Monotonicity characterizations of regular languages

Each language $L\subseteq {\Sigma }^{⁎}$ induces an infinite sequence ${\left\{Pr\right(L,n\left)\right\}}_{n=1}^{\infty }$, where for all $n\ge 1$, the value $Pr(L,n)\in [0,1]$ 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 ${\left\{Pr\right(L,n\left)\right\}}_{n=1}^{\infty }$ for a regular language L, concerned zero-one laws, density, and accumulation points. We study monotonicity of ${\left\{Pr\right(L,n\left)\right\}}_{n=1}^{\infty }$, 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 $w^{\omega }$ is in L. Thus, in the second definition, monotonicity depends not only on the length of w, but also on the words being periodic. We also study the complexity of calculating $Pr(L,n)$ for the various definitions.