On determinism versus nondeterminism for restarting automata

Abstract

A restarting automaton processes a given word by executing a sequence of local simplifications until a simple word is obtained that the automaton then accepts. Such a computation is expressed as a sequence of cycles. A nondeterministic restarting automaton M is called correctness preserving, if, for each cycle u ⊢ M c v , the string v belongs to the characteristic language L C ( M ) of M , if the string u does. Our first result states that for each type of restarting automaton X ∈ { R , RW , RWW , RL , RLW , RLWW } , if M is a nondeterministic X-automaton that is correctness preserving, then there exists a deterministic X-automaton M 1 such that the characteristic languages L C ( M 1 ) and L C ( M ) coincide. When a restarting automaton M executes a cycle that transforms a string from the language L C ( M ) into a string not belonging to L C ( M ) , then this can be interpreted as an error of M . By counting the number of cycles it may take M to detect this error, we obtain a measure for the influence that errors have on computations. Accordingly, this measure is called error detection distance. It turns out, however, that an X-automaton with bounded error detection distance is equivalent to a correctness preserving X-automaton, and therewith to a deterministic X-automaton. This means that nondeterminism increases the expressive power of X-automata only in combination with an unbounded error detection distance.