The Turing Machine as a cognitive model of human computation (original) (raw)
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Natural Computing, 2007
Turing's notion of human computability is exactly right not only for obtaining a negative solution of Hilbert's Entscheidungsproblem that is conclusive, but also for achieving a precise characterization of formal systems that is needed for the general formulation of the incompleteness theorems. The broad intellectual context reaches back to Leibniz and requires a focus on mechanical procedures; these procedures are to be carried out by human computers without invoking higher cognitive capacities. The question whether there are strictly broader notions of effectiveness has of course been asked for both cognitive and physical processes. I address this question not in any general way, but rather by focusing on aspects of mathematical reasoning that transcend mechanical procedures. Section 1 discusses Go¨del's perspective on mechanical computability as articulated in his [193?], where he drew a dramatic conclusion from the undecidability of certain Diophantine propositions, namely, that mathematicians cannot be replaced by machines. That theme is taken up in the Gibbs Lecture of 1951; Go¨del argues there in greater detail that the human mind infinitely surpasses the powers of any finite machine. An analysis of the argument is presented in Section 2 under the heading Beyond calculation. Section 3 is entitled Beyond discipline and gives Turing's view of intelligent machinery; it is devoted to the seemingly sharp conflict between Go¨del's and Turing's views on mind. Their deeper disagreement really concerns the nature of machines, and I'll end with some brief remarks on (supra-) mechanical devices in Section 4.
Philosophia: International Journal of Philosophy (2014) 15(1): 50-62
Due to his significant role in the development of computer technology and the discipline of artificial intelligence, Alan Turing has supposedly subscribed to the theory of mind that has been greatly inspired by the power of the said technology which has ev entually become the dominant framework for current researches in artificial intelligence and cognitive science, namely, computationalism or the computational theory of mind. In this essay, I challenge this supposition. In particular, I will try to show tha t there is no evidence in Turing's two seminal works that supports such a supposition. His 1936 paper is all about the notion of computation or computability as it applies to mathematical functions and not to the nature or workings of intelligence. On the other hand, while his 1950 work is about intelligence, it is, however, particularly concerned with the problem of whether intelligence can be attributed to computing machines and not of whether computationality can be attributed to human intelligence or to intelligence in general.
Turing's Analysis of Computation and Theories of Cognitive Architecture
Cognitive Science, 1998
Turing's analysis of computation is a fundamental part of the background of cognitive science. In this paper it is argued that a re-interpretation of Turing's work is required to underpin theorizing about cognitive architecture. It is claimed that the symbol systems view of the mind, which is the conventional way of understanding how Turing's work impacts on cognitive science, is deeply flawed. There is an alternative interpretation that is more faithful to Turing's original insights, avoids the criticisms made of the symbol systems approach and is compatible with the growing interest in agent-environment interaction. It is argued that this interpretation should form the basis for theories of cognitive architecture.
Is the church turing thesis a red herring for cognitive science
Society for the Study of Artificial Intelligence and Simulation of Behaviour, 2021
This paper considers whether computational formalisms beyond the Church Turing Thesis (CTT) could be helpful in understanding the mind. We argue that they may be, and that the way that the CTT has been invoked in Cognitive Science may therefore act as a Red Herring. That is, the way the CTT is invoked in Cognitive Science may mislead and perhaps contribute to premature abandonment of possibly fruitful research directions in Cognitive Science. We do not suggest some sort of "hypercomputation'. Whilst it is possible to use a rich interactive machine to implement a simple function this does not lead to new computable functions. In other words, the CTT is valid even if more sophisticated machinery is employed. It is the other direction that is the core of this paper: When considering more sophisticated computational tasks, then standard Turing machines (and their mode of operation) are not sufficient to explore the range of possibilities. The CTT is commonly interpreted as stating that the intuitive concept of computability is fully captured by Turing machines or any equivalent formalism (such as recursive functions, the lamba calculus, Post production rules, and many others). The CTT implies that if a function is (intuitively) computable, then it can be computed by a Turing machine. Conversely, if a Turing machine cannot compute a function, it is not computable by any mechanism whatsoever. We suggest an inadvertent error that has been made which is the claim that relatively simple computational formalisms like Turing Machines can do anything that more complex computional formalisms can do. To show this we present the landscape of computability within and beyond the bounds covered by the mathematical CTT. This shows that in regions of the computational landscape beyond the CTT there may be hierarchies of increasingly powerful computational formalisms. Erroneously interpreting CTT as enforcing a 'one size fits all' interpretation to computational formalisms leads to extreme reductionism that means contemporary computationalism is viewed as inadequate to explaining many phenomena related to thought and mind in living systems. Once this Red Herring interpretation for CTT is avoided this leaves the way open to exploring how richer kinds of computation that may possess many shades of expressivity can form part of Cognitive Science explanations.
