Counting and Sampling Traces in Regular Languages

Abstract

In this work, we study the problems of counting and sampling Mazurkiewicz traces that a regular language touches. Fix an alphabet $\Sigma$ and an independence relation $\mathbb{I} \subseteq \Sigma \times \Sigma$. The input consists of a regular language $L \subseteq \Sigma^*$, given by a finite automaton with $m$ states, and a natural number $n$ (in unary). For the counting problem, the goal is to compute the number of Mazurkiewicz traces (induced by $\mathbb{I}$) that intersect the $n^\text{th}$ slice $L_n = L \cap \Sigma^n$, i.e., traces that admit at least one linearization in $L_n$. For the sampling problem, the goal is to output a trace drawn from a distribution that is approximately uniform over all such traces. These tasks are motivated by bounded model checking with partial-order reduction, where an \emph{a priori} estimate of the reduced state space is valuable, and by testing methods for concurrent programs that use partial-order-aware random exploration. We first show that the counting problem is #P-hard even when $L$ is accepted by a deterministic automaton, in sharp contrast to counting words of a DFA, which is polynomial-time solvable. We then prove that the problem lies in #P for both NFAs and DFAs, irrespective of whether $L$ is trace-closed. Our main algorithmic contributions are a \emph{fully polynomial-time randomized approximation scheme} (FPRAS) that, with high probability, approximates the desired count within a prescribed accuracy, and a \emph{fully polynomial-time almost uniform sampler} (FPAUS) that generates traces whose distribution is provably close to uniform.