DTIME

Abstract: The equivalence problem for Kleene's regular expressions has several effective solutions, all of which are computationally inefficient. In [1], we showed that this inefficiency is an inherent property of the problem by showing that the problem of membership in any arbitrary context-sensitive language was easily reducible to the equivalence problem for regular expressions. We also showed that with a squaring abbreviation ( writing (E)2 for E×E) the equivalence problem for expressions required computing space exponential in the size of the expressions. In this paper we consider a number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines. The definitions of the word problems and a table summarizing their complexity appears in the next section. More detailed comments and an outline of some of the proofs follows in the remaining sections. Complete proofs will appear in the forthcoming papers [9, 10, 13]. In the final section we describe some open problems.

Abstract: A model of computation based on random access machines operating in parallel and sharing a common memory is presented. The computational power of this model is related to that of traditional models. In particular, deterministic parallel RAM's can accept in polynomial time exactly the sets accepted by polynomial tape bounded Turing machines; nondeterministic RAM's can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines. Similar results hold for other classes. The effect of limiting the size of the common memory is also considered.

Abstract: This paper deals with finite networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a “sigmoidal” scalar nonlinearity to a linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by rational nets. In particular, one can do this in linear time, and there is a net made up of about 1,000 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Furthermore, we assert a similar theorem about non-deterministic Turing Machines. Consequences for undecidability and complexity issues about nets are discussed too.