Bidimensional Turing machines as Galilean models of human computation
Even though simulation models are the dominant paradigm in cognitive science, it has been argued that Galilean models might fare better on both the description and explanation of real cognitive phenomena. The main goal of this paper is to show that the actual construction of Galilean models is clearly feasible, and well suited, for a special class of cognitive phenomena, namely, those of human computation. I will argue in particular that Turing's original formulation of the Church-Turing thesis can naturally be viewed as the core hypothesis of a new empirical theory of human computation. This theory relies on bidimensional Turing machines, a generalization of ordinary machines with one-dimensional tape to two-dimensional paper. Finally, I will suggest that this theory might become a first paradigm for a general approach to the study of cognition, an approach entirely based on Galilean models of cognitive phenomena.
Does Computation Reveal Machine Cognition?
Biosemiotics, 2013
This paper seeks to understand machine cognition. The nature of machine cognition has been shrouded in incomprehensibility. We have often encountered familiar arguments in cognitive science that human cognition is still faintly understood. This paper will argue that machine cognition is far less understood than even human cognition despite the fact that a lot about computer architecture and computational operations is known. Even if there have been putative claims about the transparency of the notion of machine computations, these claims do not hold out in unraveling machine cognition, let alone machine consciousness (if there is any such thing). The nature and form of machine cognition remains further confused also because of attempts to explain human cognition in terms of computation and to model/simulate (aspects of) human cognitive processing in machines. Given that these problems in characterizing machine cognition persist, a view of machine cognition that aims to avoid these problems is outlined. The argument that is advanced is that something becomes a computation in machines only when a human interprets it, which is a kind of semiotic causation. From this it follows that a computing machine is not engaged in a computation unless a human interprets what it is doing; instead, it is engaged in machine cognition, which is defined as a member or subset of the set of all possible mappings of inputs to outputs. The human interpretation, which is a semiotic process, gives meaning to what a machine does, and then what it does becomes a computation.
Computation in cognitive science: it is not all about Turing-equivalent computation
Studies in History and Philosophy of Science Part A, 2010
One account of the history of computation might begin in the 1930's with some of the work of Alonzo Church, Alan Turing, and Emil Post. One might say that this is where something like the core concept of computation was first formally articulated. Here were the first attempts to formalize an informal notion of an algorithm or effective procedure by which a mathematician might decide one or another logico-mathematical question. As each of these formalisms was shown to compute the same set of functions-the partial recursive functions-each of them might be described as a form of Turing-equivalent computation. This work set the cornerstone for what we might call computation theory. This history might then proceed to give pride of place to this form of computation in subsequent developments in cognitive science and in related disciplines and subdisciplines. Such a history might note that, in the 1940's, the results of this work would have been transferred into the emerging field of computer science with the design and construction of the first electronic digital computers. Here one would mention Turing again, as well as perhaps Norbert Wiener, Julian Bigelow, John von Neumann, and many others. At about the same time, this theory of computation would have been inserted into the theory of neural networks by way of Warren McCulloch and Walter Pitts's seminal work, "A Logical Calculus of the Ideas Immanent in Nervous Activity." Somewhat later, during the 1960's, Hilary
The Broad Conception of Computation
American Behavioural 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, 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. 1. Introduction Turing's famous (1936) concerns the theoretical limits of what a human mathematician can compute. As is well known, Turing was able to show, in answer to a question raised by Hilbert, that there are classes of mathematical problems whose solutions cannot be discovered by a person acting solely in accordance with an algorithm. Nowadays this paper is generally taken to have achieved rather more than Turing himself seems to have intended. It is commonly regarded as a treatment-indeed, the definitive treatment-of the limits of what machines can compute. This is curious, for it is far from obvious that the theoretical limits of human computation and the theoretical limits of machine computation need coincide. Might not some machine be able to perform computations that a human being cannot-even an idealised human being who never makes mistakes and who has unlimited resources (time, scratchpad, etc.)? This is the question I shall discuss here. I shall press the claim-a claim which, as I explain, originates with Turing himself-that there are (in the mathematical sense) computing machines capable of computing more than any Turing machine. These machines form a diverse class, some being digital in nature and some analogue. For want of a better name I will refer to such computing machines as nonclassical computers. It may in the future be possible to build such machines. Moreover, it is conceivable that the brain is a nonclassical computer. This possibility has, it seems, been widely overlooked both by proponents of traditional versions of the computational theory of mind and by opponents of the view that the brain is a computer. In order to keep the discussion as self-contained as possible, I will begin by briefly reviewing some essential concepts, in particular those of a Turing machine, a (Turing-machine-) computable number and a (Turing-machine-) computable function. A Turing machine is an idealised computing device consisting of a read/write head with a paper tape passing through it. 