MINSKY'S SMALL UNIVERSAL TURING MACHINE

1991 ◽  
Vol 02 (05) ◽  
pp. 551-562 ◽  
Author(s):  
RAPHAEL M. ROBINSON
Keyword(s):  
Turing Machine ◽  
State Machine ◽  
The Other ◽  
Turing Machines ◽  
Universal Turing Machine

Marvin L. Minsky constructed a 4-symbol 7-state universal Turing machine in 1962. It was first announced in a postscript to [2] and is also described in [3, Sec. 14.8]. This paper contains everything that is needed for an understanding of his machine, including a complete description of its operation. Minsky's machine remains one of the minimal known universal Turing machines. That is, there is no known such machine which decreases one parameter without increasing the other. However, Rogozhin [6], [7] has constructed seven universal machines with the following parameters: [Formula: see text] His 4-symbol 7-state machine is somewhat different from Minsky's, but all of his machines use a construction similar to that used by Minsky. The following corrections should be noted: First machine, for q 6 00Lq 1 read q 6 00Lq 7; second machine, for q 4 11Rq 4 read q 4 11Rq 10; last machine, for q 2 b 2 bLq 2 read [Formula: see text]. A generalized Turing machine with 4 symbols and 7 states, closely related to Minsky's, was constructed and used in [5].

Algorithms and Turing Machines

1989 ◽  
Author(s):  
Roger Penrose ◽  
Martin Gardner
Keyword(s):  
Elementary School ◽  
Turing Machine ◽  
Mathematical Knowledge ◽  
Ninth Century ◽  
The Other ◽  
Turing Machines ◽  
Universal Turing Machine ◽  
Other Hand ◽  
Mathematical Expressions ◽  
Mathematical Textbook

What Precisely is an algorithm, or a Turing machine, or a universal Turing machine? Why should these concepts be so central to the modern view of what could constitute a ‘thinking device’? Are there any absolute limitations to what an algorithm could in principle achieve? In order to address these questions adequately, we shall need to examine the idea of an algorithm and of Turing machines in some detail. In the various discussions which follow, I shall sometimes need to refer to mathematical expressions. I appreciate that some readers may be put off by such things, or perhaps find them intimidating. If you are such a reader, I ask your indulgence, and recommend that you follow the advice I have given in my ‘Note to the reader’ on p. viii! The arguments given here do not require mathematical knowledge beyond that of elementary school, but to follow them in detail, some serious thought would be required. In fact, most of the descriptions are quite explicit, and a good understanding can be obtained by following the details. But much can also be gained even if one simply skims over the arguments in order to obtain merely their flavour. If, on the other hand, you are an expert, I again ask your indulgence. I suspect that it may still be worth your while to look through what I have to say, and there may indeed be a thing or two to catch your interest. The word ‘algorithm’ comes from the name of the ninth century Persian mathematician Abu Ja’far Mohammed ibn Mûsâ alKhowârizm who wrote an influential mathematical textbook, in about 825 AD, entitled ‘Kitab al-jabr wa’l-muqabala’. The way that the name ‘algorithm’ has now come to be spelt, rather than the earlier and more accurate ‘algorism’, seems to have been due to an association with the word ‘arithmetic’. (It is noteworthy, also, that the word ‘algebra’ comes from the Arabic ‘al-jabr’ appearing in the title of his book.) Instances of algorithms were, however, known very much earlier than al-Khowârizm’s book.

scholarly journals An Informal Arithmetical Approach to Computability and Computation, II

1964 ◽  
Vol 7 (2) ◽  
pp. 183-200 ◽  
Author(s):  
Z.A. Melzak
Keyword(s):  
Turing Machine ◽  
Turing Machines ◽  
Symbol Space ◽  
Computing Device ◽  
Computing Power ◽  
Adequate Supply ◽  
Universal Turing Machine ◽  
At Will

In the first part of this paper [l] there was introduced a hypothetical computing device, the Q-machine. It was derived by abstracting from the process of calculating carried out by a man on his fingers, assuming an adequate supply of hands and the ability to grow fingers at will. The Q-machine was shown to be equal in computing power to a universal Turing machine. That is, the Q-machine could compute any number regarded as computable by any theory of computability developed so far. It may be recalled here that Turing machines were obtained by Turing [2] by abstracting from the process of calculating carried out by a man on some concrete 'symbol space' (tape, piece of paper, blackboard) by means of fixed but arbitrary symbols. Hence the contrast between the Q-machine and the Turing machines is that between arithmetical manipulation of counters and logical manipulation of symbols. In particular, one might say, loosely, that in a Turing machine, as in arithmetic, numbers are represented by signs whereas in the Q-machine, as on a counting frame, numbers represent themselves.

