There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument (...) for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation. (shrink)

We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. (...) The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second-order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime, which is thus a universal constant of that spacetime. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any spacetime there will be a countable ordinal upper bound, , on the complexity of questions in the Borel hierarchy computable in it. Introduction 1.1 History and preliminaries Hyperarithmetic Computations in MH Spacetimes 2.1 Generalising SADn regions 2.2 The complexity of questions decidable in Kerr spacetimes An Upper Bound on Computational Complexity for Each Spacetime CiteULike Connotea Del.icio.us What's this? (shrink)

Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing that it is a non-sequitur. I conclude that the Effective version (...) of the Church–Turing Thesis is unaffected by SAD computation. (shrink)

Presented here is a new result concerning the computational power of so-called SADn computers, a class of Turing-machine-based computers that can perform some non-Turing computable feats by utilising the geometry of a particular kind of general relativistic spacetime. It is shown that SADn can decide n-quantifier arithmetic but not (n+1)-quantifier arithmetic, a result that reveals how neatly the SADn family maps into the Kleene arithmetical hierarchy. Introduction Axiomatising computers The power of SAD computers Remarks regarding the concept of computability.

I use modal logic and transfinite set-theory to define metaphysical foundations for a general theory of computation. A possible universe is a certain kind of situation; a situation is a set of facts. An algorithm is a certain kind of inductively defined property. A machine is a series of situations that instantiates an algorithm in a certain way. There are finite as well as transfinite algorithms and machines of any degree of complexity (e.g., Turing and super-Turing machines and more). There (...) are physically and metaphysically possible machines. There is an iterative hierarchy of logically possible machines in the iterative hierarchy of sets. Some algorithms are such that machines that instantiate them are minds. So there is an iterative hierarchy of finitely and transfinitely complex minds. (shrink)

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

I discuss the philosophical implications that the rising new science of quantum computing may have on the philosophy of computer science. While quantum algorithms leave the notion of Turing-Computability intact, they may re-describe the abstract space of computational complexity theory hence militate against the autonomous character of some of the concepts and categories of computer science.

A recent proposal to solve the halting problem with the quantum adiabatic algorithm is criticized and found wanting. Contrary to other physical hypercomputers, where one believes that a physical process “computes” a (recursive-theoretic) non-computable function simply because one believes the physical theory that presumably governs or describes such process, believing the theory (i.e., quantum mechanics) in the case of the quantum adiabatic “hypercomputer” is tantamount to acknowledging that the hypercomputer cannot perform its task.

Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary (...) to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument. (shrink)

We claim that a recent article of P. Cotogno ([2003]) in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no ?logical incomputability? in their being computed over a wider class. Hence this ?logical incomputability? regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. This seems to lead (...) him into a further error in his analysis of the supposed conventional status of the infinite time Turing machines of Hamkins and Lewis ([2000]). Theorem 1 refutes this directly. The diagonalisation misunderstanding Infinite computation Conclusion. (shrink)

The view that the brain is a sort of computer has functioned as a theoretical guideline both in cognitive science and, more recently, in neuroscience. But since we can view every physical system as a computer, it has been less than clear what this view amounts to. By considering in some detail a seminal study in computational neuroscience, I first suggest that neuroscientists invoke the computational outlook to explain regularities that are formulated in terms of the information content of electrical (...) signals. I then indicate why computational theories have explanatory force with respect to these regularities:in a nutshell, they underscore correspondence relations between formal/mathematical properties of the electrical signals and formal/mathematical properties of the represented objects. I finally link my proposal to the philosophical thesis that content plays an essential role in computational taxonomy. (shrink)

In spite of their practical importance, the connections between technology and mathematics have not received much scholarly attention. This article begins by outlining how the technology–mathematics relationship has developed, from the use of simple aide-mémoires for counting and arithmetic, via the use of mathematics in weaving, building and other trades, and the introduction of calculus to solve technological problems, to the modern use of computers to solve both technological and mathematical problems. Three important philosophical issues emerge from this historical résumé: (...) how mathematical knowledge depends on technology, the definition of the hybrid concept of a computation, and the usefulness of mathematics in technology. Each of these issues is briefly discussed, and it is shown that in order to analyze them, we need to combine tools and ideas from both the philosophy of technology and the philosophy of mathematics. In conclusion, it is argued that much more of interest can be found in the historically and philosophically unexplored terrains of the technology–mathematics relationship. (shrink)