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, serving both as the vehicle for input and output and as a working memory for storing the results of intermediate steps of the computation. The tape is of unbounded length-for Turing's aim was to show that there are tasks that these machines are unable to perform even given unlimited working memory and unlimited time. Nevertheless, Turing required that the input inscribed on the tape should consist of a finite number of symbols. Later I discuss the effect of lifting this requirement. FIG 1 ABOUT HERE ('A Turing machine'). The read/write head is programmable. It may be helpful to think of the operation of programming as consisting of altering the head's internal wiring by means of a plugboard arrangement. To compute with the device first program it, then inscribe the input on the tape (in binary or decimal code, say), place the head over the square containing the leftmost input symbol, and set the machine in motion. Once the computation is completed the machine will come to a halt with the head positioned over the square containing the leftmost symbol of the output (or elsewhere if so programmed). The head contains a subdevice that I will call the indicator. This is a second form of working memory. The indicator can be set at a number of 'positions'. In Turing machine jargon the position of the indicator at any time is called the state of the machine at that time. To give a simple example of the indicator's function, it may be used to keep track of whether the symbol last encountered was '0' or '1': if '0', the indicator is set to its first position, and if '1', to its second position. Fundamentally there are just six types of operation that a Turing machine performs in the course of a computation. It may (i) read (i.e. identify) the symbol currently under the head; (ii) write a symbol on the square currently under the head (after first deleting the symbol already written there, if any); (iii) move the tape left one square; (iv) move the tape right one square; (v) change state; (vi) halt. These are called the primitive operations of the machine. Commercially available computers are hard-wired to perform primitive operations considerably more sophisticated than those of a Turing machine-add, multiply, decrement, store-at-address, branch, and so forth. (The precise constitution of the list of primitives varies from manufacturer to manufacturer.) However, the remarkable fact is that none of these machines can outdo a Turing machine: despite its austere simplicity a Turing machine is capable of computing anything that e.g. a to omit an instruction to halt. Computer programs that never terminate by design are commonplace. Air traffic control systems, automated teller machine networks, and nuclear reactor control systems are all examples of such. A non-terminating Turing machine program that is of importance for the present discussion consists of a list of instructions for calculating sequentially each digit of the decimal representation of π (say by using one of the standard power series expressions for π). A Turing machine that is set up to loop repeatedly through these instructions will spend all eternity writing out the decimal representation of π digit by digit, 3.14159. .. Turing called the numbers that can be written out in this way by a Turing machine the computable numbers. That is, a number is computable, in Turing's sense, if and only if there is a Turing machine that calculates in sequence each digit of the number's decimal representation. (There is nothing special about decimal representation here: I use it because everyone is familiar with it. This necessary and sufficient condition can equally well be stated with 'binary representation', for example, in place of 'decimal representation'.) Straight off, one might expect it to be the case that every number that has a decimal representation, either finite or infinite-that is, every real number-is computable. (The real numbers comprise the integers, the rational numbers-which is to say, numbers that can be expressed as a ratio of integers, for example 1 / 2 and 3 / 4-and the irrational numbers, such as √2 and π, which cannot be expressed as a ratio of integers.) For what could prevent there being, for each particular real number, a Turing machine that 'churns out' that number's decimal representation digit by digit? However, Turing did indeed show that not every real number is computable. The decimal representations of some real numbers are so completely lacking in pattern that there simply is no finite list of instructions, of the sort that can be followed by a finite-input Turing machine, for calculating the n th digit of the representation for arbitrary n. Indeed, most real numbers are like this. There are only countably many computable numbers whereas, as Cantor showed last century, there are uncountably many real numbers. (A set is countable if and only if either it is finite or its members can be put into a one-to-one correspondence with the integers.) It is common to speak of Turing machines computing functions (in the mathematical sense of 'function', not the biological). These may be functions over any objects that are capable of being Henceforward I will abbreviate 'computable in Turing's sense' by 'Computable' (and similarly 'Uncomputable'). By the unqualified expression 'computable' I will always mean 'computable by some machine'. The nonclassical nature of the computing machines described in the next section arises from the fact that addition over the real numbers, as opposed to addition over the Computable real numbers, is not a Computable function. This is easy to establish. Let z be an Uncomputable real. Many pairs of numbers will sum to z; let x and y be one such pair. Since z is not Computable nor is x+y: no Turing machine can be in the process of churning out the decimal representation of x+y. (There is, of course, the additional difficulty, previously discussed, of entering x and y as input, since at least one of them must itself be Uncomputable.) Not all functions over the Computable numbers are Computable. The task of computing the values of such a function is beyond a Turing machine, even though the function's arguments (or inputs), being Computable numbers, can be represented on the machine's tape by means of finitely many symbols. Turing gave a famous example of such a function, the so-called halting function, which I will discuss in section 3. In the meantime, it will be useful at this stage to introduce, without proof, a rather simple example of one of these functions, to be written E(x,y). Where x and y are any Computable numbers, E(x,y)=1 if and only if x=y, and E(x,y)=0 if...
Computation and cognition: Issues in the foundations of cognitive science
Behavioral and Brain Sciences, 1980
Abstract: The computational view of mind rests on certain intuitions regarding the fundamental similarity between computation and cognition. We examine some of these intuitions and suggest that they derive from the fact that computers and human organisms ...