Lower bounds on the size of sweeping automata

Abstract

Establishing good lower bounds on the complexity of languages is an important area of current research in the theory of computation. However, despite much effort, fundamental questions such as p = ? NP and L =? NL remain open. To resolve these questions it may be necessary to develop a deep combinatorial understanding of polynomial time or log space computations, possibly a formidable task. One avenue for approaching these problems is to study weaker models of computation for which the analogous problems may be easier to settle, perhaps yielding insight into the original problems. Sakoda and Sipser [3] raise the following question about finite automata: Is there a polynomial p, such that every n-state 2nfa (two-way nondeterministic finite automaton) has an equivalent p(n)-state 2dfa? They conjecture a negative answer to this. In this paper we take a step toward proving this conjecture by showing that 2nfa are exponentially more succinct than 2dfa of a certain restricted form. Call a 2dfa a sweeping automaton if it does not change its head direction except at the ends of the input tape. Thus, its head motions consist of a series of one way sweeps back and forth across the input. Our main result is that there are languages which can be accepted by an ?-state lnfa yet which require a 2"-state sweeping automaton. A number of other researchers have compared the suc-cinctness of various models of finite automata. Some of the earliest such results were obtained by Meyer and Fischer [1] who used information counting arguments to derive lower bounds. More recently, Schmidt [2] has compared nondeterministic, unambiguous nondeterministic, and deterministic one-way finite automata. He introduced the important idea of associating states with vectors over the field GF(2); this technique is also among those used here. Besides presenting the above open question, Sakoda and Sipser introduced the complete languages B? Which play a role in our lower bound proof. Berman and Lingas [4] consider this same question and prove a relationship between a variant of it and the L = ? NL question.