We have a conundrum. The physical basis of information is clearly a highly active research area. Yet the power of information theory comes precisely from separating it from the detailed problems of building physical systems to perform information processing tasks. Developments in quantum information over the last two decades seem to have undermined this separation, leading to suggestions that information is itself a physical entity and must be part of our physical theories, with resource-cost implications. We will consider a variety (...) of ways in which physics seems to a affect computation, but will ultimately argue to the contrary: rejecting the claims that information is physical provides a better basis for understanding the fertile relationship between information theory and physics. instead, we will argue that the physical resource costs of information processing are to be understood through the need to consider physically embodied agents for whom information processing tasks are performed. Doing so sheds light on what it takes for something to be implementing a computational or information processing task of a given kind. (shrink)

Black holes are extremely relativistic objects. Physical processes around them occur in a regime where the gravitational field is extremely intense. Under such conditions, our representations of space, time, gravity, and thermodynamics are pushed to their limits. In such a situation philosophical issues naturally arise. In this chapter I review some philosophical questions related to black holes. In particular, the relevance of black holes for the metaphysical dispute between presentists and eternalists, the origin of the second law of thermadynamics and (...) its relation to black holes, the problem of information, black holes and hypercomputing, the nature of determinsim, and the breakdown of predictability in black hole space-times. I maintain that black hole physics can be used to illuminate some important problems in the border between science and philosophy, either epistemology and ontology. (shrink)

In a single-iteration fake barn case, the agent correctly identifies an object of interest on the first try, despite the presence of nearby lookalikes that could have mislead her. In a multiple-iteration fake barn case, the agent first encounters several fakes, misidentifies each of them, and then encounters and correctly identifies a genuine item of interest. Prior work has established that people tend to attribute knowledge in single-iteration fake barn cases, but multiple-iteration cases have not been tested. However, some theorists (...) contend that multiple-iteration cases are more important and will elicit a strong tendency to deny knowledge. Here I report a behavioural experiment investigating knowledge judgments in multiple-iteration fake barn cases. The main finding is that people tend to attribute knowledge in these cases too. Ironically, the results indicate that the presence of fakes could prevent iterated errors from lowering knowledge attributions. The results also provide evidence that ordinary knowledge attributions are based on attributions of cognitive ability. (shrink)

Being alive throughout all history need not save you from dying, even if history extends infinitely into the past and future. Infinitely-long lives can fall short of genuine immortality and suffer all an ordinary mortal’s diminution in experience. Adapting David Lewis on time travel, Roy Sorensen imagines quasi-immortal ‘spore gods’, whose finite personal lives are distributed across infinite external time. While criticising the ‘Eleatic’ terms in which Sorensen presents spore gods, this paper argues his essential claims are correct: ‘spore god’ (...) worlds are logically and metaphysically possible. Indeed, going beyond Sorensen, spore gods are even nomologically accessible within General Relativity. (shrink)

Christopher G. Timpson provides the first full-length philosophical treatment of quantum information theory and the questions it raises for our understanding of the quantum world. He argues for an ontologically deflationary account of the nature of quantum information, which is grounded in a revisionary analysis of the concepts of information.

One of the recurrent problems in the foundations of physics is to explain why we rarely observe certain phenomena that are allowed by our theories and laws. In thermodynamics, for example, the spontaneous approach towards equilibrium is ubiquitous yet the time-reversal-invariant laws that presumably govern thermal behaviour in the microscopic level equally allow spontaneous departure from equilibrium to occur. Why are the former processes frequently observed while the latter are almost never reported? Another example comes from quantum mechanics where the (...) formalism, if considered complete and universally applicable, predicts the existence of macroscopic superpositions—monstrous Schr¨odinger cats—and these are never observed: while electrons and atoms enjoy the cloudiness of waves, macroscopic objects are always localized to definite positions. (shrink)

