Nondeterministic Syntactic Complexity

TL;DR

This paper introduces nondeterministic syntactic complexity as a new measure for regular languages, linking algebraic structures with automata theory to unify previous work on nondeterministic state-minimality.

Contribution

It defines a novel measure for regular languages and connects it to algebraic and automata-theoretic concepts, providing a unified framework for nondeterministic state-minimality.

Findings

The measure equals the minimal number of states in subatomic nondeterministic automata.

All previous structural work on nondeterministic minimality computes this measure.

The approach uses an algebraic interpretation of nondeterministic automata as semilattice-structured deterministic automata.

Abstract

We introduce a new measure on regular languages: their nondeterministic syntactic complexity. It is the least degree of any extension of the `canonical boolean representation' of the syntactic monoid. Equivalently, it is the least number of states of any subatomic nondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.