Generalized Gluon Currents and Applications in QCD

We consider the process containing two quark lines and an arbitrary number of gluons in a spinor helicity framework. A current with two off-shell gluons appears in the amplitude. We first study this modified gluon current using recursion relations. The recursion relation for the modified gluon current is solved for the case of like-helicity gluons. We apply the modified gluon current to compute the amplitude for $q \bar q \rightarrow q \bar q gg \cdots g$ in the like-helicity gluon case.Comment: 80 pages, 2 figures (appended in pictex), CLNS 91/112

On Bar Recursive Interpretations of Analysis.

PhDThis dissertation concerns the computational interpretation of analysis via proof interpretations, and examines the variants of bar recursion that have been used to interpret the axiom of choice. It consists of an applied and a theoretical component. The applied part contains a series of case studies which address the issue of understanding the meaning and behaviour of bar recursive programs extracted from proofs in analysis. Taking as a starting point recent work of Escardo and Oliva on the product of selection functions, solutions to Godel's functional interpretation of several well known theorems of mathematics are given, and the semantics of the extracted programs described. In particular, new game-theoretic computational interpretations are found for weak Konig's lemma for 01 -trees and for the minimal-bad-sequence argument. On the theoretical side several new definability results which relate various modes of bar recursion are established. First, a hierarchy of fragments of system T based on finite bar recursion are defined, and it is shown that these fragments are in one-to-one correspondence with the usual fragments based on primitive recursion. Secondly, it is shown that the so called `special' variant of Spector's bar recursion actually defines the general one. Finally, it is proved that modified bar recursion (in the form of the implicitly controlled product of selection functions), open recursion, update recursion and the Berardi-Bezem- Coquand realizer for countable choice are all primitive recursively equivalent in the model of continuous functionals.EPSR

Strong normalisation for applied lambda calculi

We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations' are. From this we derive a general normalisation theorem for applied typed lambda-calculi: If all constants have a total value, then all typeable terms are strongly normalising. We apply this result to extensions of G\"odel's system T and system F extended by various forms of bar recursion for which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis

Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers