Effectivity and reducibility with ordinal Turing machines

This article expands our work in (LNCS 9709 (2016), 225–233). By its reliance on Turing computability, the classical theory of effectivity, along with effective reducibility and Weihrauch reducibility, is only applicable to objects that are either countable or can be encoded by countable objects. We propose a notion of effectivity based on Koepke’s Ordinal Turing Machines (OTMs) that applies to arbitrary set-theoretical Π 2 -statements, along with according variants of effective reducibility and Weihrauch reducibility. As a sample application, we compare various choice principles with respect to effectivity.

An introduction to feedback Turing computability

Feedback computability is computation with an oracle that contains the correct convergence/divergence information for all computations calling that same oracle. Here we study feedback Turing computability, as well as feedback for some smaller classes of computation. We also examine some versions of parallelization of these notions.

Church’s thesis

An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A eventually produces fn as output. A function is ‘computable’ if there is an algorithm that computes it. A set S is ‘decidable’ if there is an algorithm that decides membership in S: if, given any appropriate n as input, the algorithm would output ‘yes’ if n∈S, and ‘no’ if n∉S. The notions of ‘algorithm’, ‘computable’ and ‘decidable’ are informal (or pre-formal) in that they have meaning independently of, and prior to, attempts at rigorous formulation. Church’s thesis, first proposed by Alonzo Church in a paper published in 1936, is the assertion that a function is computable if and only if it is recursive: ‘We now define the notion…of an effectively calculable function…by identifying it with the notion of a recursive function….’ Independently, Alan Turing argued that a function is computable if and only if there is a Turing machine that computes it; and he showed that a function is Turing-computable if and only if it is recursive. Church’s thesis is widely accepted today. Since an algorithm can be ‘read off’ a recursive derivation, every recursive function is computable. Three types of ‘evidence’ have been cited for the converse. First, every algorithm that has been examined has been shown to compute a recursive function. Second, Turing, Church and others provided analyses of the moves available to a person following a mechanical procedure, arguing that everything can be simulated by a Turing machine, a recursive derivation, and so on. The third consideration is ‘confluence’. Several different characterizations, developed more or less independently, have been shown to be coextensive, suggesting that all of them are on target. The list includes recursiveness, Turing computability, Herbrand–Gödel derivability, λ-definability and Markov algorithm computability.

Computability theory

The effective calculability of number-theoretic functions such as addition and multiplication has always been recognized, and for that judgment a rigorous notion of ‘computable function’ is not required. A sharp mathematical concept was defined only in the twentieth century, when issues including the decision problem for predicate logic required a precise delimitation of functions that can be viewed as effectively calculable. Predicate logic emerged from Frege’s fundamental ‘Begriffsschrift’ (1879) as an expressive formal language and was described with mathematical precision by Hilbert in lectures given during the winter of 1917–18. The logical calculus Frege had also developed allowed proofs to proceed as computations in accordance with a fixed set of rules; in principle, according to Gödel, the rules could be applied ‘by someone who knew nothing about mathematics, or by a machine’. Hilbert grasped the potential of this mechanical aspect and formulated the decision problem for predicate logic as follows: ‘The Entscheidungsproblem [decision problem] is solved if one knows a procedure that permits the decision concerning the validity, respectively, satisfiability of a given logical expression by a finite number of operations.’ Some, for example, von Neumann (1927), believed that the inherent freedom of mathematical thought provided a sufficient reason to expect a negative solution to the problem. But how could a proof of undecidability be given? The unsolvability results of other mathematical problems had always been established relative to a determinate class of admissible operations, for example, the impossibility of doubling the cube relative to ruler and compass constructions. A negative solution to the decision problem obviously required the characterization of ‘effectively calculable functions’. For two other important issues a characterization of that informal notion was also needed, namely, the general formulation of the incompleteness theorems and the effective unsolvability of mathematical problems (for example, of Hilbert’s tenth problem). The first task of computability theory was thus to answer the question ‘What is a precise notion of "effectively calculable function"?’. Many different answers invariably characterized the same class of number-theoretic functions: the partial recursive ones. Today recursiveness or, equivalently, Turing computability is considered to be the precise mathematical counterpart to ‘effective calculability’. Relative to these notions undecidability results have been established, in particular, the undecidability of the decision problem for predicate logic. The notions are idealized in the sense that no time or space limitations are imposed on the calculations; the concept of ‘feasibility’ is crucial in computer science when trying to capture the subclass of recursive functions whose values can actually be determined.

SAD computers and two versions of the Church–Turing Thesis

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.Published in British Journal for the Philosophy of Science 60.4: 765–92.