Showing papers on "Turing machine published in 1972"

On non-determinancy in simple computing devices

Abstract: This paper studies one-tape Turing machines with k read-only heads which are restricted to the original input. The main result shows that if any set accepted by such a 3-head non-deterministic Turing machine can be accepted by a deterministic Turing machine with more read-only heads, then the deterministic and non-deterministic context-sensitive languages are identical. Several related results are derived and some tantalizing open problems are discussed.

Turing machines and the spectra of first-order formulas with equality

Abstract: In this paper we show that these similarities are not accidental - that spectra and context sensitive languages are closely related, and that their open questions are merely special cases of a family of open questions which relate to the difference (if any) between deterministic and non-deterministic time-or space-bounded Turing machines.

Programming with(out) the GOTO

Abstract: A brief history of the goto controversy (retention or deletion of the goto statement) is presented. After considering some of the theoretical and practical aspects of the problem, a summary of arguments both for and against the goto is given.

Reversal-bounded multi-pushdown machines

Abstract: This paper presents several representations of the recursively enumerable (re) sets The first states that every re set is the homomorphic image of the intersection of two linear context-free languages Another states that every re set is accepted by an on-line Turing acceptor with two pushdown stores such that in every computation, each pushdown store can make at most one reversal (that is, one change from "pushing" to "popping") It is shown that this automatatheoretic representation cannot be strengthened by restricting the acceptors to be either deterministic multitape, nondeterministic one-tape, or nondeterministic multicounter acceptors An investigation of the properties of reversal-bounded computations suggests that reversal bounds are not a "natural" measure of computational complexity for multitape Turing acceptors The above results are used to obtain an independence theorem for full semi-AFLs and an undecidability result for effective families of languages

New Results for Rado's Sigma Function for Binary Turing Machines

Abstract: A computer program was written and executed to search for better lower bounds to Rado's noncomputable sigma and shift functions for binary Turing machines. Former results in this search (called by Rado the Busy Beaver logical game) are reviewed and new bounds found by this program are presented.

Computability and Decidability: An Introduction for Students of Computer Science

Abstract: 1: Sets and Functions.- 1.1. The objects.- 1.2. Ordered sequences and sets.- 1.3. Further notations and definitions concerning sets.- 1.4. Functions.- 1.5. Particular objects.- 2: Sets and Functions of Strings.- 2.1. Definitions.- 2.2. String functions.- 2.3. Further notations and definitions.- 2.4. The interpretation of strings.- 2.5. Alphabetic order.- 2.6. Enumeration of strings and n-tuples of strings.- 2.7. Enumeration functions.- 2.8. Calculating the value of the enumeration functions.- 3: Computable Functions.- 3.1. Historical background.- 3.2. The basic idea of Turing.- 3.3. Physical model.- 3.4. Formal definition of a Turing machine.- 3.5. Examples of Turing machines.- 3.6. Computable functions.- 3.7. The thesis of Turing.- 3.8. Normal Turing machines.- 4: The Universal Turing Machine.- 4.1. The string description of a Turing machine.- 4.2. The universal Turing machine.- 4.3. Discussion.- 5: Some Functions Which are Not Computable.- 5.1. The halting problem.- 5.2. The blank tape halting problem.- 5.3. The uniform halting problem.- 5.4. The equivalence problem.- 5.5. General remark.- 6: Effectively Enumerable and Decidable Sets.- 6.1. Introduction.- 6.2. Definitions.- 6.3. Effectively enumerable sets and the domain of computable functions.- 6.4. Effectively enumerable sets and the range of total computable functions.- 6.5. A set which is not effectively enumerable.- 6.6. Decidable sets versus effectively enumerable sets.- 6.7. An effectively enumerable set which is not decidable.- 6.8. Some informal comments.- Appendix 1: Bibliographical Notes.- Appendix 2: List of the Most Important Notations.- Appendix 3: List of the Most Important Concepts.

A simple hardware model of a Turing machine: its educational use

Abstract: A simple hardware model of a Turing machine has been built at Brandeis University for educational purposes. The machine was designed and built by the first author and is being used by the second to test its value as an instructional aid in the teaching of programming. This paper presents the factors that influenced the design of the model, provides data concerning its characteristics and operation, and describes its use in teaching elementary programming to undergraduates and high school students.

Relationship between Pushdown Automata and Tape-Bounded Turing Machines.

Abstract: Recall that to define a language we can either: 1. Give a set of rules (i.e. a grammar) to produce all the legal strings (sentences) of the language. 2. Provide a machine (i.e. an algorithm) to recognise all the sentences of the language. There is a close relationship between the two approaches. Commonly we define a language by giving a grammar and then base parsers (or compilers) on the corresponding machine. The machines are automata like Turing machines, but constrained in the same sense as the Chomsky hierarchy. Finite State Control input tape read head a0 a1 a2 ... Auxiliary Memory The input tape is a sequence of tokens. Each time a symbol is processed the read head advances. The auxiliary memory is usually a linear organisation (e.g. a stack). The memory alphabet is usually V t ∪V n. The finite state control can be in any one of a finite number of states. Each action of the machine may change the FSC state, change the auxiliary memory, advance to the next input symbol. The action of the machine depends on the current FSC state, the current input symbol, the current memory symbol(s). The machine starts in some particular start state (q 0), with the read head at the first input symbol (a 0), with the memory empty. A machine accepts an input string as a sentence of the language if it reaches a goal state with the input exhausted.

Turing Machines and the Mind-Body Problem

Abstract: i In his [1961] Hilary Putnam has argued that 'all the issues and puzzles that make up the traditional mind-body problem ... arise in connection with any computing system capable of answering questions about its own structure'. This shows, he believes, that such issues and puzzles are wholly linguistic or logical in character, and 'have thus nothing to do with the unique nature (if it is unique) of human subjective experience'.1 His argument makes use of the notion of a Turing machine. Such a machine consists of (a) a physical tape, scanner and printing device, along with a control and storage mechanism, and (b) a machine table which comprises a set of instructions for the machine to follow, and which directs the machine to print, erase or move the tape as required. There are thus two kinds of descriptions which can be given of a Turing Machine, the one called its logical description which concerns only the table of rules and instructions and does not presuppose any specific physical embodiment (it may be electronic relays, clerks sitting at desks, or whatever), and the other called its structural or physical description which applies to the specific system chosen to carry out the instructions of the machine table. These two types of descriptions, Putnam goes on to note, are closely analogous to the two types of descriptions typically applied to human beings, namely the psychological and the physical. Thus, the physical description of a human being corresponds to the physical (or structural) description of the machine, and the psychological description corresponds to the logical. A few words of explanation are needed for the second half of this analogy. Putnam makes the following three points in this connection:

On the speed of addition and multiplication on one-tape, off-line turing machines

Abstract: It is shown that for any e > 0 and for any sufficiently large l (1 — e)2l2/logbQ is a lower bound for the average computation time required by any one-tape, off-line Turing machine with Q internal states for implementing addition or multiplication of two consecutively written b-adic numbers (b ⩾ 2) with l digits each, where the average is taken over all pairs of numbers with l digits. Conversely: For any e > 0 and for both operations Turing machine constructions are indicated, whose computation times are smaller than (1 + e)2l2/logb Q for any sufficiently large l and for any pair of numbers with l digits each, where the number Q of internal states of the Turing machine has to be chosen large for small e.

Maze recognizing automata (Extended Abstract)

Abstract: In [2], computations of nondeterministic machines are shown to correspond to threadings of certain mazes. We briefly summarize these results and then obtain some new extentions of them. A new device called a maze-recognizing automaton is introduced. This is a type of finite-state device that crawls through mazes. The following statements are proven equivalent. (i) There is a maze-recognizing automaton which recognizes the threadable mazes. (ii) Every nondeterministic L(n) tape-bounded Turing machine is equivalent to some deterministic L(n) tape-bounded Turing machine, provided L(n) ≥ log2n.

The Universal Turing Machine

Abstract: The notion of the universal V-Turing machine is introduced and the thesis of Turing is reformulated

Quantization of molecular motion at the synapse and its consequences as contrasted with the generalized Turing machine.

Abstract: Using the most general models, we show that molecular motion concerned with transmission of the nervous impulse across the synapse is indeterminate (in the sense of Heisenberg). This occurs despite the assumption of a minimal uncertainty product and the taking into account of the nature of the random walk. Consequences of the interposition of an indeterminate process in nervous system function are detailed by investigation of the operation of a “quantized” Turing machine. Such a generalized algorithm has been taken by some investigators to be the starting point for building analogues to brain function. The brain, with a demonstrated indeterminate basis for at least one of its “levels” of operation, has capabilities or modes of operation not able to be mimed by the most general calculating engine.

Comparing complexity classes of formal languages

Abstract: A property of polynomial complete languages is extended in order to better compare various classes of formal languages defined by time- or tape-bounded turing machines.

Results of the use of a recursive function translator

Abstract: A recursive function interpreter was used to good effect in an undergraduate course in elementary automata theory. Although the functions were translated into combinations of Turing machines, the interpreter is not to be construed to be a Turing Table processor. The results in this paper reflect both the author's subjective judgment, and the compilation of statistics from a questionnaire given to the students.The primary effect was not the understanding of Turing machines or recursive functions but rather the enthusiasm for such study evidenced by the computer science students taking the course. The evaluation of the interpreter is an on-going process, extending even to students on the sophomore and junior level.This paper describes both the simulator and the effects of its use, as well as some general principles concerning such devices.