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- Kenneth Aizawa, It is Not All About Turing-Equivalent Computation.One account of the history of computation might begin in the 1930’s with some of the work of Alonzo Church, Alan Turing, and Emil Post. One might say that this is where something like the core concept of computation was first formally articulated. Here were the first attempts to formalize an informal notion of an algorithm or effective procedure by which a mathematician might decide one or another logico-mathematical question. As each of these formalisms was shown to compute the same set of functions—the partial recursive functions—each of them might be described as a form of Turing-equivalent computation. This work set the cornerstone for what we might call computation theory. This history might then proceed to give pride of place to this form of computation in subsequent developments in cognitive science and in related disciplines and subdisciplines. Such a history might note that, in the 1940’s, the results of this work would have been transferred into the emerging field of computer science with the design and construction of the first electronic digital computers. Here one would mention Turing again, as well as perhaps Norbert Wiener, Julian Bigelow, John von Neumann, and many others. At about the same time, this theory of computation would have been inserted into the theory of neural networks by way of Warren McCulloch and Walter Pitts’s seminal work, “A Logical Calculus of the Ideas Immanent in Nervous Activity.” Somewhat later, during the 1960’s, Hilary Putnam introduced Turing machine tables into the philosophy of mind as a tool for illuminating various features of the mind-body problem, eventually transforming the intellectual landscape in the metaphysics of mind. Also during the 1960’s, Turingequivalent computation would have infiltrated psychology through the influence of Chomskyan linguistics and under the rubric of information processing psychology. Further, such computation would have been integrated into the fields of cognitive science and neuroscience as they emerged during the 1970’s and 1980’s..
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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.
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| | Other links: springerlink.com dx.doi.org | Scholar | At my library | More options ... 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.
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| | Other links: bjps.oxfordjournals.org jstor.org dx.doi.org | Scholar | At my library | More options ... ``Neural computing'' is a research field based on perceiving the human brain as an information system. This system reads its input continuously via the different senses, encodes data into various biophysical variables such as membrane potentials or neural firing rates, stores information using different kinds of memories (e.g., short-term memory, long-term memory, associative memory), performs some operations called ``computation'', and outputs onto various channels, including motor control commands, decisions, thoughts, and feelings. We show a natural model of neural computing that gives rise to hyper-computation. Rigorous mathematical analysis is applied, explicating our model's exact computational power and how it changes with the change of parameters. Our analog neural network allows for supra-Turing power while keeping track of computational constraints, and thus embeds a possible answer to the superiority of the biological intelligence within the framework of classical computer science. We further propose it as standard in the field of analog computation, functioning in a role similar to that of the universal Turing machine in digital computation. In particular an analog of the Church-Turing thesis of digital computation is stated where the neural network takes place of the Turing machine.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: cs.queensu.ca springerlink.com kluweronline.com journals.kluweronline.com ingentaconnect.com dx.doi.org | Scholar | At my library | More options ...
| | Other links: cs.queensu.ca springerlink.com kluweronline.com journals.kluweronline.com ingentaconnect.com dx.doi.org | Scholar | At my library | More options ... Two very different insights motivate characterizing the brain as a computer. One depends on mathematical theory that defines computability in a highly abstract sense. Here the foundational idea is that of a Turing machine. Not an actual machine, the Turing machine is really a conceptual way of making the point that any well-defined function could be executed, step by step, according to simple 'if-you-are-in-state-P-and-have-input-Q-then-do-R' rules, given enough time (maybe infinite time) [see COMPUTATION]. Insofar as the brain is a device whose input and output can be characterized in terms of some mathematical function -- however complicated -- then in that very abstract sense, it can be mimicked by a Turning machine. Given what is known so far brains do seem to depend on cause-effect operations, and hence brains appear to be, in some formal sense, equivalent to a Turing machine [see CHURCH-TURING THESIS]. On its own, however, this reveals nothing at all of how the mind-brain actually works. The second insight depends on looking at the brain as a biological device that processes information from the environment to build complex representations that enable the brain to make predictions and select advantageous behaviors. Where necessary to avoid ambiguity, we will refer to the first notion of computation as algorithmic computation, and the second as information processing computation.
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| | Scholar | More options ... To compute is to execute an algorithm. More precisely, to say that a device or organ computes is to say that there exists a modelling relationship of a certain kind between it and a formal specification of an algorithm and supporting architecture. The key issue is to delimit the phrase of a certain kind. I call this the problem of distinguishing between standard and nonstandard models of computation. The successful drawing of this distinction guards Turing's 1936 analysis of computation against a difficulty that has persistently been raised against it, and undercuts various objections that have been made to the computational theory of mind.
