Process Realizability

We develop a notion of realizability for Classical Linear Logic based on a concurrent process calculus.Comment: Appeared in Foundations of Secure Computation: Proceedings of the 1999 Marktoberdorf Summer School, F. L. Bauer and R. Steinbruggen, eds. (IOS Press) 2000, 167-18

Natural Factors of the Medvedev Lattice Capturing IPC

Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic (IPC). However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that for every non-trivial factor of the Medvedev lattice its theory is contained in Jankov's logic, the deductive closure of IPC plus the weak law of the excluded middle. This answers a question by Sorbi and Terwijn

Unifying Functional Interpretations: Past and Future

This article surveys work done in the last six years on the unification of various functional interpretations including G\"odel's dialectica interpretation, its Diller-Nahm variant, Kreisel modified realizability, Stein's family of functional interpretations, functional interpretations "with truth", and bounded functional interpretations. Our goal in the present paper is twofold: (1) to look back and single out the main lessons learnt so far, and (2) to look forward and list several open questions and possible directions for further research.Comment: 18 page

Linear Realisability Over Nets and Second Order Quantification

We present a new realisability model based on othogonality for Linear Logic in the context of nets – untyped proof structures with generalized axiom. We show that it adequately models second order multiplicative linear logic. As usual, not all realizers are representations of a proof, but we identify specific types (sets of nets closed under bi-othogonality) that capture exactly the proofs of a given sequent. Furthermore these types are orthogonal’s of finite sets; this ensures the existence of a correctnesss criterion that runs in finite time. In particular, in the well known case of multiplicative linear logic, the types capturing the proofs are generated by the tests of Danos-Regnier, we provide - to our knowledge - the first proof of the folklore result which states ”test of a formula are proofs of its negation”