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1. PlanetMath: Turing Machine
AMS MSC, 03D10 (Mathematical logic and foundations Computability and recursion theory Turing machines and related notions)
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Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Turing machine (Topic) A Turing machine is an imaginary computing machine invented by Alan Turing to describe what it means to compute something. infinite number of cells stretching in both directions, with the tape head always located over exactly one of these cells. Each cell has one of a finite number of symbols written on it. The machine has a finite set of states , and with every move the machine can change states, change the symbol written on the current cell, and move one space left or right . The machine has a program which specifies each move based on the current state and the symbol under the current cell. The machine stops when it reaches a combination of state and symbol for which no move is defined. One state is the start state, which the machine is in at the beginning of a computation.

2. Compilability Classes
Hence, advicetaking Turing machines are closely related to non-uniform families of . Therefore, it is possible to define the notions of hardness and
Next: Reductions among KR Formalisms Up: Space Efficiency of Propositional Previous: Notations and Assumptions

Compilability Classes
We assume the reader is familiar with basic complexity classes, such as P, NP and (uniform) classes of the polynomial hierarchy [ ]. Here we just briefly introduce non-uniform classes [ ]. In the sequel, C, , etc. denote arbitrary classes of the polynomial hierarchy. We assume that the input instances of problems are strings built over an alphabet . We denote with the empty string and assume that the alphabet contains a special symbol to denote blanks. The length of a string is denoted by Definition 1 An advice is a function that takes an integer and returns a string. Advices are important in complexity theory because definitions and results are often based on special Turing machines that can determine the result of an oracle ``for free'', that is, in constant time. Definition 2 An advice-taking Turing machine is a Turing machine enhanced with the possibility to determine in constant time, where

3. 'A Madman Dreams Of Turing Machines,' By Janna Levin - The New York Times Book R
Both incompleteness and undecidability are technical notions. related Articles. First Chapter A Madman Dreams of Turing machines (September 3, 2006)
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Sunday Book Review
Obsessive-Genius Disorder

By JIM HOLT Published: September 3, 2006 It is a curious thing that the two greatest logical geniuses of the last century both killed themselves. Alan Turing died after taking a bite of an apple that was laced with cyanide. Kurt Gödel died of a paradox: he thought people were plotting to kill him by poisoning his food, so he refused to eat and consequently starved to death. Did these two men self-destruct because of their logical prowess, or in spite of it? Did they reach some tragic deduction about life that the rest of us are too dim to see? And was it just a coincidence that both were inordinately fond of the Walt Disney movie “Snow White”? Skip to next paragraph Janna Levin
By Janna Levin.

4. MathNet-Mathematical Subject Classification
03D10, Turing machines and related notions See also 68Q05. 03D15, Complexity of computation See also 68Q15. 03D20, Recursive functions and relations,

5. Sachgebiete Der AMS-Klassifikation: 00-09
03D05 Automata and formal grammars in connection with logical questions, See also {68Qxx} 03D10 Turing machines and related notions, See also {68Q05} 03D15
Sachgebiete der AMS-Klassifikation: 00-09
nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen
01-XX 03-XX 04-XX 05-XX 06-XX 08-XX
nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen

6. Turing Machines
Intuitive notion of algorithms, equals, Both Turing machine and related to Gödel s incompleteness problem is the Halting Problem for Turing machines.

7. Stephen Wolfram: A New Kind Of Science | Online
Notes for The Notion of Computation Universality in Turing machines and Other Systems related LINKS. Pages related to this note
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8. Ed Blakey - Publications (with Abstracts)
By far the most studied model of computation is the Turing machine model. in which to generalize the closely related notions of resource and complexity.
A Model-Independent Theory of Computational Complexity; Price: from Patience to Precision (and Beyond) (2007) [PDF: 485kb, 29pp.] Blakey, E. This dissertation was written during the author’s studies for the degree of Doctor of Philosophy in Computer Science (Oxford, 2006 — present), in particular for consideration during the transfer from PRS to DPhil status. It describes the author’s DPhil thesis project, including work already completed and future work. Hide abstract. Abstract. The field of computational complexity strives to categorize problems according to the cost of their solution. Whereas reasoning directly about the computational complexity of a problem seems inherently difficult, it is relatively easy to ascertain the complexity of specific methods (algorithms, analogue computers, Turing machines, etc.) that solve the problem Consequently, our sole understanding of a problem’s complexity is, in the majority of cases, gleaned via our having determined the complexity of solution methods for the problem; specifically, we have that the problem’s complexity is bounded from above by the complexity of the most efficient known solution method. In order to improve such bounds, it is desired to consider as large a set as is possible of solution methods for a problem. Each set so considered in practice, however, is likely to be of solution methods taken from a single model of computation (often that of the Turing machine). This is a necessary evil of our inability meaningfully to compare the complexity of instances of different computation models.

