Showing papers on "Turing machine published in 1971"

Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory

Abstract: Publisher Summary This chapter discusses the algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Turing's analysis gives a mathematical analysis of configurational computations. The difference between configuration computations and algorithmic procedures is twofold. Firstly, in configurational computations the objects are symbols, whereas in algorithmic procedures the objects operated on are unrestricted. Secondly, in configurational computations at each stage one has a finite configuration whose size is not restricted before computation. In algorithmic procedures, one fixes beforehand a finite number of registers to hold the objects. The chapter presents mathematical analysis of algorithmic procedures with and without counting. Important features of elementary recursion theory are discussed. Effective definitional schemes, or equivalently, generalized Turing algorithms to generalize elementary recursion theory are applied. Turing's analysis could be improved in relation to configurational computability.

Computability by probabilistic Turing machines

Abstract: In the present paper, the definition of probabilistic Turing machines is extended to allow the introduction of relative computability. Relative computable functions, predicates and sets are discussed and their operations investigated. It is shown that, despite the fact that randomness is involved, most of the conventional results hold in the probabilistic case. Various classes of ordinary functions characterizable by computable random functions are introduced, and their relations are examined. Perhaps somewhat unexpectedly, it is shown that, in some sense, probabilistic Turing machines are capable of computing any given function. Finally, a necessary and sufficient condition for an ordinary function to be partially recursive is established via computable probabilistic Turing machines.

Computational complexity and g ¨ odel's incompleteness theorem

Abstract: Given any simply consistent formal theory F of the state complexity L(S) of nite binary sequences S as computed by 3-tape-symbol Turing machines, there exists a natural number L(F ) such that L(S) >n is provable in F only if n L (F ). The proof resembles Berry's paradox, not the Epimenides nor Richard paradoxes.

Graph property recognition machines

Abstract: A study is presented of machines that accept, as inputs, finite connected embedded graphs. These are, roughly, Turing machines in which the Turing tape is replaced by such a graph.

A Note on Degrees of Self-Describing Turing Machines

Abstract: Certain automata which build other automata are considered. Each such automaton contains a Turing machine. The properties of the Turing machine, of an offspring automaton, are completely determined by certain outputs of the Turing machine of the parent automaton. This makes it possible to speak of a parent Turing machine as describing its offspring Turing machine. I t is desirable to be able to conduct an algorithmic or partial algorithmic analysis of the properties of the descendants of an arbitrary Turing machine M given only a description of M. To this end, six types of Turing machines which describe certain effectively computable distortions of themselves (self-description being a special case) are defined. The Turing machines of two of these types are also universal Turing machines. For each of the six types of Turing machines, the problem of deciding whether or not a given Turing machine is of that type is considered. The degree of unsolvability of each of these decision problems is computed. The results of these computations are used to show that the corresponding decision problems have successively better partial algorithmic solutions which can be obtained algorithmically.

Establishing lower bounds on algorithms: a survey

Abstract: Algorithms for various computations have been known and studied for centuries, but it is only recently that much theoretical attention has been devoted to the analysis of algorithms. Turing machines and recursive functions were the first approaches, but these models, which provide much interesting mathematics, do not look at the problem from a practical standpoint. In "real" computing, no one uses Turing machines to evaluate polynomials or to multiply matrices, and little of practical significance is obtained from that approach. On the other hand, recent work has led to more realistic models and, correspondingly, to more practical results. Most of the results cannot be considered to be truly practical, but, all of them were motivated by practical considerations.

A note on AFLs and bounded erasing

Abstract: F-bounded erasing operator in abstract family of language for mapping, applying to families defined by tape-bounded Turing acceptors

A Note on "Axioms" for Computational Complexity and Computation of Finite Functions

Abstract: Recent studies of computational complexity have focused on “axioms” which characterize the “difficulty of a computation” (Blum, 1967a or the measure of the “size of a program,” (Blum, 1967b and Pager, 1969) . In this paper we wish to carefully examine the consequences of hypothesizing a relation which connects measures of size and of difficulty of computation. The relation is motivated by the fact that computations are performed “a few instructions at a time” so that if one has a bound on the difficulty of a computation, one also has a bound on the “number of executed instructions.” This relation enables one to easily show that algorithms exist for finding the most efficient programs for computing finite functions. This result, which has been obtained independently for certain Turing machine measures by David Pager, contrasts sharply with results for measures of size, where it is known that no algorithm can exist for going from a finite function to the shortest program for computing it (Blum, 1967b) (Pager, 1969) . In a concluding section, which can be read independently of the above-mentioned results, some remarks are made about the desirability of using a program for computing an infinite function when one is interested in the function only on some finite domain. There is nothing deep in this paper, and we hope that a reader familiar with the rudiments of recursion theory will find this paper a simple introduction to the “axiomatic” theory of computational complexity. Such a reader might do well to begin with the concluding remarks after reading the basic definitions.

