2,286 research outputs found
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
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Existential witness extraction in classical realizability and via a negative translation
We show how to extract existential witnesses from classical proofs using
Krivine's classical realizability---where classical proofs are interpreted as
lambda-terms with the call/cc control operator. We first recall the basic
framework of classical realizability (in classical second-order arithmetic) and
show how to extend it with primitive numerals for faster computations. Then we
show how to perform witness extraction in this framework, by discussing several
techniques depending on the shape of the existential formula. In particular, we
show that in the Sigma01-case, Krivine's witness extraction method reduces to
Friedman's through a well-suited negative translation to intuitionistic
second-order arithmetic. Finally we discuss the advantages of using call/cc
rather than a negative translation, especially from the point of view of an
implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS),
201
Analysis and Verification of Service Interaction Protocols - A Brief Survey
Modeling and analysis of interactions among services is a crucial issue in
Service-Oriented Computing. Composing Web services is a complicated task which
requires techniques and tools to verify that the new system will behave
correctly. In this paper, we first overview some formal models proposed in the
literature to describe services. Second, we give a brief survey of verification
techniques that can be used to analyse services and their interaction. Last, we
focus on the realizability and conformance of choreographies.Comment: In Proceedings TAV-WEB 2010, arXiv:1009.330
Branched covers of the sphere and the prime-degree conjecture
To a branched cover between closed, connected and orientable surfaces one
associates a "branch datum", which consists of the two surfaces, the total
degree d, and the partitions of d given by the collections of local degrees
over the branching points. This datum must satisfy the Riemann-Hurwitz formula.
A "candidate surface cover" is an abstract branch datum, a priori not coming
from a branched cover, but satisfying the Riemann-Hurwitz formula. The old
Hurwitz problem asks which candidate surface covers are realizable by branched
covers. It is now known that all candidate covers are realizable when the
candidate covered surface has positive genus, but not all are when it is the
2-sphere. However a long-standing conjecture asserts that candidate covers with
prime degree are realizable. To a candidate surface cover one can associate one
Y -> X between 2-orbifolds, and in a previous paper we have completely analyzed
the candidate surface covers such that either X is bad, spherical, or
Euclidean, or both X and Y are rigid hyperbolic orbifolds, thus also providing
strong supporting evidence for the prime-degree conjecture. In this paper,
using a variety of different techniques, we continue this analysis, carrying it
out completely for the case where X is hyperbolic and rigid and Y has a
2-dimensional Teichmueller space. We find many more realizable and
non-realizable candidate covers, providing more support for the prime-degree
conjecture.Comment: Some slips in the first version have been corrected, and a reference
to the omitted proofs now fully available online has been added; 44 pages, 14
figure
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