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The Simplest Non-Regular Deterministic Context-Free Language
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Petr Jančar 
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46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2021-08-18
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Acknowledgements
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)
https://doi.org/10.4230/LIPIcs.MFCS.2021.63
Abstract
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
Keywords
- deterministic context-free language
- truth-table reduction
- Mealy automaton
- pushdown automaton
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References
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