Physical Church-Turing Thesis

THE CHURCH-TURING THESIS (CTT) underlies tantalizing open questions concerning the fundamental place of computing in the physical universe. For example, is every physical system computable? Is the universe essentially computational in nature? What are the implications for computer science of recent speculation about physical uncomputability? Does CTT place a fundamental logical limit on what can be computed, a computational "barrier" that cannot be broken, no matter how far and in what multitude of ways computers develop? Or could new types of hardware, based perhaps on quantum or relativistic phenomena, lead to radically new computing paradigms that do breach the Church-Turing barrier, in which the uncomputable becomes computable , in an upgraded sense of "com-putable"? Before addressing these questions , we first look back to the 1930s to consider how Alonzo Church and Alan Turing formulated, and sought to justify , their versions of CTT. With this necessary history under our belts, we then turn to today's dramatically more powerful versions of CTT.

«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.»

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.

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.

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.

I argue that the importance of hypercomputation for the philosophy of mathematics has not yet been recognized.  In the article I explore a thought experiment, in which we have a hypercomputational device H at our disposal. This means, in particular, that I assume the negation of the physical version of Church-Turing thesis. I discuss it in the context of the problem of understanding and explanation in mathematics - and its possible impact on the notion of mathematical knowledge.