9. The Invalid Results Of Paul Davies
The Refutation of Paul Davies Turing Machine Hypothesis mathematics and Turing computation, which is related to computer programming notions.
The Refutation of Paul Davies' Turing Machine Hypothesis and All That This Hypothesis Implies by Robert A. Herrmann Ph. D. 28 March 2000, revised 23 DEC 2006. Introduction Paul Davies is the author of the [I] "The Mind of God: The Scientific Basis for a Rational World." Davies is a "Templeton Prize" winner. This prize is supposed to be given for a significant "Progress in Religion." However, Davies' work is a "giant leap backwards" rather than an advance. Indeed, this refutation of Davies' hypothesis indicates the gross errors that are continually being made in the awarding of this prize. personal evidence that an actual supernatural deity exists - for if he had such evidence, he would know immediately that such a hypothesis is false. What Davies does, as with other individuals who only approach God through the essence of philosophy, is to present linguistic descriptions for claimed Divine attributes, a "paper and pencil" description. Although, I also present such descriptions, I also freely admit that without an "indescribable" personal relation with God, as say expressed by 1 John 2:27, one has no means to verify that any such description for a Divine attribute is correct. Without such verification, one can expect error to be prevalent. Formal Verses Informal Davies' basic misunderstanding is relative to "formal" logical deduction as done by a mathematical logician, if at all, and the art of "informal" mathematical deduction which are the usual thinking processes one associates with a "mathematician." It appears as if Davies has never created new mathematics since he would know that it is more of an art form than a science. In [1], I discuss these facts and that there appears to be no fully describable set of rules or processes that I "informally" apply to know in advance that a particular mathematical "theorem" can be successfully established. I simply "know intuitively," in most cases, when a particular statement is probably provable as a mathematical "fact," and whether or not I can successfully establish the result. This I have done a few thousand times. The concepts that Davies uses to come to his conclusion that the "laws of nature" are "Turning-computable" (to be further explained) are relative to

10. Can Turing Machines Capture Everything We Can Compute?
The notion of Turingcomputable function is no clearer (not more non-constructive, .. Mathematica and related systems I. Translated by Elliott Mendelson.
Index Main essay Can Turing machines capture everything we can compute? Bhupinder Singh Anand A .pdf file of this essay before the current update is available at If we define classical foundational concepts constructively, and introduce non-algorithmic effective methods into classical mathematics, then we can bridge the chasm between truth and provability, and define computational methods that are not Turing computable. Contents 1. Introduction 2. Non-constructivity in classical theory 3. Constructive definitions 4. Effective solvability of the Halting problem ... 5. Is there a case for an Arithmetical Provability Thesis? Introduction In a short opinion paper, “Computation Beyond Turing Machines” Peter Wegner and Dina Goldin advance the thesis that: “A paradigm shift is necessary in our notion of computational problem solving, so it can provide a complete model for the services of today's computing systems and software agents.” We note that Wegner and Goldin’s arguments, in support of their thesis, seem to reflect an extraordinarily eclectic view of mathematics, combining both acceptance of, and frustration at, the standard interpretations and dogmas of classical mathematical theory: i ...Turing machines are inappropriate as a universal foundation for computational problem solving, and ... computer science is a fundamentally non-mathematical discipline.

11. Research Laboratory For Logic And Computation, GC CUNY
The principle that Turing machines are formal versions of algorithms and that The first part introduces basic categorical notions, such as categories,
Research Laboratory for Logic and Computation

FALL 2003
Tuesday, 2pm - 4pm, room 4421
December 2 talk
Yegor Bryukhov. Type Theory for a practicing mathematician.

Abstract: continue
December 2 talk
Yegor Bryukhov. Type Theory for a practicing mathematician.

Abstract: In this talk we will follow R.Constable's paper "Naive Computational Type Theory" which in turn "follows" the book of Paul Halmos "Naive Set Theory". This paper gives some new perspectives in Type Theory, including a new meaning of openness of Type Theory. We'll start from the fundamentals: "what is type", "propositions as types", and then go to type-theoretic analogues of a set, subset, pair, union, intersection, function, relation, etc. We will consider two meanings of this popular statement "Type Theory is open-ended", one is old and the other one is new. They are related but the new one is much deeper. It shows that the Type Theory is VERY different from the Set Theory. Time permitting we'll discuss dependent intersection (a relatively new result by Alexei Kopylov) and records. November 25 talk Walter Dean (GC and Rutgers). From Church's Thesis to Extended Church's Thesis.

12. Computational Complexity: A Conceptual Perspective [Goldreich]
Drafts (and additional related texts) are available from HERE. . 1.2.2 Computational Tasks; 1.2.3 Uniform Models (Algorithms) Turing machines,
Computational Complexity: A Conceptual Perspective
Oded Goldreich
To be published in April 2008.
Currently, being copyedited by the publisher. Publisher: Cambridge University Press
Below is the book's tentative preface and Organization . Drafts (and additional related texts) are available from HERE Last updated: Nov. 2007.
Preface (tentative, Jan. 2007)
The strive for efficiency is ancient and universal, as time and other resources are always in shortage. Thus, the question of which tasks can be performed efficiently is central to the human experience. A key step towards the systematic study of the aforementioned question is a rigorous definition of the notion of a task and of procedures for solving tasks. These definitions were provided by computability theory, which emerged in the 1930's. This theory focuses on computational tasks, and considers automated procedures (i.e., computing devices and algorithms) that may solve such tasks. In focusing attention on computational tasks and algorithms, computability theory has set the stage for the study of the computational resources (like time) that are required by such algorithms. When this study focuses on the resources that are necessary for any algorithm that solves a particular task (or a task of a particular type), the study becomes part of the theory of Computational Complexity (also known as Complexity Theory). Complexity Theory is a central field of the theoretical foundations of Computer Science. It is concerned with the study of the

13. : A Madman Dreams Of Turing Machines
I just finished reading A Madman Dreams of Turing machines by Janna Levin. washing clean his confusion, his muddled notions, and his breath.
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14. Panmere - Rosennean Complexity And Other Interests » Turing Machines
In May 2007, the Wolfram 2,3 Turing Machine Research Prize was established . rigorous definitions of the informal notions of “effective calculability”.

