A Close Analysis of the Subset Construction

TL;DR

This paper investigates the computational difficulty of estimating the size of deterministic automata derived from nondeterministic automata, proving PSPACE-hardness results and introducing subset complexity as a practical bounding measure.

Contribution

It establishes the PSPACE-hardness of estimating DFA size from NFA and introduces subset complexity to predict exponential blow-up in subset construction.

Findings

Computing NFA state complexity is PSPACE-hard.

Deciding exponential increase in DFA size is PSPACE-hard.

Subset complexity can be efficiently bounded using matrix properties.

Abstract

Given a nondeterministic finite-state automaton (NFA), we aim to estimate the size of an equivalent deterministic finite-state automaton (DFA). We demonstrate that computing the state complexity of an NFA within polynomial precision is PSPACE-hard. Furthermore, we also demonstrate that it is PSPACE-hard to decide whether the classical subset construction will yield an equivalent DFA with an exponential increase in the number of states. This result implies that making any a prior estimate of the running time of the subset construction is inherently difficult. To address this, and to enable forecasting of such exponential blow-up in certain special cases, we introduce the notion of subset complexity, which provides an upper bound on the size of the DFA produced by the subset construction. We show that the subset complexity can be efficiently bounded above using the cyclicity and rank of the transition matrices of the NFA. This yields a sufficient condition for identifying NFAs that can be efficiently determinized via the subset construction.