# The structure and complexity of minimal NFA's over a unary alphabet

## Abstract

Many difficult open problems in theoretical computer science center around nondeterminism. We study the fundamental problem of converting a given deterministic finite automaton (DFA) to a minimal nondeterministic finite automaton (NFA). Despite extensive work on finite automata, this fundamental problem has remained open. Recently, in [Ji91] we studied this problem and showed that this (and related) problems are computationally *hard*. Here we study the restriction of this problem to the case when the input DFA is over a one-letter alphabet. Even in this restricted case the problem is computationally *hard* even though our evidence of *hardness is* different from (and is weaker than) the standard ones such as NP-hardness. Let A → B denote the problem of converting a finite automaton of type A to a minimal finite automaton of type B. Our main result is that DFA → NFA, when the input is a unary cyclic DFA (a DFA whose graph is a simple cycle), is in NP but not in P unless NP ⊑ DTIME(*n*^{O(log n)}). Our work was also motivated by the problem of finding structurally simple ‘normal forms’ of NFA's over a unary alphabet. We present some normal forms for *minimal* NFA's over a unary alphabet and present an application to lower bounds on the size complexity of an NFA. In fact, the normal form result is used in a nontrivial manner to show the NP membership result stated above.

## Keywords

Normal Form Start State Vertex Cover Regular Language Finite Automaton## Preview

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