15. Turing Machines (Stanford Encyclopedia Of Philosophy)
An apparently more radical reformulation of the notion of Turing machine allows .. related Entries. Church, Alonzo ChurchTuring Thesis computability
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Turing Machines
First published Thu Sep 14, 1995; substantive revision Fri Nov 5, 2004 Turing machines, first described by Alan Turing in (Turing 1937), are simple abstract computational devices intended to help investigate the extent and limitations of what can be computed. Turing, writing before the invention of the modern digital computer, was interested in the question of what it means to be computable. Intuitively a task is computable if one can specify a sequence of instructions which when followed will result in the completion of the task. Such a set of instructions is called an effective procedure , or algorithm , for the task. This intuition must be made precise by defining the capabilities of the device that is to carry out the instructions. Devices with different capabilities may be able to complete different instruction sets, and therefore may result in different classes of computable tasks (see the entry on computability and complexity Turing proposed a class of devices that came to be known as Turing machines. These devices lead to a formal notion of computation that we will call

16. Continuous Turing Machines
(5) Can there exist a Universal Continuous Turing Machine? This is related to the notion that a CTM computes a result iff it generates a function over
Continuous Turing Machines
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17. Universal Turing Machine - Wikipedia, The Free Encyclopedia
Davis makes a case that Turing s ACE computer anticipated the notions of . Other smaller universal Turing machines have since been found.
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Universal Turing machine
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This article is a supplement to the article Turing machine
Alan Turing 's "universal computing machine" (alternately "universal machine", "machine U", "U") is the name given by him (1936-1937) to his model of an all-purpose "a-machine" (computing machine) that could "run" any arbitrary (but well-formed) sequence of instructions called "quintuples". This model is considered by some (e.g., Davis (2000)) to be the origin of the "stored program computer" used by John von Neumann (1946) for his "Electronic Computing Instrument" that now bears von Neumann's name: the von Neumann architecture This machine as a model of computation is now called the "Universal Turing machine".
edit Introduction
Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet. In that sense it behaves like a computer with a fixed program. However, we can encode the action table of any Turing machine in a string. Thus we can construct a Turing machine that expects on its tape a string describing an action table followed by a string describing the input tape, and computes the tape that the encoded Turing machine would have computed. Turing described such a construction in some detail in his 1936 paper.

18. Brains: Classical Computation And Hypercomputation At The 2006 Eastern APA
To maintain contact with the epistemological notion of computation, Following me, Oron Shagrir discussed accelerating Turing machines, i.e.,
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On Mind and Related Matter
Monday, January 02, 2006
Classical Computation and Hypercomputation at the 2006 Eastern APA
On Wednesday, December 28, 2006, at the Eastern APA in NYC, we held our session on classical computation and hypercomputation. (For some background, see previous posts.) From my point of view, it went roughly as follows.
In my presentation, I argued that in discussions of the Physical Church-Turing Thesis (Physical CT), we need to distinguish between what I called a bold and a modest version. According to the bold version, which is popular in physics and philosophy of physics circles, everything that can be “physically done” is computable by Turing machines. I argued that this thesis (including its more precise formulations) is both too strong (i.e., it is falsified by genuine random processes and by a liberal use of real numbers) and not related to the original notion of computation that led to work on computability theory and CT in the first place. The original notion was the epistemological notion of what problems of a certain kind can be solved in a reliable way.
To maintain contact with the epistemological notion of computation, I argued that we need to formulate a modest version of Physical CT, according to which everything that can be “physically computed” can be computed by Turing machines. By “physically computed”, I mean a process that can be used by a human observer to solve problems defined over strings of digits. In other words, modest Physical CT is true if and only if it is impossible to build a genuine hypercomputer, i.e., a device that can be used by a human observer to compute arbitrary values of a function that cannot be computed by Turing machines. Since it is presently unknown whether genuine hypercomputers can be built (though it seems unlikely that they can), the truth value of modest Physical CT remains to be determined (though the thesis is quite plausible).

19. A Modest Expansion Of The Scope Of The Church-Turing Thesis « Apperceptual
Turing developed the idea of Turing machines while working on Hilbert’s to me that this assumption is very closely related to the notion of “reality”.
Apperception: the process whereby perceived qualities of an object are related to past experience. Attributes and Relations: Redder than Red Democracy 2.0
A Modest Expansion of the Scope of the Church-Turing Thesis
The Church-Turing thesis is that every function that would naturally be regarded as computable can be computed by a Turing machine . This thesis cannot be proven rigorously, because it relates an informal notion (naturally be regarded as computable) to a formal notion (Turing machine). However, there is very strong evidence for the Church-Turing thesis, since every proposed formal treatment of computation ( Church Lambda calculus Markov algorithms , the Game of Life , and even quantum computers ) has been shown to be equivalent to Turing machines. real reality , in this context? That question is what I want to explore here. Turing developed the idea of Turing machines while working on Hilbert Entscheidungsproblem (decision problem). Informally, Hilbert wondered whether it was possible to automate mathematics. Could there be an automatic procedure that could take any mathematical theorem as input and produce as output a proof that the theorem is true or that it is false? Turning showed that there could be no such procedure. More precisely, he showed that there are mathematical questions that Turing machines cannot answer. A Turing machine is essentially a very abstract model of a mathematician Several people have argued that the physical universe may be a Turing machine ( Wolfram Poundstone Deutsch ). Others have argued that the human mind may be a Turing machine (

