Thomas Powell

On Spector's bar recursion

We show that Spectors “restricted” form of bar recursion is sufficient (over system T) to define Spectors search functional. This new result is then used to show that Spectors restricted form of bar recursion is in fact as general as the supposedly more general form of bar recursion. Given that these two forms of bar recursion correspond to the (explicitly controlled) iterated products of selection function and quantifiers, it follows that this iterated product of selection functions is T-equivalent to the corresponding iterated product of quantifiers.

System T and the Product of Selection Functions

We show that the finite product of selection functions (for all finite types) is primitive recursively equivalent to Goedels higher-type recursor (for all finite types). The correspondence is shown to hold for similar restricted fragments of both systems: The recursor for type level n+1 is primitive recursively equivalent to the finite product of selection functions of type level n. Whereas the recursor directly interprets induction, we show that other classical arithmetical principles such as bounded collection and finite choice are more naturally interpreted via the product of selection functions.

A Game-Theoretic Computational Interpretation of Proofs in Classical Analysis

It has been shown by Escardo and the first author that a functional interpretation of proofs in analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in sequential games. We argue that this result has genuine practical value by interpreting some well-known theorems of mathematics and demonstrating that the product gives these theorems a natural computational interpretation that can be clearly understood in game theoretic terms.

A functional interpretation with state

We present a new variant of Gödels functional interpretation in which extracted programs, rather than being pure terms of system T, interact with a global state. The purpose of the state is to store relevant information about the underlying mathematical environment. Because the validity of extracted programs can depend on the validity of the state, this offers us an alternative way of dealing with the contraction problem. Furthermore, this new formulation of the functional interpretation gives us a clear semantic insight into the computational content of proofs, and provides us with a way of improving the efficiency of extracted programs.

Bar recursion over finite partial functions

Abstract We introduce a new, demand-driven variant of Spectors bar recursion in the spirit of the Berardi–Bezem–Coquand functional of [4] . The recursion takes place over finite partial functions u, where the control parameter ω, used in Spectors bar recursion to terminate the computation at sequences s satisfying ω ( s ˆ ) | s | , now acts as a guide for deciding exactly where to make bar recursive updates, terminating the computation whenever ω ( u ˆ ) ∈ dom ( u ) . We begin by exploring theoretical aspects of this new form of recursion, then in the main part of the paper we show that demand-driven bar recursion can be directly used to give an alternative functional interpretation of classical countable choice. We provide a short case study as an illustration, in which we extract a new bar recursive program from the proof that there is no injection from N → N to N , and compare this with the program that would be obtained using Spectors original variant. We conclude by formally establishing that our new bar recursor is primitive recursively equivalent to the original Spector bar recursion, and thus defines the same class of functionals when added to Godels system T .

Gödel's functional interpretation and the concept of learning

In this article we study Gödel’s functional interpretation from the perspective of learning. We define the notion of a learning algorithm, and show that intuitive realizers of the functional interpretation of both induction and various comprehension schemas can be given in terms of these algorithms. In the case of arithmetical comprehension, we clarify how our learning realizers compare to those obtained traditionally using bar recursion, demonstrating that bar recursive interpretations of comprehension correspond to ‘forgetful’ learning algorithms. The main purpose of this work is to gain a deeper insight into the semantics of programs extracted using the functional interpretation. However, in doing so we also aim to better understand how it relates to other interpretations of classical logic for which the notion of learning is inbuilt, such as Hilbert’s epsilon calculus or the more recent learning-based realizability interpretations of Aschieri and Berardi.

On the Computational Content of Termination Proofs

Given that a program has been shown to terminate using a particular proof, it is natural to ask what we can infer about its complexity. In this paper we outline a new approach to tackling this question in the context of term rewrite systems and recursive path orders. From an inductive proof that recursive path orders are well-founded, we extract an explicit realiser which bounds the derivational complexity of rewrite systems compatible with these orders. We demonstrate that by analysing our realiser we are able to derive, in a completely uniform manner, a number of results on the relationship between the strength of path orders and the bounds they induce on complexity.