Counting and Sampling Traces in Regular Languages

Abstract

In this work, we study the fundamental problems of counting and sampling traces that a regular language touches. Formally, one fixes the alphabet $\Sigma$ and an independence relation $\mathbb{I} \subseteq \Sigma \times \Sigma$. The computational problems we address take as input a regular language $L$ over $\Sigma$, presented as a finite automaton with $m$ states, together with a natural number $n$ (presented in unary). For the counting problem, the output is the number of Mazurkiewicz traces (induced by $\mathbb{I}$) that intersect the nth slice $L_n = L \cap \Sigma^n$ of $L$, i.e., traces that have at least one linearization in $L_n$. For the sampling problem, the output is a trace drawn from a distribution that is approximately uniform over all such traces. These problems are motivated by applications such as bounded model checking based on partial-order reduction, where an a priori estimate of the size of the state space can significantly improve usability, as well as testing approaches for concurrent programs that use partial-order-aware random sampling, where uniform exploration is desirable for effective bug detection. We first show that the counting problem is #P-hard even when the automaton accepting the language $L$ is deterministic, which is in sharp contrast to the corresponding problem for counting the words of a DFA, which is solvable in polynomial time. We then show that the counting problem remains in the class $\#{\sf P}$ for both NFAs and DFAs, independent of whether $L$ is trace-closed. Finally, our main contributions are a fully polynomial-time randomized approximation scheme (FPRAS) that, with high probability, estimates the desired count within a specified accuracy parameter, and a fully polynomial-time almost uniform sampler (FPAUS) that generates traces while ensuring that the distribution induced on them is approximately uniform with high probability.