Turing Machines

2017 ◽  
Author(s):  
Paul J. Nahin
Keyword(s):  
Turing Machine ◽  
Magnetic Tape ◽  
Input Data ◽  
Finite State Machine ◽  
State Machine ◽  
Memory Storage ◽  
Turing Machines ◽  
Storage Medium ◽  
Linear Sequence ◽  
Finite State

This chapter discusses Turing machines. A Turing machine is the combination of a sequential, finite-state machine plus an external read/write memory storage medium called the tape (think of a ribbon of magnetic tape). The tape is a linear sequence of squares, with each square holding one of several possible symbols. The Turing machine's power to compute comes from its tape, for two reasons. First, Turing was the first to conceive of the idea of a stored program that could be changed by the operation of the machine itself. The program, and its input data, exist together on the tape as sequences of symbols. Second, because of the arbitrarily long length of the tape, a Turing machine has the ability to “remember” what has happened in the arbitrarily distant past.

scholarly journals Reputation in the Cyberworld

2013 ◽  
Vol 19 ◽  
pp. 4-11
Author(s):  
Michael Eldred
Keyword(s):  
Turing Machine ◽  
Turing Machines ◽  
Universal Turing Machine ◽  
Digital Era

The article explores the socio-ontological foundations of the phenomenon of reputation in the context of today’s ever-encroaching cyberworld. The categories of whoness and value are essential for understanding reputation ontologically. The cyberworld itself has only become historically possible through the Cartesian mathematical cast of being and its digital refinement in the Universal Turing Machine. From one perspective, the cyberworld is an endless concatenation of Turing machines. It is, however, also a matrix in which bit-strings circulate that have a decisive impact on who anybody is held to be by others, i.e. on their reputation. The game of striving to be esteemed as who you are thus assumes a new complexion in the digital era.

scholarly journals Teorias da Aleatoriedade

2004 ◽  
Vol 11 (2) ◽  
pp. 75-98
Author(s):  
Carlos A. P. Campani ◽  
Paulo Blauth Menezes
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Random Sequence ◽  
The Other ◽  
Turing Machines ◽  
Intuitive Concept ◽  
Theory Of Information ◽  
Definition Of

This work is a survey about the definition of “random sequence”. We emphasize the definition of Martin-Löf and the definition based on incompressibility (Kolmogorov complexity). Kolmogorov complexity is a profound and sofisticated theory of information and randomness based on Turing machines. These two definitions solve all the problems of the other approaches, satisfying our intuitive concept of randomness, and both are mathematically correct. Furthermore, we show the Schnorr’s approach, that includes a requisite of effectiveness (computability) in his definition. We show the relations between all definitions in a critical way. Keywords: randomness, Kolmogorov complexity, Turing machine, computability, probability.

scholarly journals On Turing Machines Deciding According to the Shortest Computations

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 304
Author(s):  
Florin Manea
Keyword(s):  
Computational Complexity ◽  
Turing Machine ◽  
Polynomial Time ◽  
Point Of View ◽  
The Other ◽  
Input Word ◽  
Turing Machines ◽  
Other Hand

In this paper we propose and analyse from the computational complexity point of view several new variants of nondeterministic Turing machines. In the first such variant, a machine accepts a given input word if and only if one of its shortest possible computations on that word is accepting; on the other hand, the machine rejects the input word when all the shortest computations performed by the machine on that word are rejecting. We are able to show that the class of languages decided in polynomial time by such machines is PNP[log]. When we consider machines that decide a word according to the decision taken by the lexicographically first shortest computation, we obtain a new characterization of PNP. A series of other ways of deciding a language with respect to the shortest computations of a Turing machine are also discussed.

scholarly journals Undecidability in quantum thermalization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Naoto Shiraishi ◽  
Keiji Matsumoto
Keyword(s):  
Turing Machine ◽  
Thermal Equilibrium ◽  
General Theorem ◽  
Nearest Neighbour ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Universal Turing Machine ◽  
Many Body Systems ◽  
Neighbour Interaction

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

Sign in / Sign up

Export Citation Format

Share Document