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| | Other links: springerlink.com jstor.org | Scholar | At my library | More options ... This paper deals with the question: what are the key requirements for a physical system to perform digital computation? Time and again cognitive scientists are quick to employ the notion of computation simpliciter when asserting basically that cognitive activities are computational. They employ this notion as if there was or is a consensus on just what it takes for a physical system to perform computation, and in particular digital computation. Some cognitive scientists in referring to digital computation simply adhere to Turing’s notion of computability . Classical computability theory studies what functions on the natural numbers are computable and what mathematical problems are undecidable. Whilst a mathematical formalism of computability may perform a methodological function of evaluating computational theories of certain cognitive capacities, concrete computation in physical systems seems to be required for explaining cognition as an embodied phenomenon . There are many non-equivalent accounts of digital computation in physical systems. I examine only a handful of those in this paper: (1) Turing’s account ; (2) The triviality “account”; (3) Reconstructing Smith’s account of participatory computation ; (4) The Algorithm Execution account . My goal in this paper is twofold. First, it is to identify and clarify some of the underlying key requirements mandated by these accounts. I argue that these differing requirements justify a demand that one commits to a particular account when employing the notion of computation in regard to physical systems. Second, it is to argue that despite the informative role that mathematical formalisms of computability may play in cognitive science, they do not specify the relationship between abstract and concrete computation.
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| | Other links: dx.doi.org | Scholar | At my library | More options ... What''s computation? The received answer is that computation is a computer at work, and a computer at work is that which can be modelled as a Turing machine at work. Unfortunately, as John Searle has recently argued, and as others have agreed, the received answer appears to imply that AI and Cog Sci are a royal waste of time. The argument here is alarmingly simple: AI and Cog Sci (of the Strong sort, anyway) are committed to the view that cognition is computation (or brains are computers); butall processes are computations (orall physical things are computers); so AI and Cog Sci are positively silly.I refute this argument herein, in part by defining the locutions x is a computer and c is a computation in a way that blocks Searle''s argument but exploits the hard-to-deny link between What''s Computation? and the theory of computation. However, I also provide, at the end of this essay, an argument which, it seems to me, implies not that AI and Cog Sci are silly, but that they''re based on a form of computation that is well beneath human persons.
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| | Other links: springerlink.com | Scholar | At my library | More options ... A version of the Church-Turing Thesis states that every effectively realizable physical system can be defined by Turing Machines (‘Thesis P’); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non-well-founded, 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 affected. Therefore, Thesis P is not essentially different from the standard Church-Turing Thesis. 1 Introduction 2 Computability and incomputability 3 The physical interpretation of the Church-Turing Thesis 4 Supertasks and infinite computation 5 Computation on non-well-founded domains 6 Analog computation 7 Quantum computation 8 Retrocausal computation 9 Conclusions.
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| | Other links: bjps.oupjournals.org jstor.org dx.doi.org | Scholar | At my library | More options ... It has been argued that neural networks and other forms of analog computation may transcend the limits of Turing-machine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turing-machine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing in or inspired by natural systems). This points to the need for new models of computation addressing issues orthogonal to those that have occupied the traditional theory of computation.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: citeseer.ist.psu.edu portal.acm.org cs.utk.edu cs.queensu.ca springerlink.com kluweronline.com ingentaconnect.com dx.doi.org | Scholar | At my library | More options ...
| | Other links: citeseer.ist.psu.edu portal.acm.org cs.utk.edu cs.queensu.ca springerlink.com kluweronline.com ingentaconnect.com dx.doi.org | Scholar | At my library | More options ... One account of the history of computation might begin in the 1930's with some of the work of Alonzo Church, Alan Turing, and Emil Post. One might say that this is where something like the core concept of computation was first formally articulated. Here were the first attempts to formalize an informal notion of an algorithm or effective procedure by which a mathematician might decide one or another logico-mathematical question. As each of these formalisms was shown to compute the same set of functions—the partial recursive functions—each of them might be described as a form of Turing-equivalent computation. This work set the cornerstone for what we might call computation theory. This history might then proceed to give pride of place to this form of computation in subsequent developments in cognitive science and in related disciplines and subdisciplines. Such a history might note that, in the 1940's, the results of this work would have been transferred into the emerging field of computer science with the design and construction of the first electronic digital computers. Here one would mention Turing again, as well as perhaps Norbert Wiener, Julian Bigelow, John von Neumann, and many others. At about the same time, this theory of computation would have been inserted into the theory of neural networks by way of Warren McCulloch and Walter Pitts's seminal work, “A Logical Calculus of the Ideas Immanent in Nervous Activity.” Somewhat later, during the 1960's, Hilary Putnam introduced Turing machine tables into the philosophy of mind as a tool for illuminating various features of the mind-body problem, eventually transforming the intellectual landscape in..
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| | Other links: centenary.edu dx.doi.org | Scholar | At my library | More options ... Discussion of Kenneth Aizawa, It is not all about Turing-equivalent computation
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