Kreisel's Work on the Philosophy of Mathematics-I. Realism

Abstract: Publisher Summary This chapter reviews Kreisel's work on the philosophy of mathematics. The data of foundations consist of the mathematical experience of the working mathematician; the general problem of foundations is to analyze the experience as a whole. Crude formalism holds that mathematics consists of assertions of the form: a concretely given configuration has been constructed by means of a given mechanical rule. No general statements about such configurations belong to mathematics. The positivist doctrine considers informal derivations either as unreliable or as irrelevant to mathematics. Realism is the assumption that there are basic elements (sets and the membership relation) with the properties assumed in the cumulative hierarchy; that is, the existential assumptions of set theory are valid. Strong realism is committed to the existence of sets, some containing an infinite number of members. Rejection of formalism has led to the development of systems of logic that cannot be represented mechanically, as by a Turing machine.

On restricted turing computability

Abstract: In this paper we continue the study, begun by Hartmanis and Stearns [1], of multitape Turing machines which compute infinite binary sequences. Theorems are proved concerning the hierarchy of complexity classes of binary sequences which is obtained by bounding simultaneously both the time T(n) and the amount of tape L(n) which a Tnring machine may use to compute its output sequence; among these is a result on the maximum increase necessary in the time and tape bounding functions T(n) and L(n) in order to obtain a strictly larger class of sequences. The proofs are constructive throughout, and some open problems are posed at the end of the paper. Section 1. We assume that the definition of the classical one-tape Turing machine is known (cf. [2] or [3]). Definition 1. In this paper, by a multitape Turing machine, or briefly, machine 3-, we shall mean a Turing machine having k > 0 two-way (potentially) infinite work tapes and one one-way (e.g., right) (potentially) infinite output tape, upon each of which there is a single read-write head. Each operation of the machine 3 \" is determined by the current internal state qi of Y , by the symbols swj(1 < j < k) scanned by the heads on the k work tapes, and by the symbol s, scanned by the head on the output tape. In a single operation, ~-overprints each of the symbols sw~ with a new symbol s ' : overprints the symbol s, with a new symbol s~, moves each of the heads on the work tapes (independently) either one square to the right or one square to the left or leaves it in the same position, moves the head on the output tape one square to the right or leaves it where it is, and enters a new internal state q , At any moment, we shall say that ~rhas printed the finite output sequence ~1~2. . .% if this sequence appears on the output tape starting at the leftmost square, and the head on the output tape is scanning the first square to the right of those containing this sequence.

Records of turing machines

Abstract: Suppose a Turing machine is equipped with an extra tape. At each step of a computation being performed, it prints symbol read move symbol symbol printed on a square of the extra tape. It then moves the extra tape one square to the left. This procedure yields arecord of the computation.

Some Methodological Aspects of a System Theory

Abstract: The purpose of this lecture is two-fold. In the first part the role of the mathe­matical notion of a functional is emphasized and it is shown to be the underlying notion in straightforward practice in more or less formal systems constructions. In particular it is argued that in many cases the simple conception of a system as a set of input-output relations is inadequate.

The complexity of skewlinear tuple languages and o-regular languages

Abstract: Skewlinear tuple grammars are introduced as generalization of even linear and k -linear context-free grammars. The family of skewlinear tuple languages coincides with the family of languages generated by o -regular expressions. Many questions which are unsolvable for context-free languages and hence for tuple languages are solvable for skewlinear tuple languages. Every skewlinear tuple language can be recognized by a deterministic one-tape Turing machine of time complexity T ( r ) = r 2 and by a deterministic one-tape Turing machine with two-way input tape of tape complexity L ( r ) = log r .

The Immortality Problem for Non-Erasing Turing Machines

Abstract: Publisher Summary In this chapter M is considered a Turing machine (TM). An instantaneous description (ID) of M is a triple 〈q,X,n〉 where q∈K, X∈ ∑∞ and n≥1. describes that M is in state q with the read-write head scanning square no. n and that the tape T contains X. M is to stop if M tries to go off the tape at the left end. M is called a non-writing TM if it contains no write-instructions. The immortality problem (IP) associated with a set of TM-s is the problem of deciding, for a given TM in the set, whether or not there exists an immortal ID. It is shown that IP for non-erasing TM-s is decidable, if the tape is allowed to contain ultimately periodic words. If, however, the tape is restricted to contain only a finite number of non-blanks, then the IP for the set of nonerasing TM-s is recursively undecidable (of degree 0").

Inessential Errors in Sequential Machines

Abstract: Hartmanis and Stearns defined the concept of an inessential error in their study of errors in sequential machines and represented such errors by means of an error partition Π E. Although they showed that ΠE could not be determined using only the usual partition pair algebras, they did not provide a means by which it could be determined. The purpose of this note is to develop an algorithm for the determination of ΠE for a given machine.