On the Complexity and Decidability of Some Problems Involving Shuffle

TL;DR

This paper investigates the computational complexity and decidability of problems involving the shuffle operation on formal languages, establishing NP-completeness for some problems and polynomial-time algorithms for others, with various language classes.

Contribution

It provides new complexity classifications and decidability results for shuffle-related problems across different automata and language classes.

Findings

Three shuffle problems are NP-complete.

Polynomial-time algorithms are found for subset problems involving shuffle.

Several closure properties of shuffle are established.

Abstract

The complexity and decidability of various decision problems involving the shuffle operation are studied. The following three problems are all shown to be $NP$-complete: given a nondeterministic finite automaton (NFA) $M$, and two words $u$ and $v$, is $L(M)$ not a subset of $u$ shuffled with $v$, is $u$ shuffled with $v$ not a subset of $L(M)$, and is $L(M)$ not equal to $u$ shuffled with $v$? It is also shown that there is a polynomial-time algorithm to determine, for $NFA$s $M_1, M_2$ and a deterministic pushdown automaton $M_3$, whether $L(M_1)$ shuffled with $L(M_2)$ is a subset of $L(M_3)$. The same is true when $M_1, M_2,M_3$ are one-way nondeterministic $l$-reversal-bounded $k$-counter machines, with $M_3$ being deterministic. Other decidability and complexity results are presented for testing whether given languages $L_1, L_2$ and $R$ from various languages families satisfy $L_1$ shuffled with $L_2$ is a subset of $R$, and $R$ is a subset of $L_1$ shuffled with $L_2$. Several closure results on shuffle are also shown.