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.

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 Turing's 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 thatcompute the uncomputable. (shrink)

Can mind be modeled as a Turing machine? If you find such questions irrelevant, e.g. because the subject is already exhausted, then you need not read the book Mind versus Computer (Gams et al., 1991). If, on the other hand, you do find such questions relevant, then perhaps you need not read Dunlop's review of the book (Dunlop, 2000). (...).

In the centenary year of Turing’s birth, a lot of good things are sure to be written about him. But it is hard to find something new to write about Turing. This is the biggest merit of this article: it shows how von Neumann’s architecture of the modern computer is a serendipitous consequence of the universal Turing machine, built to solve a logical problem.

In the field of computability and algorithmicity, there have recently been two essays that are of great interest: Peter Slezak's "Descartes's Diagonal Deduction," and David Deutsch's "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer." In brief, the former shows that Descartes' Cogito argument is structurally similar to Godel's proof that there are statements true but cannot be proven within a formal system such as Principia Mathematica, while Deutsch provides strong arguments for believing that the universe can (...) be represented as a Turing machine. King contends that the conjoining of Slezak's analysis with Deutsch's provides a perspective from which it is possible to argue that a scientific theology can be taken a little more seriously at present than in the past. , , , , In the field of computability and algorithmicity, there have recently been two essays that are of great interest: Peter Slezak's "Descartes's Diagonal Deduction," and David Deutsch's "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer." In brief, the former shows that Descartes' Cogito argument is structurally similar to Godel's proof that there are statements true but cannot be proven within a formal system such as Principia Mathematica, while Deutsch provides strong arguments for believing that the universe can be represented as a Turing machine. King contends that the conjoining of Slezak's analysis with Deutsch's provides a perspective from which it is possible to argue that a scientific theology can be taken a little more seriously at present than in the past. (shrink)

The properties of Turing’s famous ‘universal machine’ has long sustained functionalist intuitions about the nature of cognition. Here, I show that there is a logical problem with standard functionalist arguments for multiple realizability. These arguments rely essentially on Turing’s powerful insights regarding computation. In addressing a possible reply to this criticism, I further argue that functionalism is not a useful approach for understanding what it is to have a mind. In particular, I show that the difficulties (...) involved in distinguishing implementation from function make multiple realizability claims untestable and uninformative. As a result, I conclude that the role of Turing machines in philosophy of mind needs to be reconsidered. (shrink)

I propose to consider the question, "Can machines think?" This should begin with definitions of the meaning of the terms "machine" and "think." The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous, If the meaning of the words "machine" and "think" are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the (...) answer to the question, "Can machines think?" is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. The new form of the problem can be described in terms of a game which we call the 'imitation game." It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart front the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either "X is A and Y is B" or "X is B and Y is A." The interrogator is allowed to put questions to A and B. We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?". (shrink)

As interpreted by Pattee, von Neumann’s Theory of Self-Reproducing Automata has proved to be a useful tool for understanding some of the difficulties and paradoxes of molecular biosemiotics. But is its utility limited to molecular systems or is it more generally applicable within biosemiotics? One way of answering that question is to look at the Theory as a model for one particular high-level biosemiotic activity, human language. If the model is not useful for language, then it certainly cannot be generally (...) useful to biosemiotics. Beginning with the Universal Turing Machine and continuing with von Neumann’s Theory and Pattee’s interpretation, the properties of universality, programmability, underspecification, complementarity of description/construction, and open-ended evolutionary potential are shown to be usefully applicable to language, thus opening a new line of inquiry in biosemiotics. (shrink)

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for (...) the concept of a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers. (shrink)

Animals, including humans, are usually judged on what they could become, rather than what they are. Many physical and cognitive abilities in the ‘animal kingdom’ are only acquired (to a given degree) when the subject reaches a certain stage of development, which can be accelerated or spoilt depending on how the environment, training or education is. The term ‘potential ability’ usually refers to how quick and likely the process of attaining the ability is. In principle, things should not be different (...) for the ‘machine kingdom’. While machines can be characterised by a set of cognitive abilities, and measuring them is already a big challenge, known as ‘universal psychometrics’, a more informative, and yet more challenging, goal would be to also determine the potential cognitive abilities of a machine. In this paper we investigate the notion of potential cognitive ability for machines, focussing especially on universality and intelligence. We consider several machine characterisations (non-interactive and interactive) and give definitions for each case, considering permanent and temporal potentials. From these definitions, we analyse the relation between some potential abilities, we bring out the dependency on the environment distribution and we suggest some ideas about how potential abilities can be measured. Finally, we also analyse the potential of environments at different levels and briefly discuss whether machines should be designed to be intelligent or potentially intelligent. (shrink)

