Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity

TL;DR

This paper establishes tight bounds on the size of finite automata representing sub- and superword closures of context-free grammars, and analyzes the computational complexity of related inequivalence problems, providing new insights into automata descriptional complexity.

Contribution

It proves tight exponential bounds on automata size for subword closures of CFGs and shows NP-completeness of inequivalence problems for these automata, advancing understanding of their complexity.

Findings

NFA size for subword closure is bounded by 2^{O(n)}

Deterministic automata can reach double-exponential size

Inequivalence problem for sub- or superword-closed NFAs is NP-complete

Abstract

We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of $2^{\mathcal{O}(n)}$ on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size $n$. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of $2^{2^{\mathcal{O}(n)}}$ following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub- or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs.