Church-Turing Thesis

A myth has arisen concerning Turing's 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 underc...

The burgeoning field of digital physics is based on the fact that physical processes are thoroughly computable, with the laws of nature acting as algorithms taking the present state of a physical system as input and producing the next state as output. In this paper I will be concerned with one of the most fundamental problems in digital physics: the problem of the hardware or – more generally – of the computing platform, i.e. the pre-existing environment that facilitates the process of computation. If physical processes are computations, if the entire universe is computational, what then is the "cosmic computer" underlying the universe, what is the hardware or platform on which the computations run? In this paper I will argue for an idealist solution to this 'platform problem' in digital physics, i.e. a solution that crucially involves self-consciousness as the ontological foundation of reality. Here Royce's mathematical model of absolute self-consciousness will prove very useful. Focusing on the recursive structure of self-consciousness (i.e. its awareness of itself, and its awareness of that awareness, and its awareness of the awareness of its awareness, and so on), Royce argued that self-consciousness exhibits the same recursion that defines the natural number system N. This then allows us to describe absolute self-consciousness as an awareness of N and thereby also of all possible mappings from N to N – i.e. an awareness of all possible algorithms, including the complex computation that constitutes our universe. Our universe can then be – tentatively, of course – explained as that complex computation that maximizes the absolute's self-consciousness insofar as it is the universe that is most conducive to the evolution of life and intelligence.

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We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language.

We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'.

We then adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective.

We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways.

We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N.

We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences.

In many philosophical discussions, it is assumed that the computational explanation of the mind implies that it is being explained as a Universal Turing Machine (UTM). The reason why it is being proposed as a model of the mind is that it is the standard model of computation, and that is a universal machine – i.e., any other digital computer might be simulated by the UTM. So if the mind is a computer that is not able to compute everything that the UTM can, the UTM might still simulate it.�In recent years, several criticisms to such a proposal have been raised. First, it was argued that the UTM requires an infinite tape, which is something that is impossible physically (or impossible as a physical part of the brain). Second, it was argued that the UTM is not a good candidate for explaining the intricacies of the human mind as it has a completely different architecture: so, while the set of functions computed by the machine could be the same as the one computed by the mind, it would differ dramatically in terms of speed and space requirements. Third, some have claimed that there are physical systems capable of hyper-Turing computation, so the UTM might not be the strongest model of computation available. I focus on the epistemic question: how could one tell that a physical system is a UTM? I will distinguish two senses in which one could say that a physical system “is” a UTM: (1) when a physical system has a function that could be simulated with a UTM (functional sense); (2) when a physical system is a mechanism best described as a UTM (mechanistic sense). I will enumerate conditions that must be fulfilled to qualify a physical system to be a UTM in both senses and show differences between the two. It will turn out that in the functional sense, many physical systems might be UTMs, whereas in the mechanistic sense, the set will be much restricted. I will also discuss the problem of finiteness: it seems that for many algorithms, the tape length is restricted anyway (especially if the restriction of the input value range is the part of the algorithm), so the finiteness might not be the biggest problem. Moreover, in the mechanistic sense, one does not select the model of computation simply by picking the strongest model possible, so the hyper-Turing models must also match the mental mechanisms as strictly as any other descriptions of mechanisms that we posit in science.�The mind does not seem to be a UTM in the mechanistic sense at all because of its architecture; and proper computational explanations in cognitive science require that the architecture is matched strictly. I will discuss the problem of the required level of detail in mechanistic computational explanations: it transpires that the philosophically popular UTM is not a good candidate for a scientific model of the mind, even if we accept the standard computational theory of mind.

Este trabajo se centra en la discusión sobre la disyunción de Gödel, así como las posiciones antimecanicistas que surgen de los teoremas de Gödel, en el escenario actual de la pluralidad de la lógica. El trabajo está motivado por dos ideas, en primer lugar, que lo que prueban las posiciones antimecanicistas de Lucas y Penrose no es la afirmación de que la mente humana no puede ser capturada por ninguna máquina, sino que no puede ser capturada por el tipo de cálculo que lleva una máquina de Turing . Segundo, aunque los resultados limitantes de Gödel son de gran importancia en el campo de la aritmética clásica, cuando pensamos en la mente matemática humana, las paradojas e inconsistencias entran en escena y nos motivan a cambiar la lógica. Por lo tanto, sostengo que la ruta paraconsistente es aceptable en esta discusión.

Gli algoritmi sono metodi per la soluzione di problemi. Possiamo caratterizzare un problema mediante i dati di cui si dispone all'inizio e dei risultati che si vogliono ottenere: risolvere un problema significa ottenere in uscita i risultati desiderati a partire da un certo insieme di dati presi in ingresso. I dati in ingresso vengono anche detti (valori in) input ei risultati in uscita (valori in) output. Possiamo assumere che ciascun problema consista di un insieme di casi particolari, o istanze.

«Adam Olszewski (1999) propone que la Tesis de Church-Turing puede ser usada para refutar el platonismo matemático1. Para ello postula una máquina que al lanzar una moneda define una función que ―computa el valor (0 o 1) para la moneda n que se haya lanzado y dice que dicha función es efectivamente computable pero no Turing-computable.

Entonces, dado que según el platonismo matemático (PM), toda función de enteros positivos a enteros positivos ya existe, existiría una función efectivamente computable pero no Turing-computable (computable por una máquina de Turing).

Esto contradice a la Tesis de Church-Turing (CT) que dice que toda función efectivamente computable es Turing-computable. Por contraposición, si CT es verdadera, entonces PM es falso. Rafał Urbaniak (2011) critica este argumento desafiando la afirmación de que lo que esta máquina de hecho realiza sea una computación. Revisaré dicha crítica y detallaré algunos puntos de la misma. Además introduciré la noción de Tesis de Church-Turing Física.»

I show—contrary to common beliefs tolerated by the “bosses”—that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but ω-inconsistent; that a sound finitary interpretation of ...

There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation.

In his famous paper, An Unsolvable Problem of Elementary Number Theory , Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his general philosophical thoughts on mathematics.

I show--contrary to common beliefs tolerated by the 'bosses'--that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but omega-inconsistent; that a sound finitary interpretation of PA is definable in terms of Turing-computability; and that PA cannot be consistently extended to ZF.

We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church-Turing thesis, but nevertheless may be a counterexample to Gandy’s thesis.

There is something distressing in the fact that this book, coauthored
by a reputable logician, published by a reputable press and
favorably reviewed by reputable reviewers, is nevertheless so marred
that it cannot begin to serve its avowed purpose. The claims made by
the authors, publishers and reviewers amount to an elaborate and
cruel hoax. Where the reader is lead to expect clarity, insight and
rationality instead he finds himself in a largely indecipherable swamp
scattered with bizarre little islands of Kafkaesque puzzles and Alicein-
Wonderland meaning shifts.

I show--contrary to common beliefs tolerated by the 'bosses'--that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but omega-inconsistent; that a sound finitary interpretation of PA is definable in terms of Turing-computability; and that PA cannot be consistently extended to ZF.

We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_1, x_2) is representable in the first-order Peano Arithmetic PA by a formula [F(x_1, x_2, x_3)] which is algorithmically verifiable, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence.

This paper develops my (BJPS 2009) criticisms of the philosophical significance of a certain sort of infinitary computational process, a hyperloop. I start by considering whether hyperloops suggest that "effectively computable" is vague (in some sense). I then consider and criticise two arguments by Hogarth, who maintains that hyperloops undermine the very idea of effective computability. I conclude that hyperloops, on their own, cannot threaten the notion of an effective procedure.

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

What are the limits of physical computation? In his 'Church's Thesis and Principles for Mechanisms', Turing's student Robin Gandy proved that any machine satisfying four idealised physical 'principles' is equivalent to some Turing machine. Gandy's four principles in effect define a class of computing machines ('Gandy machines'). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines?  Gandy himself suggests that the relationship is identity. We do not share this view. We will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing-machine computable.

Recent work on hypercomputation has raised new objections against the Church-Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These barriers suggest several ways to defend a Physical version of the Church-Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing that it is a non-sequitur. I conclude that the Effective version of the Church-Turing Thesis is unaffected by SAD computation.

We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm

G. Priest's anti-consistency argument (Priest 1979, 1984, 1987) and J. R.
Lucas's anti-mechanist argument (Lucas 1961, 1968, 1970, 1984) both appeal to G6del
incompleteness. By way of refuting them, this paper defends the thesis of quartet compatibility,
viz., that the logic of the mind can simultaneously be GSdel incomplete, consistent,
mechanical, and recursion complete (capable of all means of recursion). A representational
approach is pursued, which owes its origin to works by, among others, J. Myhill
(1964), P. Benacerraf (1967), J. Webb (1980, 1983) and M. Arbib (1987). It is shown
that the fallacy shared by the two arguments under discussion lies in misidentifying two
systems, the one for which the GSdel sentence is constructable and to be proved, and
the other in which the GSdel sentence in question is indeed provable. It follows that the
logic of the mind can surpass its own GSdelian limitation not by being inconsistent or
non-mechanistic, but by being capable of representing stronger systems in itself; and so
can a proper machine. The concepts of representational provability, representational
maximality, formal system capacity, etc., are discussed.

... Minds and Machines 12: 303–326, 2002. ... The keying is performed by counting variables to n. Indexing depends on oracular information: the Turing index for the template (see Section 4) used to build the Mn and the first n items from the oracle are sent to an internal ...