The Magic Number Problem for Subregular Language Families

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.

💡 Research Summary

The paper addresses the “magic number problem,” which asks whether for every pair of integers n and α satisfying n ≤ α ≤ exp(2,n) there exists an n‑state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has exactly α states. In the general regular language setting this problem was solved in the affirmative: no “magic numbers” (values of α that cannot be realized) exist, as shown in reference