20. The Myth Of The Turing Machine
The Myth of the Turing Machine The Failings of Functionalism and related Theses .. Turing machine equivalence, I have argued, is not a robust notion in Myth.central.jetai.1ce.nofields
The Myth of the Turing Machine
The Failings of Functionalism and Related Theses February, 2002 Submitted to JETAI Chris Eliasmith Dept. of Philosophy University of Waterloo
The properties of Turing’s famous ‘universal machine’ has long sustained functionalist intuitions about the nature of cognition. Here, I show that there is a logical problem with standard functionalist arguments for multiple realizability. These arguments rely essentially on Turing’s powerful insights regarding computation. In addressing a possible reply to this criticism, I further argue that functionalism is not a useful approach for understanding what it is to have a mind. In particular, I show that the difficulties involved in distinguishing implementation from function make multiple realizability claims untestable and uninformative. As a result, I conclude that the role of Turing machines in philosophy of mind needs to be reconsidered.
The Myth of the Turing Machine
The Failings of Functionalism and Related Theses
1. Introduction

21. Section 5.7 From "Hilbert's Tenth Problem" By Yuri MATIYASEVICH
For each such modification of the notion of Turing machine, one can introduce and pose the question of how that concept is related to Diophantine sets.
Section 5.7 from the book
written by Yuri MATIYASEVICH 5.7 Church's Thesis In the previous sections of this chapter we have obtained two, no doubt remarkable, results. Namely, we have established that the class of Diophantine sets is identical to the class of Turing semidecidable sets and Hilbert's Tenth Problem is Turing undecidable. However, these two rather technical results raise a number of new questions. While the definition of Diophantine set is quite natural, that of Turing semidecidable set is burdened by numerous technical details, many of which seem arbitrary. For example, we could use binary rather than unary notation to represent numbers. The tape could be infinite in both directions instead of in only one direction. Instead of a single head, there might be several, each executing its own set of instructions while sharing information about their respective scanned cells. Moreover, there might be several tapes. In fact, the memory need not even be linear; it could, for example, take the form of a plane divided into square cells. For each such modification of the notion of Turing machine, one can introduce a corresponding concept of semidecidability and pose the question of how that concept is related to Diophantine sets. The reformulation of Hilbert's Tenth Problem used in Section 5.6 could be criticized as well, since it is based on a very special method for coding Diophantine equations. It would be more natural to write on the tape the number of unknowns, the degree, and the coefficients of an equation in unary or some positional notation. Hilbert did not impose any restrictions on the desired method for solving the Tenth Problem. Thus, if for some appropriate notation for polynomials, someone had succeeded in constructing a decision machine, that would certainly have provided a positive solution of Hilbert's Tenth Problem. So to what extent can the Turing undecidability established in Section 5.6 be considered as constituting a negative solution?

22. Peter Suber, "Turing Machines I"
Turing machines are one of the earliest and most intuitive ways to make precise the naive notion of effective computability. All Turingcomputable functions
Turing Machines I Peter Suber Philosophy Department Earlham College What is a Turing Machine? A Turing machine is a simple but powerful computer. It is useful in thinking about the nature and limits of computability because its method of computation is about as simple as can be imagined. Important theoretical results about what can be computed that are expressed in the terms of Turing machines, therefore, are clearer to intuition than the same results expressed in other terms. Turing machines were conceived by Alan Turing (1912-1954) in his important paper, "On Computable Numbers, with an Application to the Entscheidungsproblem," Proceedings of the London Mathematical Society , 2d Series, 42 (1936) 230-65. Turing machines are one of the earliest and most intuitive ways to make precise the naive notion of effective computability. All Turing-computable functions are effectively computable; the important and indemonstrable converse (that all intuitively computable functions are Turing computable) is asserted by Church's Thesis. My exposition is based on those of George S. Boolos and Richard Jeffrey

23. JSTOR Computation Finite And Infinite Machines.
The second part consisting of Chapters 5 to 11 covers Turing machines and various computability concepts. After introducing the notion of an effective pro<99:CFAIM>2.0.CO;2-8

24. Wegner OOPSLA'95, Interection Vs Turing Machines (was: So-called Turing-Equivale
Interaction machines, defined by extending Turing machines with input to the hypothesis of Church and Turing that the intuitive notion of computation
Wegner OOPSLA'95, Interection vs Turing Machines (was: so-called Turing-Equivalence)
Massimo Dentico
Sun, 29 Aug 1999 12:36:39 +0200 equivalent power to a UTM" repeated over and over again as a bad excuse for bad programming language design. I question the meaningfulness of your term "Turing-equivalence". What definition of it do you use, if any? meaningful *in presence of interactions with the external world (including users)*? I think that on this subject the paper of Peter Wegner is illuminating. Follows a quotation of the abstract and of the paragraph 2.3 because the original text is quite long (67 pages) and in this way you could have by yourselves an idea of the content and you could decide if you want to read it entirely. However, I hope to don't annoy anyone with this long citation. Sorry for my horrible English, any correction is wellcome. ================================================================== OOPSLA Tutorial Notes, October 1995 Tutorial Notes: Models and Paradigms of Interaction Peter Wegner, Brown University, September 1995 (

