Boolean Circuit Complexity of Regular Languages

TL;DR

This paper introduces BC-complexity, a new measure for DFA complexity, revealing that while it can vary exponentially among automata with the same states, most DFAs have BC-complexity near the maximum, impacting automata minimization.

Contribution

It defines BC-complexity as an alternative to state complexity and analyzes its behavior, including exponential differences and the typical high complexity of most DFAs.

Findings

BC-complexity can differ exponentially among DFAs with same states

Minimization can exponentially increase BC-complexity

Most DFAs have BC-complexity close to the maximum

Abstract

In this paper we define a new descriptional complexity measure for Deterministic Finite Automata, BC-complexity, as an alternative to the state complexity. We prove that for two DFAs with the same number of states BC-complexity can differ exponentially. In some cases minimization of DFA can lead to an exponential increase in BC-complexity, on the other hand BC-complexity of DFAs with a large state space which are obtained by some standard constructions (determinization of NFA, language operations), is reasonably small. But our main result is the analogue of the "Shannon effect" for finite automata: almost all DFAs with a fixed number of states have BC-complexity that is close to the maximum.