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The Broad Conception Of Computation
 American Behavioral Scientist
, 1997
"... A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, somet ..."
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A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'ChurchTuring thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of nonclassical computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a numb...
On Alan Turing's Anticipation Of Connectionism
, 1996
"... It is not widely realised that Turing was probably the first person to consider building computing machines out of simple, neuronlike elements connected together into networks in a largely random manner. Turing called his networks `unorganised machines'. By the application of what he described as ' ..."
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It is not widely realised that Turing was probably the first person to consider building computing machines out of simple, neuronlike elements connected together into networks in a largely random manner. Turing called his networks `unorganised machines'. By the application of what he described as 'appropriate interference, mimicking education' an unorganised machine can be trained to perform any task that a Turing machine can carry out, provided the number of 'neurons' is sufficient. Turing proposed simulating both the behaviour of the network and the training process by means of a computer program. We outline Turing's connectionist project of 1948.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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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.
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, 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). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Effectiveness ∗
, 2011
"... We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. Efficiency is doing things right; effectiveness is doing the right things. —Peter F. Drucker ..."
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We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. Efficiency is doing things right; effectiveness is doing the right things. —Peter F. Drucker
On the Optimality of RAM Complexity
"... We demonstrate that the programs of any classical (as opposed to parallel or interactive) computation model or programming language that satisfies natural postulates of effectiveness (which specialize Gurevich’s Sequential Postulates)—regardless of the data structures it employs—can be simulated by ..."
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We demonstrate that the programs of any classical (as opposed to parallel or interactive) computation model or programming language that satisfies natural postulates of effectiveness (which specialize Gurevich’s Sequential Postulates)—regardless of the data structures it employs—can be simulated by a random access machine (RAM) with only constant overhead. In essence, the effectiveness postulates assert the following: states can be represented as logical structures; transitions depend on a fixed finite set of terms (those referred to in the algorithm); basic operations can be programmed from constructors; and transitions commute with isomorphisms. Complexity for any domain is measured in terms of constructor operations. It follows that algorithmic lower bounds for the RAM model hold (up to a constant factor determined by the algorithm in question) for any and all effective classical models of computation, whatever its control structures and data structures. This substantiates the polynomialtime version of the classical extended ChurchTuring Thesis (the Invariance Thesis), namely that every effective classical algorithm can be polynomially simulated by a Turing machine.