25. Project MUSE
It will be perhaps apparent that the way I am describing the operations of the Turing machine is related to the Derridean use of the term undecidability.
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Undecidability: The History and Time of the Universal Turing Machine
Configurations - Volume 4, Number 3, Fall 1996, pp. 359-379
The Johns Hopkins University Press
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26. Alan M. Turing, Or Alan Mathison Turing (English Mathematician) -- Britannica O
invention of Turing machine (in automata theory Nature and origin of modern all argued for this concept (and certain equivalent notions), thereby.
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Alan M. Turing, or Alan Mathison Turing (English mathematician)
A selection of articles discussing this topic.
Main article: Alan M. Turing
British mathematician and logician, who made major contributions to mathematics, cryptanalysis, logic, philosophy, and biology and to the new areas later named computer science, cognitive science, artificial intelligence, and artificial life.
development of Turing test
invention of Turing machine contribution to:
  • algorithms
    ...procedure that never ends (a condition known as the halting problem). In an unsuccessful effort to ascertain at least which propositions are unsolvable, the English mathematician and logician Alan Turing rigorously defined the loosely understood concept of an algorithm. Although Turing ended up proving that there must exist undecidable propositions, his description of the essential...

27. Theory-bk-preface.html
It starts by considering the notion of strings, and the role that strings have Abstract computing machines, called Turing transducers, are introduced as
next tail up
Computations are designed to solve problems. Programs are descriptions of computations written for execution on computers. The field of computer science is concerned with the development of methodologies for designing programs, and with the development of computers for executing programs. It is therefore of central importance for those involved in the field that the characteristics of programs, computers, problems, and computation be fully understood. Moreover, to clearly and accurately communicate intuitive thoughts about these subjects, a precise and well-defined terminology is required. This book explores some of the more important terminologies and questions concerning programs, computers, problems, and computation. The exploration reduces in many cases to a study of mathematical theories, such as those of automata and formal languages; theories that are interesting also in their own right. These theories provide abstract models that are easier to explore, because their formalisms avoid irrelevant details. Organized into seven chapters, the material in this book gradually increases in complexity. In many cases, new topics are treated as refinements of old ones, and their study is motivated through their association to programs.

28. [FOM] Simple Turing Machines, Universality, Encodings, Etc.
Such a machine cannot hope to be emulated by a Turing machine with blank tape in allowing the notion of an Arecursive function and the development of a
[FOM] Simple Turing machines, Universality, Encodings, etc.
Vaughan Pratt pratt at
Fri Nov 2 00:59:24 EDT 2007 The overall point I'm making is that universality is a hard concept to pin down, and I suspect that there may never be a definition that fits everyone's intuition as to whether a system should be considered universal or not. Likewise with defining what information should be included as part of a system, or group of systems, and what is part of its initial condition. More information about the FOM mailing list

29. Philosophical Preliminaries :: Cubicle Muses
This discovery inspired the ChurchTuring thesis every computer or notion of computation can be simulated by a Turing machine. Many other notions of
Cubicle Muses
Philosophical Preliminaries
William Taysom on Sun, 09 May 2004
In which William rambles about the philosophy of computers, computability, magic, and the internet. What makes a good interface? Why would the subject of computer interfaces be worthy of philosophical investigation? After all, one hardly thinks of the interface to a car peddles and steering wheel as a philosophical subject. What makes computers any different? A car lets you go from one place to another. It can hold passengers, store goods. You have music and air-conditioning to make the trip pleasant, seat belts and airbags to make the trip safe. A car is an isolated chamber (good for private conversations), an object of beauty, discussion, and something difficult to live without. Likewise, a computer lets you get from one place to another. Not so much physically (though they might help ) as intellectually. It can hold information, store data. You have music, and it is something difficult to live without. Computers are a lot like cars. With a car, you cruise the highway. With a computer, you cruise the Information Highway. Highways are made of asphalt or concrete. But what is the Information Highway made of? Hmm. I'm not entirely sure. However, I do know what it's good for: information. Let's go ask

30. Algorithmic Information Theory - Scholarpedia
A closely related notion is the probability that a universal computer outputs some A prefix Turing machine has a separate input tape which it reads from
Algorithmic information theory
From Scholarpedia
Marcus Hutter (2007), Scholarpedia, 2(3):2519. revision #13277 [ cite this article
Curator: Marcus Hutter, Australian National University
This article is a brief guide to the field of algorithmic information theory (AIT), its underlying philosophy, and the most important concepts. AIT arises by mixing information theory and computation theory to obtain an objective and absolute notion of information in an individual object, and in so doing gives rise to an objective and robust notion of randomness of individual objects. This is in contrast to classical information theory that is based on random variables and communication, and has no bearing on information and randomness of individual objects. After a brief overview, the major subfields, applications, history, and a map of the field are presented.
Algorithmic information theory (AIT) is the information theory of individual objects, using

31. Reductive And Playful Taxonomy: Would Duchamp Desire A Turing Machine
In order to grasp this notion fully we will need to consider computers . The concept of the Universal Turing Machine4 (essentially a computer with the
Reductive and Playful Taxonomy: Would Duchamp Desire a Turing Machine?
, I hope to look at the dichotomies in their thinking from a kind of taxonomical perspective. Many of the texts which concern themselves with digital arts, multimedia and related disciplines set out in the first instance to ascertain what is unique about the artefacts associated with this territory. It is as if the authors hope to produce an essential set of defining characteristics that will serve to place their subject matter and to simultaneously capture it. This approach would seem to fixate upon the aspects of language that are primarily concerned with naming and labelling and in this sense they neglect some wider linguistic possibilities. In light of this, the attempt to define digital arts becomes an interesting vehicle for a consideration of Wittgenstein's comments concerning family resemblance Until Wittgenstein, much of the history of philosophy had been preoccupied with the process of definition. Many philosophical questions seemed to be of the form 'what is x ?' The x in question is a variable that may have stood for concepts as seemingly diverse as self, goodness or, in the case of aesthetics, art. Much philosophical debate centred upon expounding definitions of a concept, showing the inadequacy of existing definitions or attempting to formulate more inclusive/exclusive alternatives. We see echoes of this strategy in the more contemporary debates attempting to ascertain the nature of intelligence, mental illness and perhaps more pertinent to this context, both information and digital arts.

