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24
Hypercomputation: computing more than the Turing machine
, 2002
"... In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which al ..."
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Cited by 31 (5 self)
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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw out some interesting consequences of hypercomputation for several different disciplines. I begin with a succinct introduction to the classical theory of computation and its place amongst some of the negative results of the 20 th Century. I then explain how the ChurchTuring Thesis is commonly misunderstood and present new theses which better describe the possible limits on computability. Following this, I introduce ten different hypermachines (including three of my own) and discuss in some depth the manners in which they attain their power and the physical plausibility of each method. I then compare the powers of the different models using a device from recursion theory. Finally, I examine the implications of hypercomputation to mathematics, physics, computer science and philosophy. Perhaps the most important of these implications is that the negative mathematical results of Gödel, Turing and Chaitin are each dependent upon the nature of physics. This both weakens these results and provides strong links between mathematics and physics. I conclude that hypercomputation is of serious academic interest within many disciplines, opening new possibilities that were previously ignored because of long held misconceptions about the limits of computation.
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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Cited by 21 (0 self)
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
Natural computation and nonTuring models of computation
 Theoretical Computer Science
, 2004
"... We propose certain nonTuring models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of applicatio ..."
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Cited by 18 (9 self)
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We propose certain nonTuring models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are illsuited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics. Key words: analog computation, analog computer, biocomputation, computability, computation on reals, continuous computation, formal system, hypercomputation,
The many forms of hypercomputation
 Applied Mathematics and Computation
, 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
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Cited by 16 (0 self)
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This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.
How much can analog and hybrid systems be proved (super)Turing
 Applied Mathematics and Computation
, 2006
"... Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can ..."
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Cited by 5 (1 self)
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Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or superTuring. 1
Emergence is coupled to scope, not level
 Complexity
, 2007
"... Since its application to systems, emergence has been explained in terms of levels of observation. This approach has led to confusion, contradiction, incoherence and at times mysticism. When the idea of level is replaced by a framework of scope, resolution and state, this confusion is dissolved. We f ..."
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Cited by 5 (0 self)
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Since its application to systems, emergence has been explained in terms of levels of observation. This approach has led to confusion, contradiction, incoherence and at times mysticism. When the idea of level is replaced by a framework of scope, resolution and state, this confusion is dissolved. We find that emergent properties are determined by the relationship between the scope of macrostate and microstate descriptions. This establishes a normative definition of emergent properties and emergence that makes sense of previous descriptive definitions of emergence. In particular, this framework sheds light on which classes of emergent properties are epistemic and which are ontological, and identifies fundamental limits to our ability to capture emergence in formal systems. 1
Constraints on Hypercomputation, in
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
"... “To infinity, and beyond!”, Buzz Lightyear, Toy Story, Pixar, 1995. Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to ..."
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Cited by 3 (2 self)
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“To infinity, and beyond!”, Buzz Lightyear, Toy Story, Pixar, 1995. Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to bounded or discrete, computation. Thus, such schemes may include the deployment of actualised rather than potential infinities of physical resources, or of physical representations of real numbers to arbitrary precision. Here, we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability. 1
Abstract SuperTasks, Accelerating Turing Machines and Uncomputability
"... Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To sh ..."
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Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To show this, I analyze the reasoning that leads to Thomson's paradox, point out that the paradox rests on a conflation of different perspectives of accelerating processes, and conclude that the same conflation underlies the claim that accelerating Turing machines can solve the halting problem.