B. Jack Copeland

What is computation

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 Turings 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.

Beyond the universal Turing machine

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

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 π 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 Searles Chinese room argument.

The genesis of possible worlds semantics

This article traces the development of possible worlds semantics through the work of: Wittgenstein, 1913–1921; Feys, 1924; McKinsey, 1945; Carnap, 1945–1947; McKinsey, Tarski and Jónsson, 1947–1952; von Wright, 1951; Becker, 1952; Prior, 1953–1954; Montague, 1955; Meredith and Prior, 1956; Geach, 1960; Smiley, 1955–1957; Kanger, 1957; Hintikka, 1957; Guillaume, 1958; Binkley, 1958; Bayart, 1958–1959; Drake, 1959–1961; Kripke, 1958–1965.

The Turing Test

Turings test has been much misunderstood. Recently unpublished material by Turing casts fresh light on his thinking and dispels a number of philosophical myths concerning the Turing test. Properly understood, the Turing test withstands objections that are popularly believed to be fatal.

On Alan Turing's anticipation of connectionism

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 Turings connectionist project of 1948.

The Broad Conception of Computation

A myth has arisen concerning Turings article 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 minds 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 or even states this claim (nor does Church). The author describes a number of notional machines, both analog 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 or 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 toward a new class of cognitive models.

Hypercomputation: philosophical issues

A survey of the field of hypercomputation, including discussion of four a priori objections to the possibility of hypercomputation. An exegesis of Turings pre- and post-war writings on the mind is given, and Turings views on the scope of machines are discussed.

Do Accelerating Turing Machines Compute the Uncomputable

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.

What Turing Did after He Invented the Universal Turing Machine

Alan Turing anticipated many areas of current research incomputer and cognitive science. This article outlines his contributionsto Artificial Intelligence, connectionism, hypercomputation, andArtificial Life, and also describes Turings pioneering role in thedevelopment of electronic stored-program digital computers. It locatesthe origins of Artificial Intelligence in postwar Britain. It examinesthe intellectual connections between the work of Turing and ofWittgenstein in respect of their views on cognition, on machineintelligence, and on the relation between provability and truth. Wecriticise widespread and influential misunderstandings of theChurch–Turing thesis and of the halting theorem. We also explore theidea of hypercomputation, outlining a number of notional machines that“compute the uncomputable.”