Oron Shagrir

Physical Hypercomputation and the Church–Turing Thesis

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 Gandys thesis.

Why we view the brain as a computer

The view that the brain is a sort of computer has functioned as a theoretical guideline both in cognitive science and, more recently, in neuroscience. But since we can view every physical system as a computer, it has been less than clear what this view amounts to. By considering in some detail a seminal study in computational neuroscience, I first suggest that neuroscientists invoke the computational outlook to explain regularities that are formulated in terms of the information content of electrical signals. I then indicate why computational theories have explanatory force with respect to these regularities:in a nutshell, they underscore correspondence relations between formal/mathematical properties of the electrical signals and formal/mathematical properties of the represented objects. I finally link my proposal to the philosophical thesis that content plays an essential role in computational taxonomy.

Effective Computation by Humans and Machines

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 Gandys 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.

Marr on Computational-Level Theories*

According to Marr, a computational-level theory consists of two elements, the what and the why. This article highlights the distinct role of the Why element in the computational analysis of vision. Three theses are advanced: (a) that the Why element plays an explanatory role in computational-level theories, (b) that its goal is to explain why the computed function (specified by the What element) is appropriate for a given visual task, and (c) that the explanation consists in showing that the functional relations between the representing cells are similar to the “external” mathematical relations between the entities that these cells represent.

Two Dogmas of Computationalism

This paper challenges two orthodox theses: (a) that computational processes must be algorithmic; and (b) that all computed functions must be Turing-computable. Section 2 advances the claim that the works in computability theory, including Turings analysis of the effective computable functions, do not substantiate the two theses. It is then shown (Section 3) that we can describe a system that computes a number-theoretic function which is not Turing-computable. The argument against the first thesis proceeds in two stages. It is first shown (Section 4) that whether a process is algorithmic depends on the way we describe the process. It is then argued (Section 5) that systems compute even if their processes are not described as algorithmic. The paper concludes with a suggestion for a semantic approach to computation.

Global supervenience, coincident entities and anti-individualism

Theodore Sider distinguishes two notions of global supervenience: strong global supervenience and weak global supervenience. He then discusses some applications to general metaphysical questions. Most interestingly, Sider employs the weak notion in order to undermine a familiar argument against coincident distinct entities. In what follows, I reexamine the two notions and distinguish them from a third, intermediate, notion (intermediate global supervenience). I argue that (a) weak global supervenience is not an adequate notion of dependence; (b) weak global supervenience does not capture certain assumptions about coincidence relations; (c) these assumptions are better accommodated by the stronger notion of intermediate global supervenience; (d) intermediate global supervenience, however, is also not an adequate notion of dependence; and (e) strong global supervenience is an adequate notion of dependence. It also fits in with anti-individualism about the mental. It does not, however, serve to rebut arguments against coincident entities.

Multiple Realization, Computation and the Taxonomy of Psychological States

The paper criticizes standard functionalist arguments for multiple realization. It focuses on arguments in which psychological states are conceived as computational, which is precisely where the multiple realization doctrine has seemed the strongest. It is argued that a type-type identity thesis between computational states and physical states is no less plausible than a multiple realization thesis. The paper also presents, more tentatively, positive arguments for a picture of local reduction.

Structural Representations and the Brain

In Representation Reconsidered, William Ramsey suggests that the notion of structural representation is posited by classical theories of cognition, but not by the ‘newer accounts’ (e.g. connectionist modeling). I challenge the assertion about the newer accounts. I argue that the newer accounts also posit structural representations; in fact, the notion plays a key theoretical role in the current computational approaches in cognitive neuroscience. The argument rests on a close examination of computational work on the oculomotor system. 1 Introduction 2 Structural Representation 3 Ramseys Argument 4 A Brief Review of Computational Work on Oculomotor Memory 5 The Memory Network as Structural Representation 6 Objection 1: The Memory States are not Internal Representations 7 Objection 2: The Memory States are not Real Representations 8 Summary 1 Introduction 2 Structural Representation 3 Ramseys Argument 4 A Brief Review of Computational Work on Oculomotor Memory 5 The Memory Network as Structural Representation 6 Objection 1: The Memory States are not Internal Representations 7 Objection 2: The Memory States are not Real Representations 8 Summary

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.

Super-tasks, accelerating Turing machines and uncomputability

Accelerating Turing machines are devices with the same computational structure as Turing machines (TM), but able to perform super-tasks. We ask whether performing super-tasks alone produces more computational power; for example, whether accelerating TM can solve the halting problem. We conclude that this is not the case. No accelerating TM solves the halting problem. The argument rests on an analysis of the reasoning that leads to Thomsons paradox. The key point is that the paradox rests on a conflation of different perspectives of accelerating processes. This leads to concluding that the same conflation underlies the claim that accelerating TM can solve the halting problem.