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# Monotonicity Characterizations of Regular Languages

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## Cite As

Yoav Feinstein and Orna Kupferman. Monotonicity Characterizations of Regular Languages. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.26

## Abstract

Each language L ⊆ Σ^* induces an infinite sequence Pr(L,n)_{n=1}^∞, where for all n ≥ 1, the value Pr(L,n) ∈ [0,1] is the probability of a word of length n to be in L, assuming a uniform distribution on the letters in Σ. Previous studies of Pr(L,n)_{n=1}^∞ for a regular language L, concerned zero-one laws, density, and accumulation points. We study monotonicity of Pr(L,n)_{n=1}^∞, possibly in the limit. We show that monotonicity may depend on the distribution of letters, study how operations on languages affect monotonicity, and characterize classes of languages for which the sequence is monotonic. We extend the study to languages L of infinite words, where we study the probability of lasso-shaped words to be in L and consider two definitions for Pr(L,n). The first refers to the probability of prefixes of length n to be extended to words in L, and the second to the probability of word w of length n to be such that w^ω is in L. Thus, in the second definition, monotonicity depends not only on the length of w, but also on the words being periodic.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Formal languages and automata theory
##### Keywords
• Regular Languages
• Probability
• Monotonicity
• Automata

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