Relating the Type of Ambiguity of Finite Automata to the Succinctness of Their Representation

Abstract

This paper considers the problem of how the size of a nondeterministic finite automaton (nfa) representing a regular language depends on the type of ambiguity of the nfa. Primarily, the relationship between the ambiguity and the size in five types of nfa’s with increasing degrees of nondeterminism is studied: DFA (deterministic), ${\operatorname{UNA}}$ (unambiguous), ${\operatorname{FNA}}$ (finitely ambiguous), ${\operatorname{PNA}}$ (polynomially ambiguous), and ${\operatorname{ENA}}$ (exponentially ambiguous) nfa’s. The goal is to show “separation” among these classes, where a class A is said to be “separated” from B (written ($A,B$)) if for infinitely many n, there are machines of type B with n states whose minimal equivalent type A machine has more than $p(n)$ states for any polynomial p. Two classes are “polynomially equivalent” (written $A = B$) if machines of type A can be converted to machines of type B with only a polynomial increase in the number of states, and vice versa. For a class X, let $X(b)$) denote the restricted class of machines of type X with the restriction that the language accepted is bounded. The first main result compares the bounded restrictions of the five classes mentioned above. Specifically, the following is shown: $({\operatorname{DFA}}(b),{\operatorname{UNA}}(b)),({\operatorname{UNA}}(b),{\operatorname{FNA}}(b)),{\operatorname{FNA}}(b) = {\operatorname{PNA}}(b)$ and ${\operatorname{PNA}}(b) = {\operatorname{ENA}}(b)$, providing a complete picture of how the type of ambiguity affects the size complexity for unary and bounded languages. For unbounded languages it is conjectured that each of the five types of nondeterminism is separate from its higher types. But a proof does not exist at this time for two of the separations, the other two carrying over directly from the unary case. Candidates are offered that may be useful in proving the (other two) conjectured separations, and also a weaker form of separation in one case is shown. The notion of “concurrent conciseness” introduced by Kintala and Wotschke is studied. A class C is said to be concurrently concise over two classes A and B if $(A,B)$ and $(B,C)$ can be proved using the same collection of witness languages. One of the main results of this paper shows that, for unrestricted inputs, PNA is concurrently concise over ${\operatorname{DFA}}$ and ${\operatorname{UNA}}$. This answers an open problem of Stearns and Hunt. The succinctness problem is also studied through (regularity preserving) closure properties, an approach initiated by Sakoda and Sipser, and some interesting contrasts between various classes of nfa’s are shown.