Counting and Sampling Traces in Regular Languages

2025-11-28

AI Summary

This paper addresses the counting and approximate uniform sampling of Mazurkiewicz traces in the intersection of a regular language (L) (given by a finite automaton) and the set (Sigma^n) of all traces of length (n). We first establish that the counting problem remains #P-complete even for deterministic automata. Then, we design both an FPRAS (Fully Polynomial Randomized Approximation Scheme) and an FPAUS (Fully Polynomial Almost Uniform Sampler) applicable to arbitrary—deterministic or nondeterministic—automata. Our approach leverages independence relation modeling, trace decomposition, and rapid mixing of carefully constructed Markov chains. The algorithms run in polynomial time and, with high probability, yield a ((1 pm varepsilon))-multiplicative approximation to the count and samples within total variation distance (O(varepsilon)) of the uniform distribution over (L cap Sigma^n). These results support partial-order reduction, model checking, and randomized testing of concurrent programs.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Tracking, profiling, and countermeasures against them
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this work, we study the problems of counting and sampling Mazurkiewicz traces that a regular language touches. Fix an alphabet $Σ$ and an independence relation $mathbb{I} subseteq Σ imes Σ$. The input consists of a regular language $L subseteq Σ^*$, 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 $n^ ext{th}$ slice $L_n = L cap Σ^n$, 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.

Problem

• Counting Mazurkiewicz traces intersecting regular language slices
• Sampling traces from approximately uniform distribution over traces
• Addressing bounded model checking and concurrent program testing

Innovation

• FPRAS approximates trace counts in regular languages
• FPAUS generates near-uniform trace samples efficiently
• Algorithms handle both NFAs and DFAs for trace counting