Turing machines with finite memory

Abstract

Motivated by our earlier work on Turing computability via neural networks (1992, 2001) and the results by Maass et al. (1997, 1998) on the limit of what can be actually computed by neural networks when noise (or limited precision on the weights) is considered, we introduce the notion of definite Turing machine (DTM) and investigate some of its properties. We associate to every Turing machine (TM) a finite state automaton (FSA) responsible for the next state transition and action (output and head movement). A DTM is TM in which its FSA is definite. A FSA is definite (of order k>0) if its present state can be uniquely determined by the last k inputs. We show that DTM are strictly less powerful than TM but are capable to compute all simple functions. The corresponding notion of finite-memory Turing machine is shown to be computationally equivalent to Turing machine.