A fundamental problem in artificial intelligence is that nobody really knows what intelligence is. The problem is especially acute when we need to consider artificial systems which are significantly different to humans. In this paper we approach this problem in the following way: we take a number of well known informal definitions of human intelligence that have been given by experts, and extract their essential features. These are then mathematically formalised to produce a general measure of intelligence for arbitrary machines. (...) We believe that this equation formally captures the concept of machine intelligence in the broadest reasonable sense. We then show how this formal definition is related to the theory of universal optimal learning agents. Finally, we survey the many other tests and definitions of intelligence that have been proposed for machines. (shrink)

The aim of the paper is to present the underlying reason of the unsolved symbol grounding problem. The Church-Turing Thesis states that a physical problem, for which there is an algorithm of solution, can be solved by a Turing machine, but machine operations neglect the semantic relationship between symbols and their meaning. Symbols are objects that are manipulated on rules based on their shapes. The computations are independent of the context, mental states, emotions, or feelings. The (...) symbol processing operations are interpreted by the machine in a way quite different from the cognitive processes. Cognitive activities of living organisms and computation differ from each other, because of the way they act in the real word. The result is the problem of mutual understanding of symbol grounding. (shrink)

In their recent paper “Do Accelerating Turing Machines Compute the Uncomputable?” Copeland and Shagrir draw a distinction between a purist conception of Turing machines, according to which these machines are purely abstract, and Turing machine realism according to which Turing machines are spatio-temporal and causal “notional" machines. In the present response to that paper we concede the realistic aspects of Turing’s own presentation of his machines, pointed out by Copeland and Shagrir, but argue that (...) Turing's treatment of symbols in the course of that presentation opens the door for later purist conceptions. Also, we argue that a purist conception of Turing machines plays an important role not only in the analysis of the computational properties of Turing machines, but also in the philosophical debates over the nature of their realization. (shrink)

A paradigm of scientific discovery is defined within a first-order logical framework. It is shown that within this paradigm there exists a formal scientist that is Turing computable and universal in the sense that it solves every problem that any scientist can solve. It is also shown that universal scientists exist for no regular logics that extend first-order logic and satisfy the Löwenheim-Skolem condition.

A. M. Turing has bequeathed us a conceptulary including 'Turing, or Turing-Church, thesis', 'Turing machine', 'universal Turing machine', 'Turing test' and 'Turing structures', plus other unnamed achievements. These include a proof that any formal language adequate to express arithmetic contains undecidable formulas, as well as achievements in computer science, artificial intelligence, mathematics, biology, and cognitive science. Here it is argued that these achievements hang together and have prospered well in the (...) 50 years since Turing's death. (shrink)

Abstract Philosophical discussion of Alan Turing’s writings on intelligence has mostly revolved around a single point made in a paper published in the journal Mind in 1950. This is unfortunate, for Turing’s reflections on machine (artificial) intelligence, human intelligence, and the relation between them were more extensive and sophisticated. They are seen to be extremely well-considered and sound in retrospect. Recently, IBM developed a question-answering computer (Watson) that could compete against humans on the game show Jeopardy! There (...) are hopes it can be adapted to other contexts besides that game show, in the role of a collaborator of, rather than a competitor to, humans. Another, different, research project --- an artificial intelligence program put into operation in 2010 --- is the machine learning program NELL (Never Ending Language Learning), which continuously ‘learns’ by ‘reading’ massive amounts of material on millions of web pages. Both of these recent endeavors in artificial intelligence rely to some extent on the integration of human guidance and feedback at various points in the machine’s learning process. In this paper, I examine Turing’s remarks on the development of intelligence used in various kinds of search, in light of the experience gained to date on these projects. (shrink)

Turing wrote that the “guiding principle” of his investigation into the possibility of intelligent machinery was “The analogy [of machinery that might be made to show intelligent behavior] with the human brain.” [10] In his discussion of the investigations that Turing said were guided by this analogy, however, he employs a more far-reaching analogy: he eventually expands the analogy from the human brain out to “the human community as a whole.” Along the way, he takes note of an (...) obvious fact in the bigger scheme of things regarding human intelligence: grownups were once children; this leads him to imagine what a machine analogue of childhood might be. In this paper, I’ll discuss Turing’s child-machine, what he said about different ways of educating it, and what impact the “bringing up” of a child-machine has on its ability to behave in ways that might be taken for intelligent. I’ll also discuss how some of the various games he suggested humans might play with machines are related to this approach. (shrink)

This paper presents an analysis of three major contests for machine intelligence. We conclude that a new era for Turing’s test requires a fillip in the guise of a committed sponsor, not unlike DARPA, funders of the successful 2007 Urban Challenge.