32. THEORY OF COMPUTATION Syllabus For The Course CS 5315, Spring 2007
Universal Turing machine. Can anyone really beat Church? Define classes P, NP, the notions of polynomial time reduction, NPhardness,
THEORY OF COMPUTATION Syllabus for the course CS 5315 , Spring 2007 Instructor: Vladik Kreinovich, office COMP 215, email Class time: TR 3:00-4:20 pm, COMP 308 Office hours: TR 8:30-9:00 am, 12-12:30, 2:30-3, 4:30-5 pm, or by appointment Prerequisite: CS 3350 (Automata) MAIN OBJECTIVES:
  • to teach the foundations of computing, and
  • to enable students to prove (not only to program).
  • Turing's snakelike machine: not very fancy, but it can compute anything (you just wait and wait and wait, ...). Finally: something purely theoretical (and not real machines): recursive functions . Church's bold statement: if anyone can compute anything on any machine, I can compute it on a Turing snake! Universal Turing machine. Can anyone really beat Church? We'll discuss the attempts (Gandi, Kreisel, etc) if time allows.
  • You are accustomed to the fact that everything is computable, and if your program does not work, that means a bad grade. Finally! Only in this course! Computational problems that cannot be solved! (and so you get a bad grade, if your program solves them - just kidding).
  • If a program requires a billion years to finish its computations, only a crazy theoretician can call it an algorithm. So, to sound more reasonable, we will talk about computational complexity, realistic (polynomial-time) computations, P and NP, NC, limitations on space and on the number of processors, etc. "P=NP?" as a challenge to mankind. Will science ever stop? Again, we will find here lots of undecidability results. And maybe, as a project, you will be able to prove that some problem that you were planning to solve is undecidable.
  • 33. The Smallest Computer In The World
    In principle you could build a physical Turing machine, of computable function, and all turned out to be completely equivalent to Turing s notion.
    Devlin's Angle
    November 2007
    The Smallest Computer in the World
    Long before there were iPod Nanos and laptops, in fact, long before before Bill Gates was even born, a British mathematician proved that it was possible to build computing machines that could be programmed to carry out any calculation that could ever arise. The mathematician was Alan Turing, and the theoretical device he invented in the 1930s is nowadays called a Turing machine. (There is not just one Turing machine; rather they are an entire class of hypothetical computing devices. The Wikipedia entry at provides a good introduction for anyone not familiar with the concept.) The second world war gave Turing an opportunity to put his theory into practice, and he spent the war years at the British secret code breaking facility in Bletchley Park, building the real computer that was used to break the German Enigma code, a major turning point in the war. The computer built at Bletchley Park, like all computers since, is much more complicated than Turing's theoretical machine. In principle you could build a physical Turing machine, and in fact several people have done so, but they are not useful; it could take tens or hundreds of years to program them to carry out useful calculations, and for some problems even longer to find the answer. They may be simple, but they are not practical. Their importance is that they help us understand what computation is and how it works. (For instance, the security of your credit card number when you order something online depends crucially on mathematics involving Turing machines.)

    34. Research
    Emil Post, Alonzo Church as well as Alan Turing showed there exist certain To this end I will study issues related to theorystructure (the notion of a
    Home Members Events Writings Research Studenten
    Adaptive logics are a topic of intensive research in the Centre. A provisional home page (with bibliography and bib-file) was created for them.
    Below is a list of recent projects , inversely ordered by termination date.
    Contextual and formal-logical approach to scientific problem solvingprocesses
    Promotors: Diderik Batens, Erik Weber, Joke Meheus and Kristof De Clercq
    Funding Agency: Special Research Fund of Ghent University
    Interdisciplinarity, causation and explanatory pluralism in the biomedical sciences.
    Promotors: Erik Weber and Jeroen Van Bouwel
    Funding agency: Fund for Scientific Research-Flanders
    Researcher: Leen De Vreese
    Explanations of diseases can result from a biological, psychological or social science approach. This project examines how these approaches relate to each other in the biomedical sciences. Should they be integrated? Are they, on the other hand, incompatible? Etc. We will address these questions by comparing the causal concepts, methodology and forms of explanation applied in these different approaches. A model-based approach to problems in the philosophy of social science.