Quantum Information Theory and the Foundations of Quantum Mechanics is a conceptual analysis of one of the most prominent and exciting new areas of physics, providing the first full-length philosophical treatment of quantum information theory and the questions it raises for our understanding of the quantum world. -/- Beginning from a careful, revisionary, analysis of the concepts of information in the everyday and classical information-theory settings, Christopher G. Timpson argues for an ontologically deflationary account of the nature of quantum information. (...) Against what many have supposed, quantum information can be clearly defined (it is not a primitive or vague notion) but it is not part of the material contents of the world. Timpson's account sheds light on the nature of nonlocality and information flow in the presence of entanglement and, in particular, dissolves puzzles surrounding the remarkable process of quantum teleportation. In addition it permits a clear view of what the ontological and methodological lessons provided by quantum information theory are; lessons which bear on the gripping question of what role a concept like information has to play in fundamental physics. Topics discussed include the slogan 'Information is Physical', the prospects for an informational immaterialism (the view that information rather than matter might fundamentally constitute the world), and the status of the Church-Turing hypothesis in light of quantum computation. -/- With a clear grasp of the concept of information in hand, Timpson turns his attention to the pressing question of whether advances in quantum information theory pave the way for the resolution of the traditional conceptual problems of quantum mechanics: the deep problems which loom over measurement, nonlocality and the general nature of quantum ontology. He marks out a number of common pitfalls to be avoided before analysing in detail some concrete proposals, including the radical quantum Bayesian programme of Caves, Fuchs, and Schack. One central moral which is drawn is that, for all the interest that the quantum information-inspired approaches hold, no cheap resolutions to the traditional problems of quantum mechanics are to be had. (shrink)

We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy's thesis.

A recent attempt to compute a (recursion‐theoretic) noncomputable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion‐theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from.

Analogicity in computer science is understood in two, not mutually exclusive ways: 1) with regard to the continuity feature (of data or computations), 2) with regard to the analogousness feature (i.e. similarity between certain natural processes and computations). Continuous computations are the subject of three methodological questions considered in the paper: 1a) to what extent do their theoretical models go beyond the model of the universal Turing machine (defining digital computations), 1b) is their computational power greater than that of the (...) universal Turing machine, 1c) under what conditions are continuous computations realizable in practice? The analogue-analogical computations lead to two other issues: 2a) in what sense and to what extent their accuracy depends on the adequacy of certain theories of empirical sciences, 2b) are there analogue-analogical computations in nature that are also continuous? The above issues are an important element of the philosophical discussion on the limitations of contemporary computer science. (shrink)

Which function is computed by a Turing machine will depend on how the symbols it manipulates are interpreted. Further, by invoking bizarre systems of notation it is easy to define Turing machines that compute textbook examples of uncomputable functions, such as the solution to the decision problem for first-order logic. Thus, the distinction between computable and uncomputable functions depends on the system of notation used. This raises the question: which systems of notation are the relevant ones for determining whether a (...) function is computable? These are the acceptable notations. I argue for a use criterion of acceptability: the acceptable notations for a domain of objects D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} are the notations that we can use for the usual D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-activities. Acceptable notations for natural numbers are ones that we can count with. Acceptable notations for logical formulas are the ones that we can use in inference and logical analysis. And so on. This criterion makes computability a relative notion—whether a function is computable depends on which notations are acceptable, which is relative to our interests and abilities. I argue that this is not a problem, however, since there is independent reason for taking the extension of computable function to be relative to contingent facts. Similarly, the fact that the use criterion makes a mathematical distinction depend on practical concerns is also not a problem because it dovetails with similar phenomena in other areas of computability theory, namely the roles of notation in computation over real numbers and of Extended Church’s Thesis in complexity theory. (shrink)

This article shows a clear sense in which general relativity allows for a type of ‘machine’ that can bring about a spacetime structure suitable for the implementation of ‘supertasks’. 1Introduction2Preliminaries3Malament–Hogarth Spacetimes4Machines5Malament–Hogarth Machines6Conclusion.

A survey of the field of hypercomputation, including discussion of a variety of objections.

