejecting every other word of the same length as x, and no automaton of fewer states has this property. Importantly, and in contrast to the classical notion, the witnessing automaton may have multiple paths of computation accepting x. We denote this measure of complexity by A_{Ne}, and study a class of languages of low A_{Ne}-complexity defined as L_q = {x ∈ Σ^* : A_{Ne}(x) < q|x|}, which is parameterised by rationals q ∈ (0,1/2) (generalising a class of sets first studied by Kjos-Hanssen). We show that for every q ∈ (0,1/2), this class is neither context-free nor recognisable by certain Boolean circuits. In the process, we answer an open question of Kjos-Hanssen quantifying the complexity of L_{1/3} in terms of Boolean circuits, and also prove the Shannon effect for A_{Ne}.

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Joey Chen, Bjørn Kjos-Hanssen, Ivan Koswara, Linus Richter, and Frank Stephan. Languages of Words of Low Automatic Complexity Are Hard to Compute. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.24

Author Details

Joey Chen
  • Department of Mathematics, National University of Singapore, Singapore
Bjørn Kjos-Hanssen
  • Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, HI, USA
Ivan Koswara
  • School of Computing, National University of Singapore, Singapore
Linus Richter
  • Department of Mathematics, National University of Singapore, Singapore
Frank Stephan
  • Department of Mathematics, National University of Singapore, Singapore
  • School of Computing, National University of Singapore, Singapore

Funding

  • Kjos-Hanssen, Bjørn: This work was partially supported by a grant from the Simons Foundation (#704836 to Bjørn Kjos-Hanssen).
  • Richter, Linus: This work was fully supported by Singapore Ministry of Education grant MOE-000538-01.
  • Stephan, Frank: This work was partially supported by Singapore Ministry of Education grant MOE-000538-01.

Acknowledgements

Parts of this work have appeared in the first author’s Bachelor’s thesis submitted to the National University of Singapore.

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