    35. IngentaConnect On Kolmogorov Complexity In The Real Turing Machine Setting
    We extend the notion of Kolmogorov complexity to the computational model of real Turing machines. Following the standard lines of definition we show an
    var tcdacmd="dt";

    36. CS 3172
    This module covers two related areas (1) the notion of complexity classes for We will introduce the notion of a Turing Machine as our underlying model.
    CS 3172: Advanced Algorithms, 2nd Semester 2002
    This course is officially announced here Dates (confirmed)
    • The course starts on Monday, January 28th and ends on Friday, June 7th
      Each lecture is scheduled from 1011 am. We start at 10 am.
    • Part I (Complexity Classes) is given by part II (Special Algorithms) by David Rydeheard
    • Each part consists of 8 lectures, plus 2 lectures reserved for example classes.
      For each part, one week (2 lectures) is given as additional time for the homeworks.
      This means that there are no lectures on Feb. 25th/ May 3rd and April 29th/ May 3rd.
    • Example classes for Part I are on March 4/8 and for Part II on May 4/8.
    • The exam (closed notes, closed book) takes place in Math 1.10 from 9.45 to 11.45 on 22nd May.
      Check out new stuff ....
    This module provides an advanced course in algorithms, assuming the student already knows simple algorithms for some of the common computational tasks, and can reason about correctness of algorithms and also derive their complexity. This module covers two related areas: (1) the notion of complexity classes for algorithms and algorithmic tasks, including the famous P/NP problem, and (2) a range of advanced algorithms which illustrate these complexity classes and also the structure, origins and correctness of algorithms.
    The module will consist of 18 lectures and the remaining 6 lectures'-worth of time will be devoted to exercises that develop and extend the practical side of the subject. Both the lectured material and the practical exercises are examinable.

    37. Machines/Automata
    A Turing Machine (TM) can recognize this set. The Turing Machine can write This thesis is the proposal that the informal notion of a function that can
    Machines that follow Instructions - But So What? We have arrived at a rather minimal agent who can understand and carry out very simple instructions. (If you want to read Turing's development of these ideas, click here But can this agent be instructed to solve any really interesting problems? Only a very small subset of problems? All problems? ... Well, we can define several interestingly different agents by only slightly altering the way they are constructed. The simplest machine or automaton that we think of as a computational device is called a finite state machine (FSM). This machine can read from its input tape, but most significantly, it can not write on the tape. So what, you might say. Well, think of yourself doing some long division problem, or shopping for a big party, or doing the cryptarithmetic problem that we did earlier. Now, what if you couldn't write anything down to aid yourself in carrying out the computation required? It turns out that if you can't write to a memory, then certain types of functions can not be computed using this limited instruction set/agent. Exactly what these limits are is a bit more subtle than you probably can imagine. So don't trust your intuitions. But, do explore the idea of an FSM a bit further by perusing some of the further information provided below.

    38. Theoretical Computer Science And You
    The information theoretic notion of Entropy is a generalization of the Omega is equal to the probability that a randomly chosen Turing Machine will halt
    The information theoretic notion of Entropy is a generalization of the physical notion. There are many ways to describe Entropy. It is a measure of the randomness of a random variable. It is also a measure of the amount of information a random variable or stochastic process contains. It is also a lower bound on the amount a message can be compressed. And finally it is the average number of yes/no questions that need to be asked about an random entity to determine its value.
    Entropy and related Information Theoretic metrics are used extensively in Artificial Intelligence applications that do stochastic modeling such as speech recognition, pattern recognition, medical diagnotics and financial modeling. They are useful not only because they measure randomness but also because they can be used to determine questions with the highest information content and show the best way to decrease Entropy through successive querrys.
    Another example of the use of Entropy is in compression. We know the Entropy of English is around 1 bit per letter which means that when we communicate, the redundancy factor is around 26. This suggests that we should be able to compress English to a high degree which in fact we can do. If we ever reach compression rates of 26, we should stop, pat ourselves on the back, and move on to other problems because we can do no better.
    Let's look briefly at the formula for Entropy through an example. The problem we consider is how many times a deck of cards must be shuffled to ensure complete randomness. First we present the equation for Entropy: it is the sum over all values of a rv of the probability of that value times the log of that prob(i.e. p(x)logp(x)). This equation can be derived from first principles of the properties of information.

    39. Theory Of Computing 2007/2008 (FUB MSc In Computer Science) - Lectures
    a nonR.E. language the diagonalization languages; a non-recursive language the universal language; Universal Turing machines; the notion of reduction
    Free University of Bolzano/Bozen
    Faculty of Computer Science
    Master of Science in Computer Science
    Theory of Computing
    Lectures A.Y. 2007/2008
    Prof. Diego Calvanese
    Teaching material
    Introduction to Automata Theory, Languages, and Computation (3rd edition). J.E. Hopcroft, R. Motwani, J.D. Ullman. Addison Wesley, 2007. Lecture Notes for Theory of Computing . Diego Calvanese. 2007. Made available as scanned pages in pdf. Exercises on Theory of Computing . Will be made available as scanned pages in pdf.
    Week Topics Monday
    (lecture) Wednesday
    (lecture) Wednesday
    (exercise) Extra
    Oct. 8
    Course introduction Course introduction,
    basic notions about sets
    Lec 1,2
    Basic notions about relations, functions, languages
    Lec 3,4
    Formal proofs Exer 1 Oct. 15 Undecidability Undecidable problems Lec 5,6 The Turing Machine Lec 7,8 Turing Machines Exer 2 Oct. 22 (Extended) Turing Machines Programming techniques for TMs Lec 9,10 Multitape and nondeterministic TMs Lec 11,12 Nondeterministic TMs and extensions of TMs Exer 3 Oct. 29 Decidability and undecidability Church-Turing Thesis Lec 13,14

    40. In Defense Of Math At Work
    Alas, this notion or at least one very close to it is one I affirm; The concept of being a computer seems primary to that of Turing machine or
    Date Prev Date Next Thread Prev Thread Next ... Thread Index
    In Defense of Math at Work
    IN DEFENSE OF COMPUTATION AS MATH AT WORK Selmer Bringsjord http://www.rpi.edi/~brings

