156 research outputs found
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
Strong Normalization for HA + EM1 by Non-Deterministic Choice
We study the strong normalization of a new Curry-Howard correspondence for HA
+ EM1, constructive Heyting Arithmetic with the excluded middle on
Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda
calculus plus an operator ||_a which represents, from the viewpoint of
programming, an exception operator with a delimited scope, and from the
viewpoint of logic, a restricted version of the excluded middle. We give a
strong normalization proof for the system based on a technique of
"non-deterministic immersion".Comment: In Proceedings COS 2013, arXiv:1309.092
A functional interpretation for nonstandard arithmetic
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles
Interactive Realizability for Classical Peano Arithmetic with Skolem Axioms
Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is an approximation of some Skolem functions. The realizer interacts with the environment, which may provide a counter-proof, a counterexample invalidating the current construction of the realizer. But the realizer is always able to turn such a negative outcome into a positive information, which consists in some new piece of knowledge learned about the mentioned Skolem functions. The aim of this work is to extend Interactive realizability to a system which includes classical first-order Peano Arithmetic with Skolem axioms. For witness extraction, the learning capabilities of realizers will be exploited according to the paradigm of learning by levels. In particular, realizers of atomic formulas will be update procedures in the sense of Avigad and thus will be understood as stratified-learning algorithms
Knowledge Spaces and the Completeness of Learning Strategies
We propose a theory of learning aimed to formalize some ideas underlying
Coquand's game semantics and Krivine's realizability of classical logic. We
introduce a notion of knowledge state together with a new topology, capturing
finite positive and negative information that guides a learning strategy. We
use a leading example to illustrate how non-constructive proofs lead to
continuous and effective learning strategies over knowledge spaces, and prove
that our learning semantics is sound and complete w.r.t. classical truth, as it
is the case for Coquand's and Krivine's approaches
- …