Monotonicity Characterizations of Regular Languages
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.
Regular Languages
Probability
Monotonicity
Automata
Theory of computation~Formal languages and automata theory
26:1-26:19
Regular Paper
Supported by the Israel Science Foundation, Grant 2357/19, and by the European Research Council, Advanced Grant ADVANSYNT.
Yoav
Feinstein
Yoav Feinstein
School of Engineering and Computer Science, Hebrew University, Jerusalem, Israel
Orna
Kupferman
Orna Kupferman
School of Engineering and Computer Science, Hebrew University, Jerusalem, Israel
https://orcid.org/0000-0003-4699-6117
10.4230/LIPIcs.FSTTCS.2023.26
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Yoav Feinstein and Orna Kupferman
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