Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability.
Fifty years ago this month[[June]], in the Computing Machine Laboratory at Manchester University, the world's first electronic stored-program computer performed its first calculation. The tiny program, stored on the face of a cathode ray tube, was just 17 instructions long. Electronic engineers Freddie Williams and Tom Kilburn built the Manchester computer in accordance with fundamental ideas explained to them by Max Newman, professor of mathematics at Manchester. The computer fell sideways out of research that nobody could have guessed would have (...) any practical application. The initial idea germinated thirteen years earlier in the head of Alan Turing, who was working on a recherché problem in mathematical logic. While thinking about this problem Turing dreamed up an abstract machine, nowadays known simply as the 'universal Turing machine' and which, as he put it, would compute 'all numbers which could naturally be regarded as computable'. The machine consisted of a memory in.. (shrink)
The tense tree method extends Jeffrey’s well-known formulation of classical propositional logic to tense logic (Jeffrey 1991).1 Tense trees combine pure tense logic with features of Prior’s U-calculi (where ‘U’ is the earlier-than relation; see Prior 1967 and the Introduction to this volume). The tree method has a number of virtues: trees are well suited to computational applications; semantically, the tree systems presented here are no less illuminating than model theory; the metatheory associated with tree formulations is often more tractable (...) than that required in a model-theoretic setting; and last but not least the tree method is ideal for pedagogical purposes. (shrink)
Ignoring the temporal dimension, an object such as a railway tunnel or a human body is a three-dimensional whole composed of three-dimensional parts. The four-dimensionalist holds that a physical object exhibiting identity across time—Descartes, for example—is a four-dimensional whole composed of 'briefer' four-dimensional objects, its temporal parts. Peter van Inwagen (1990) has argued that four-dimensionalism cannot be sustained, or at best can be sustained only by a counterpart theorist. We argue that different schemes of individuation of temporal parts are available, (...) which undermines van Inwagen's argument. (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.
The tape is divided into squares, each square bearing a single symbol—'0' or '1', for example. This tape is the machine's general-purpose storage medium: the machine is set in motion with its input inscribed on the tape, output is written onto the tape by the head, and the tape serves as a short-term working memory for the results of intermediate steps of the computation. The program governing the particular computation that the machine is to perform is also stored on the (...) tape. A small, fixed program that is 'hard-wired' into the head enables the head to read and execute the instructions of whatever program is on the tape. The machine's atomic operations are very simple—for example, 'move left one square', 'move right one square', 'identify the symbol currently beneath the head', 'write 1 on the square that is beneath the head', and 'write 0 on the square that is beneath the head'. Complexity of operation is achieved by the chaining together of large numbers of these simple atoms. Any universal Turing machine can be programmed to carry out any calculation that can be performed by a human mathematician working with paper and pencil in accordance with some algorithmic method. This is what is meant by calling these machines 'universal'. (shrink)
In his PhD thesis (1938) Turing introduced what he described as 'a new kind of machine'. He called these 'O-machines'. The present paper employs Turing's concept against a number of currently fashionable positions in the philosophy of mind.
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 'Church-Turing 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 number of foundational arguments that are commonly rehearsed in cognitive science, and gesture towards a new class of cognitive models. (shrink)
It is not widely realised that Turing was probably the first person to consider building computing machines out of simple, neuron-like 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. (shrink)