Descriptional Complexity of Winning Sets of Regular Languages

Marcus, Pierre; Törmä, Ilkka

doi:10.1007/978-3-030-62536-8_11

Cited by 2 publications

(2 citation statements)

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“…If X is a subshift, then the smallest hereditary subshift containing X is called the hereditary closure of X. In the earlier papers [28,23,18], the word downward-closed was used, but we opt to use hereditary which seems more common in symbolic dynamics [13,14].…”

Section: Preliminariesmentioning

confidence: 99%

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Automatic winning shifts

Peltomäki,

Salo

2021

Preprint

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To each one-dimensional subshift X, we may associate a winning shift W(X) which arises from a combinatorial game played on the language of X. Previously it has been studied what properties of X does W(X) inherit. For example, X and W(X) have the same factor complexity and if X is a sofic subshift, then W(X) is also sofic. In this paper, we develop a notion of automaticity for W(X), that is, we propose what it means that a vector representation of W(X) is accepted by a finite automaton.Let S be an abstract numeration system such that addition with respect to S is a rational relation. Let X be a subshift generated by an S-automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of W(X) (which follows from X having sublinear factor complexity), then W(X) is accepted by a finite automaton, which can be effectively constructed from the description of X. We provide an explicit automaton when X is generated by certain automatic words such as the Thue-Morse word.

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Section: Preliminariesmentioning

confidence: 99%

“…Additionally, it was shown that W(X) has a substitutive structure when X is a subshift generated by a marked uniform substitution. In [18], Marcus and Törmä study the descriptional complexity of W(X) when X is regular. A concept equivalent to winning sets has been introduced in the context of set systems in [2].…”

Section: Introductionmentioning

confidence: 99%