What we have learnt in the last six or seven decades about virtual machinery, as a result of a great deal of science and technology, enables us to offer Darwin a new defence against critics who argued that only physical form, not mental capabilities and consciousness could be products of evolution by natural selection. The defence compares the mental phenomena mentioned by Darwin’s opponents with contents of virtual machinery in computing systems. Objects, states, events, and processes in virtual machinery which (...) we have only recently learnt how to design and build, and could not even have been thought about in Darwin’s time, can interact with the physical machinery in which they are implemented, without being identical with their physical implementation, nor mere aggregates of physical structures and processes. The existence of various kinds of virtual machinery depends on complex webs of causal connections involving hardware and software structures, events and processes, where the specification of such causal webs requires concepts that cannot be defined in terms of concepts of the physical sciences. That indefinability, plus the possibility of various kinds of self-monitoring within virtual machinery, seems to explain some of the allegedly mysterious and irreducible features of consciousness that motivated Darwin’s critics and also more recent philosophers criticising AI. There are consequences for philosophy, psychology, neuroscience and robotics. (shrink)

Recent advances in neuroscience lead to a wider realm for philosophy to include the science of the Darwinian-evolved computational brain, our inner world producing organ, a non-recursive super- Turing machine combining 100B synapsing-neuron DNA-computers based on the genetic code. The whole system is a logos machine offering a world map for global context, essential for our intentional grasp of opportunities. We start from the observable contrast between the chaotic universe vs. our orderly inner world, the noumenal cosmos. (...) So far, philosophy has been rehearsing our thoughts, our human-internal world, a grand painting of the outer world, how we comprehend subjectively our experience, worked up by the logos machine, but now we seek a wider horizon, how humans understand the world thanks to Darwinian evolution to adapt in response to the metaphysical gap, the chasm between the human animal and its environment, shaping the organism so it can deal with its variable world. This new horizon embraces global context coded in neural structures that support the noumenal cosmos, our inner mental world, for us as denizens of the outer environment. Kant’s inner and outer senses are fundamental ingredients of scientific philosophy. Several sections devoted to Heidegger, his lizard debunked, but his version of the metaphysical gap & his doctrine of the logos praised. Rorty and others of the behaviorist school discussed also. (shrink)

In this paper I discuss the topics of mechanism and algorithmicity. I emphasise that a characterisation of algorithmicity such as the Turing machine is iterative; and I argue that if the human mind can solve problems that no Turing machine can, the mind must depend on some non-iterative principle — in fact, Cantor's second principle of generation, a principle of the actual infinite rather than the potential infinite of Turing machines. But as there has been (...) theorisation that all physical systems can be represented by Turing machines, I investigate claims that seem to contradict this: specifically, claims that there are noncomputable phenomena. One conclusion I reach is that if it is believed that the human mind is more than a Turing machine, a belief in a kind of Cartesian dualist gulf between the mental and the physical is concomitant. (shrink)

The importance of the Stability Problem in neurocomputing is discussed, as well as the need for the study of infinite networks. Stability must be the key ingredient in the solution of a problem by a neural network without external intervention. Infinite discrete networks seem to be the proper objects of study for a theory of neural computability which aims at characterizing problems solvable, in principle, by a neural network. Precise definitions of such problems and their solutions are given. Some consequences (...) are explored, in particular, the neural unsolvability of the Stability Problem for neural networks. (shrink)

Putnam construed the aim of Carnap’s program of inductive logic as the specification of a “universal learning machine,” and presented a diagonal proof against the very possibility of such a thing. Yet the ideas of Solomonoff and Levin lead to a mathematical foundation of precisely those aspects of Carnap’s program that Putnam took issue with, and in particular, resurrect the notion of a universal mechanical rule for induction. In this paper, I take up the question whether the (...) Solomonoff–Levin proposal is successful in this respect. I expose the general strategy to evade Putnam’s argument, leading to a broader discussion of the outer limits of mechanized induction. I argue that this strategy ultimately still succumbs to diagonalization, reinforcing Putnam’s impossibility claim. (shrink)

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that (...) the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the L ζ -stables. It also implies that the machines devised are "Σ 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)

Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this kind can be used to understand (a) a Turing machine receiving extra computational power from a physical process, or (b) an experimenter modelled as a Turing machine performing (...) a test of a known physical theory T. Our earlier work was based upon experiments in Newtonian mechanics. Here we extend the scope of the theory of experimental oracles beyond Newtonian mechanics to electrical theory. First, we specify an experiment that measures resistance using a Wheatstone bridge and start to classify the computational power of this experimental oracle using non-uniform complexity classes. Secondly, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on our ability to measure resistance by the Wheatstone bridge. The connection between the algorithm and physical test is mediated by a protocol controlling each query, especially the physical time taken by the experimenter. In our studies we find that physical experiments have an exponential time protocol, this we formulate as a general conjecture. Our theory proposes that measurability in Physics is subject to laws which are co-lateral effects of the limits of computability and computational complexity. (shrink)

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down $\zeta$, the least ordinal not the length of any eventual output of an Infinite Time Turing machine ; using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural (...) ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the L$_\zeta$-stables. It also implies that the machines devised are "$\Sigma_2$ Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)

The diagonalization argument of Putnam denies the possibility of a universal learning machine. Yet the proposal of Solomonoff and Levin promises precisely such a thing. In this paper I discuss how their proposed measure function manages to evade Putnam's diagonalization in one respect, only to fatally fall prey to it in another.

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, (...) I present a more mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable. (shrink)

``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. (shrink)

In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw out some interesting consequences of hypercomputation for several different disciplines.