In 1987, Roy Sorensen coined the term “parahistory” to denote the hypothetical study of evidence retrieved via time travel. Parahistory would thus stand to history rather as parapsychology does to psychology; studying data (in this case artefacts) that are obtained in ways unrecognised by orthodox science. This paper considers future-derived parahistorical artefacts. Past/future asymmetries threaten irresolvable problems in calibrating future objects' periods, in dating future artefacts and insulating them from causal loops. In turn, causal loop objects at best cannot be (...) non-arbitrarily dated at all and at worst, cannot even be artefacts. Even if there can be evidence for travel from the future, any such evidence that appeals to future artefacts faces pervasive difficulties and is very unlikely to be compelling. (shrink)

This article defends a modest version of the Physical Church-Turing thesis (CT). Following an established recent trend, I distinguish between what I call Mathematical CT—the thesis supported by the original arguments for CT—and Physical CT. I then distinguish between bold formulations of Physical CT, according to which any physical process—anything doable by a physical system—is computable by a Turing machine, and modest formulations, according to which any function that is computable by a physical system is computable by a Turing machine. (...) I argue that Bold Physical CT is not relevant to the epistemological concerns that motivate CT and hence not suitable as a physical analog of Mathematical CT. The correct physical analog of Mathematical CT is Modest Physical CT. I propose to explicate the notion of physical computability in terms of a usability constraint, according to which for a process to count as relevant to Physical CT, it must be usable by a finite observer to obtain the desired values of a function. Finally, I suggest that proposed counterexamples to Physical CT are still far from falsifying it because they have not been shown to satisfy the usability constraint. (shrink)

In quantum computing, where algorithms exist that can solve computational problems more efficiently than any known classical algorithms, the elimination of errors that result from external disturbances or from imperfect gates has become the ...

1. The Physical Church-Turing Thesis. Physicists often interpret the Church-Turing Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact that universal computers are as powerful in their computational capabilities as any physically realizable system can be, that they can simulate any physical system . . . (...) No physically implementable procedure could then shortcut a computationally irreducible process. (Wolfram 1985) Wolfram’s thesis consists of two parts: (a) Any physical system can be simulated (to any degree of approximation) by a universal Turing machine (b) Complexity bounds on Turing machine simulations have physical significance. For example, suppose that the computation of the minimum energy of some system of n particles takes at least exponentially (in n) many steps. Then the relaxation time of the actual physical system to its minimum energy state will also take exponential time. (shrink)

We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.

Accelerating Turing machines have attracted much attention in the last decade or so. They have been described as “the work-horse of hypercomputation” (Potgieter and Rosinger 2010: 853). But do they really compute beyond the “Turing limit”—e.g., compute the halting function? We argue that the answer depends on what you mean by an accelerating Turing machine, on what you mean by computation, and even on what you mean by a Turing machine. We show first that in the current literature the term (...) “accelerating Turing machine” is used to refer to two very different species of accelerating machine, which we call end-stage-in and end-stage-out machines, respectively. We argue that end-stage-in accelerating machines are not Turing machines at all. We then present two differing conceptions of computation, the internal and the external, and introduce the notion of an epistemic embedding of a computation. We argue that no accelerating Turing machine computes the halting function in the internal sense. Finally, we distinguish between two very different conceptions of the Turing machine, the purist conception and the realist conception; and we argue that Turing himself was no subscriber to the purist conception. We conclude that under the realist conception, but not under the purist conception, an accelerating Turing machine is able to compute the halting function in the external sense. We adopt a relatively informal approach throughout, since we take the key issues to be philosophical rather than mathematical. (shrink)

This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. 2005 Elsevier Inc. All rights reserved.

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) in chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turing thesis) related to the possible “solution of supertasks,” and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for physical applications are discussed: (...) Canlorian “naive” (i.e., nonaxiomatic) set theory, contructivism, and operationalism. In the author's opinion, an attitude of “suspended attention” (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same time, physicists should be open to “bizarre” or “mindboggling” new formalisms, which need not be operationalizable or testable at the lime of their creation, but which may successfully lead to novel fields of phenomenology and technology. (shrink)

The aim of this paper is to present some basic notions of the theory of quantum computing and to compare them with the basic notions of the classical theory of computation. I am convinced, that the results of quantum computation theory (QCT) are not only interesting in themselves, but also should be taken into account in discussions concerning the nature of mathematical knowledge. The philosophical discussion will however be postponed to another paper. QCT seems not to be well-known among philosophers (...) (at least not to the degree it deserves), so the aim of this paper is to provide the necessary technical preliminaries presented in a way accessible to the general philosophical audience. (shrink)