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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally e ..."
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Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, wa ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
Physical Hypercomputation and the Church–Turing Thesis
, 2003
"... 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 ..."
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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.
The challenge of characterizing operations in the mechanisms underlying behavior
 Journal of the Experimental Analysis of Behavior
, 2005
"... Neuroscience and cognitive science seek to explain behavioral regularities in terms of underlying mechanisms. An important element of a mechanistic explanation is a characterization of the operations of the parts of the mechanism. The challenge in characterizing such operations is illustrated by an ..."
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Neuroscience and cognitive science seek to explain behavioral regularities in terms of underlying mechanisms. An important element of a mechanistic explanation is a characterization of the operations of the parts of the mechanism. The challenge in characterizing such operations is illustrated by an example from the history of physiological chemistry in which some investigators tried to characterize the internal operations in the same terms as the overall physiological system while others appealed to elemental chemistry. In order for biochemistry to become successful, researchers had to identify a new level of operations involving operations over molecular groups. Existing attempts at mechanistic explanation of behavior are in a situation comparable to earlier approaches to physiological chemistry, drawing their inspiration either from overall psychology activities or from lowlevel neural processes. Successful mechanistic explanations of behavior require the discovery of the appropriate component operations. Such discovery is a daunting challenge but one on which success will be beneficial to both behavioral scientists and cognitive and neuroscientists. Key words: mechanistic explanation, operations, laws, levels of organization, connectionism,
Theory of one tape linear time Turing machines
 Proc. 30th SOFSEM Conference on Current Trends in Theory and Practice of Computer Science, Lecture Notes in Computer Science, Vol.2932, pp.335–348
, 2004
"... Abstract. A theory of onetape lineartime Turing machines is quite different from its polynomialtime counterpart. This paper discusses the computational complexity of onetape Turing machines of various machine types (deterministic, nondeterministic, reversible, alternating, probabilistic, countin ..."
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Abstract. A theory of onetape lineartime Turing machines is quite different from its polynomialtime counterpart. This paper discusses the computational complexity of onetape Turing machines of various machine types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in time O(n), where the running time of a machine is defined as the height of its computation tree. We also address a close connection between onetape lineartime Turing machines and finite state automata. §1. Model of Computation: Turing Machines. We use a standard definition of an offline Turing machine. Of special interest is a onetape Turing machine (abbreviated 1TM) M = (Q, Σ, Γ, δ, q0, qacc, qrej), where Q is a finite set of (internal) states, Σ is a nonempty finite input alphabet 3, Γ is a finite tape alphabet including Σ, q0 in Q is an initial state, qacc and qrej in Q are an accepting state and a rejecting state, respectively, and δ is a transition function. Different transition functions δ give rise to various types of 1TMs described in
Alan Turing and the Mathematical Objection
 Minds and Machines 13(1
, 2003
"... Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet accord ..."
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Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection ” to his view that machines can think. Logicomathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. Key words: artificial intelligence, ChurchTuring thesis, computability, effective procedure, incompleteness, machine, mathematical objection, ordinal logics, Turing, undecidability The ‘skin of an onion ’ analogy is also helpful. In considering the functions of the mind or the brain we find certain operations which we can express in purely mechanical terms. This we say does not correspond to the real mind: it is a sort of skin which we must strip off if we are to find the real mind. But then in what remains, we find a further skin to be stripped off, and so on. Proceeding in this way, do we ever come to the ‘real ’ mind, or do we eventually come to the skin which has nothing in it? In the latter case, the whole mind is mechanical (Turing, 1950, p. 454–455). 1.
Mental mechanisms: What are the operations
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"... An important goal of the cognitive sciences is to explain mental phenomena in terms of mechanisms. A mechanistic explanation requires characterizing the operations of the mechanism’s parts. The challenge in characterizing such operations is illustrated by an example from biology in which some invest ..."
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An important goal of the cognitive sciences is to explain mental phenomena in terms of mechanisms. A mechanistic explanation requires characterizing the operations of the mechanism’s parts. The challenge in characterizing such operations is illustrated by an example from biology in which some investigators tried to characterize the internal operations in the same terms as the overall physiological system while others appealed to elemental chemistry. Before biochemistry became successful, researchers had to identify operations at a new level of organization—operations over molecular groups. Existing attempts at mechanistic explanation in cognitive science are in a situation comparable to the earlier stage in the biological example, drawing their inspiration either from overall psychological activities or from lowlevel neural processes, neither of which is likely to provide a successful account of the operations in mental mechanisms.
Explanation: Mechanism, Modularity, and Situated Cognition 1
"... The situated cognition movement has emerged in recent decades (although it has roots in psychologists working earlier in the 20 th century including Vygotsky, Bartlett, and Dewey) largely in reaction to an approach to explaining cognition that tended to ignore the context in which cognitive activiti ..."
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The situated cognition movement has emerged in recent decades (although it has roots in psychologists working earlier in the 20 th century including Vygotsky, Bartlett, and Dewey) largely in reaction to an approach to explaining cognition that tended to ignore the context in which cognitive activities typically occur. Fodor’s (1980) account of the research strategy of methodological solipsism, according to which only representational states within the mind are viewed as playing causal roles in producing cognitive activity, is an extreme characterization of this approach. (As Keith Gunderson memorably commented when Fodor first presented this characterization, it amounts to reversing behaviorism by construing the mind as a white box in a black world). Critics as far back as the 1970s and 1980s objected to many experimental paradigms in cognitive psychology as not being ecologically valid; that is, they maintained that the findings only applied to the artificial circumstances created in the laboratory and did not generalize to real world settings (Neisser, 1976; 1987). The situated cognition movement, however, goes much further than demanding ecologically valid experiments—it insists that an agent’s cognitive activities are inherently embedded and supported by dynamic interactions with the agent’s body and features of its environment. Sometimes advocates of a situated approach to cognition present their position in an extreme manner that sets the situated approach in opposition to the attempts in cognitive science and cognitive neuroscience to understand the mechanisms within the mind/brain that underlie
Computation vs. Information Processing: Why Their Difference Matters to Cognitive Science
"... Since the cognitive revolution, it’s become commonplace that cognition involves both computation and information processing. Is this one claim or two? Is computation the same as information processing? The two terms are often used interchangeably, but this usage masks important differences. In this ..."
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Since the cognitive revolution, it’s become commonplace that cognition involves both computation and information processing. Is this one claim or two? Is computation the same as information processing? The two terms are often used interchangeably, but this usage masks important differences. In this paper, we distinguish information processing from computation and examine some of their mutual relations, shedding light on the role each can play in a theory of cognition. We recommend that theorists of cognition be explicit and careful in choosing 1 notions of computation and information and connecting them together. Much confusion can be avoided by doing so.
Church Without Dogma: Axioms for computability
"... Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to pre ..."
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Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses. To investigate effective calculability is to analyze processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (or discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. Representing human and machine computations by discrete dynamical systems, the boundedness and locality conditions can be captured through axioms for Turing computors and Gandy machines; models of