Minimal Useful Size of Counters for (Real-Time) Multicounter Automata

Abstract

We show that, for nondeterministic and alternating machines with weak space bounds, the minimal space that is required for accepting a nonregular language by real-time or one-way multicounter automata is \((\log n)^{\varepsilon }\!\). The same space is required for two-way multicounter automata, independent of whether they are deterministic, nondeterministic, or alternating, and of whether they work with strong or weak space bounds. On the other hand, for deterministic, nondeterministic, and alternating machines with strong space bounds, and also for deterministic machines with weak space bounds, we show that the minimal space required for accepting a nonregular language by real-time or one-way multicounter automata is \(n^{\varepsilon }\!\). All these bounds hold both for unary and general nonregular languages. Here \(\varepsilon \) represents an arbitrarily small—but fixed—real positive constant; the “space” refers to the values stored in the counters, rather than to the lengths of their binary representation.