The Simplest Non-Regular Deterministic Context-Free Language

Authors Petr Jančar , Jiří Šíma

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Author Details

Petr Jančar
  • Dept. of Computer Science, Faculty of Science, Palacký University Olomouc, Czech Republic
Jiří Šíma
  • Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic


J. Šíma also thanks Martin Plátek for his intensive collaboration at the first stages of this research.

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Petr Jančar and Jiří Šíma. The Simplest Non-Regular Deterministic Context-Free Language. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 63:1-63:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We introduce a new notion of 𝒞-simple problems for a class 𝒞 of decision problems (i.e. languages), w.r.t. a particular reduction. A problem is 𝒞-simple if it can be reduced to each problem in 𝒞. This can be viewed as a conceptual counterpart to 𝒞-hard problems to which all problems in 𝒞 reduce. Our concrete example is the class of non-regular deterministic context-free languages (DCFL'), with a truth-table reduction by Mealy machines. The main technical result is a proof that the DCFL' language L_# = {0^n1^n ∣ n ≥ 1} is DCFL'-simple, and can be thus viewed as one of the simplest languages in the class DCFL', in a precise sense. The notion of DCFL'-simple languages is nontrivial: e.g., the language L_R = {wcw^R∣ w ∈ {a,b}^*} is not DCFL'-simple. By describing an application in the area of neural networks (elaborated in another paper), we demonstrate that 𝒞-simple problems under suitable reductions can provide a tool for expanding the lower-bound results known for single problems to the whole classes of problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Transducers
  • deterministic context-free language
  • truth-table reduction
  • Mealy automaton
  • pushdown automaton


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