On simulation cost of unary limited automata

Abstract

A k-limited automaton is a linear bounded automaton that may rewrite each tape cell only in the first k visits, where k ≥ 0 is a fixed constant. It is known that these automata accept context-free languages only. We investigate the descriptional complexity of deterministic limited automata accepting unary languages. Since these languages are necessarily regular, we study the cost in the number of states when a klimited automaton is simulated by finite automata. For the conversion of a 4n-state 1-limited automaton into one-way or two-way deterministic or nondeterministic finite automata a lower bound of n ・ F(n) states is shown, where F denotes Landau’s function. So, even the ability deterministically to rewrite any cell only once gives an enormous descriptional power. For the simulation cost for removing the ability to rewrite each cell k ≥ 1 times, that is, the cost for the simulation of (sweeping) klimited automata by deterministic finite automata, we obtain a lower bound of n ・ F(n)k. A polynomial upper bound is shown for the simulation by two-way deterministic finite automata, where the degree of the polynomial is quadratic in k. If the k-limited automaton is rotating, the upper bound reduces to O(nk+1). A lower bound of Ω(nk+1) is derived even for nondeterministic two-way finite automata. So, for rotating klimited automata, the trade-off for the simulation is tight in the order of magnitude.