Showing papers by "Juris Hartmanis published in 1968"

Computational Complexity of One-Tape Turing Machine Computations

Abstract: The quantitative aspects of one-tape Turing machine computations are considered. It is shown, for instance, that there exists a sharp time bound which must be reached for the recognition of nonregular sets of sequences. It is shown that the computation time can be used to characterize the complexity of recursive sets of sequences, and several results are obtained about this classification. These results are then applied to the recognition speed of context-free languages and it is shown, among other things, that it is recursively undecidable how much time is required to recognize a nonregular context-free language on a one-tape Turing machine. Several unsolved problems are discussed.

On the Recognition of Primes by Automata

Abstract: A study of the problem of recognizing the set of primes by automata is presented. A simple algebraic condition is derived which shows that neither the set of primes nor any infinite subset of the set of primes can be accepted by a pushdown or finite automaton.In view of this result an interesting open problem is to determine the “weakest” automaton which can accept the set of primes. It is shown that the linearly bounded automaton can accept the set of primes, and it is conjectured that no automaton whose memory grows less rapidly can recognize the set of primes. One of the results shows that if this conjecture is true, it cannot be proved by the use of arguments about the distribution of primes, as described by the Prime Number Theorem. Some relations are established between two classical conjectures in number theory and the minimal rate of memory growth of automata which can recognize the set of primes.

Structure of undecidable problems in automata theory

Abstract: The purpose of this paper is to gain a better understanding of the structure of undecidable problems in automata theory by investigating the degree of unsolvability of these problems This is achieved by using Turing machines with oracles to define when one undecidable problem can be reduced to another and to establish an infinite hierarchy of (equivalent) undecidable problems This hierarchy is then used to classify well-known undecidable problems about various families of automata and formal languages and to study the relations between these problems This approach reveals a well defined structuring of the undecidable problems and permits a more systematic study of these problems and their relation to various families of automata

Tape reversal complexity hierarchies

Abstract: Studies in computational complexity have proceeded both by dealing with measures of complexity in the abstract [1] and by investigating properties of specific 'papameters of a computation. In the case where the computing device considered is a Turing machine, the measures of time [6,9], memory space [10], and number of storage tapes [8] have all been considered. In this paper we discuss the measure of tape reversals on a Turing machine acceptor with one working tape. This measure yields a striking difference in "speed up" properties between the case in which the Turing machine is equipped with a separate one-way tape ("on-line") and that in which the input is placed initiallyon the working tape of the Turing machine (Hoff-linen). In spite of this difference, however, the on-line and offline reversal complexity classes can be shown to coincide whenever the number of reversals permitted is at least as great as the length of the input. The discussion in this paper will be very informal. References [3] and [5] will contain more formal treatments of some of the results to be presented here. An on-line Turing machine (NTM) is defined as a 2-tape Turing machine in which one tape is reserved solely for input and can be moved only in one direction. Thus, an NTM g has a finite state set S = {sl,s2, ... ,sn} , a finite alphabet