The Magic Number Problem for Subregular Language Families

TL;DR

This paper explores the existence of minimal nondeterministic finite automata whose equivalent minimal deterministic automata have a specific number of states, focusing on subregular language families and identifying the presence of only trivial magic numbers.

Contribution

It extends the analysis of the magic number problem to various subregular language families, showing that only trivial magic numbers exist in these cases.

Findings

No non-trivial magic numbers for subregular languages.

Trivial magic numbers are the only ones that exist for these language families.

Partial results indicate certain finite language state counts are non-magic.

Abstract

We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has alpha states, for all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n). A number alpha not satisfying this condition is called a magic number (for n). It was shown in [11] that no magic numbers exist for general regular languages, while in [5] trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, combinational languages, star-free, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.