    41. Springer Online Reference Works
    The notion of a Turing machine can be used for making precise the general idea of an algorithm in a given alphabet, as follows. By a Turing algorithm in an

    Encyclopaedia of Mathematics
    Article referred from
    Article refers to
    Turing machine
    The name attached to abstract computers (cf. Computer, abstract ) of a specific type. The concept of a machine of such a kind originated in the middle of the 's from A.M. Turing as the result of an analysis carried out by him of the actions of a human being carrying out some or other calculations in accordance with a plan worked out in advance, that is, carrying out successive transformations of complexes of symbols. This analysis, in turn, was carried out by him with the aim of solving the then urgent problem of finding a precise mathematical equivalent for the general intuitive idea of an algorithm . In the course of development of the theory of algorithms (cf. Algorithms, theory of ), there emerged a number of modifications of the original definition of Turing. The version given here goes back to E. Post ; in this form the definition of a Turing machine has achieved widespread popularity (the Turing machine has been described in detail, for example, in and A Turing machine is conveniently represented as an automatically-functioning system capable of being in a finite number of internal states and endowed with an infinite external memory, called a tape. Two of the states are distinguished, the initial state and the final state. The tape is divided into cells and is unbounded to the left and to the right. Any letter of some finite

    42. The Turing Machine
    At various points in his paper, he argued that his notion of computability, as embodied in the Turing Machine, is equivalent to what is computable by humans
    Here is an essay I wrote on the Turing Machine. I had a lot of fun doing research for this paper and I ended up reading books in many different areas in Computers and Mathematics. I also felt that the first and final paragraph tie together in a very eloquent manner. The Turing Machine
    In his play Hamlet , Shakespeare provides an interesting description of humanity: "What a piece of work is man! How noble in reason, how infinite in faculties, in form and moving how express and admirable, in action how like an angel, in apprehension how like a god!” An intriguing question that one may ask is where does this leave the machines we create? Perhaps at the beginning of this century, one would be able to obtain an almost unanimous affirmation of the superiority of the human brain over the machine. The very question was ridiculous. In 1937, a British mathematician named Alan Matison Turing dreamed up a very simple machine that he contended was in some ways our equal. This device came to be known as the Turing Machine.
    At the 1928 International Congress, the great mathematician David Hilbert posed three questions. Was mathematics consistent, complete and decidable? At the time, Hilbert believed that the answer to all three questions was in the affirmative. Kurt Gödel, a Czech mathematician, soon stunned the mathematical community by showing that the answer to Hilbert’s first two questions was in the negative. In 1935, Turing attended a lecture course given by the distinguished Cambridge topologist M. H. A. Newman that ended with a treatment of Gödel’s proof that no system of axioms for arithmetic could be both consistent and complete. Turing learned that the third question posed by Hilbert was still unresolved. This was the question of the

    The classical computation model is based on the notion of a potentially infinite computing device (like a Turing machine or an idealized Pascal machine).
    THE UNIVERSITY OF MICHIGAN COMPUTING RESEARCH LABORATORY1 RECONSIDERING TURING'S THESIS (TOWARD MORE REALISTIC SEMANTICS OF PROGRAMS) Yurl Gurevlch CRL-TR-36-84 September 1984 Room 1079, East Engineering Building Ann Arbor, Michigan 48109 USA Tel: (313) 783-8000'Any opinions, findings, and conclusions or recommendations expressed in this publication are those or the authors and do not necessarily reflect the views of the funding agency. Reconsidering Turing's thesis (Toward more realistic semantics of programs) Yuri Ourevich1 Department of Electrical Engineering and Computer Science The University of Michigan, Ann Arbor, Ml 48109 Abstract. The classical computation model is based on the notion of a potentially infinite computing device (like a Turing machine or an idealized Pascal machine). We propose here an alternative computation model which explicitly recognizes finiteness of computers. The new operational semantics is especially appropriate in the case of algorithms sensitive to the bounds on machine resources (like algorithms for operating systems). 1Supported In part by NSF grant MCS83-0 022 UNIVERl OF MICIGAN (1 3026 89, 331

    44. Bounded Queries In Recursion Theory
    This notion is considered as a model of computation which extends the usual model of Turing machine to the power of asking questions called queries
    Bounded Queries in Recursion Theory by William I. Gasarch and Georgia A. Martin Progress in Computer Science and Applied Logic: Vol. 16 ISBN 0-8176-3966-7
    In recursion theory one considers functions which can be computed by an algorithm. Computational complexity theory is dedicated to the study of the difficulty of computations based on the notion of a measure of computational complexity in terms of the amount of some resources a program uses in a specific computation. An important measure of the complexity of a computable function is the time needed to compute it. Other resources, such as space , have also been considered. The object of the book is to classify functions which are not calculable from the point of view of their difficulty , in a quantitative way. For this, a new notion of complexity that is quantitative is introduced such that it expresses the level of difficulty of a function (such as the Turing degree). This work is a reflection of the contribution of the authors to the foundation and the development of a new direction of research in computational complexity theory. An oracle Turing machine is defined as a Turing machine together with an extra tape, an extra head to be used for reading that tape, and a mechanism to move the extra head and to overwrite characters on the extra tape. This notion is considered as a model of computation which extends the usual model of Turing machine to the power of asking questions - called

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