717 research outputs found
Dialectica Interpretation with Marked Counterexamples
Goedel's functional "Dialectica" interpretation can be used to extract
functional programs from non-constructive proofs in arithmetic by employing two
sorts of higher-order witnessing terms: positive realisers and negative
counterexamples. In the original interpretation decidability of atoms is
required to compute the correct counterexample from a set of candidates. When
combined with recursion, this choice needs to be made for every step in the
extracted program, however, in some special cases the decision on negative
witnesses can be calculated only once. We present a variant of the
interpretation in which the time complexity of extracted programs can be
improved by marking the chosen witness and thus avoiding recomputation. The
achieved effect is similar to using an abortive control operator to interpret
computational content of non-constructive principles.Comment: In Proceedings CL&C 2010, arXiv:1101.520
Functional interpretation and inductive definitions
Extending G\"odel's \emph{Dialectica} interpretation, we provide a functional
interpretation of classical theories of positive arithmetic inductive
definitions, reducing them to theories of finite-type functionals defined using
transfinite recursion on well-founded trees.Comment: minor corrections and change
Computational interpretations of analysis via products of selection functions
We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions
Metastability in the Furstenberg-Zimmer tower
According to the Furstenberg-Zimmer structure theorem, every
measure-preserving system has a maximal distal factor, and is weak mixing
relative to that factor. Furstenberg and Katznelson used this structural
analysis of measure-preserving systems to provide a perspicuous proof of
Szemer\'edi's theorem. Beleznay and Foreman showed that, in general, the
transfinite construction of the maximal distal factor of a separable
measure-preserving system can extend arbitrarily far into the countable
ordinals. Here we show that the Furstenberg-Katznelson proof does not require
the full strength of the maximal distal factor, in the sense that the proof
only depends on a combinatorial weakening of its properties. We show that this
combinatorially weaker property obtains fairly low in the transfinite
construction, namely, by the th level
A proof of strong normalisation using domain theory
Ulrich Berger presented a powerful proof of strong normalisation using
domains, in particular it simplifies significantly Tait's proof of strong
normalisation of Spector's bar recursion. The main contribution of this paper
is to show that, using ideas from intersection types and Martin-Lof's domain
interpretation of type theory one can in turn simplify further U. Berger's
argument. We build a domain model for an untyped programming language where U.
Berger has an interpretation only for typed terms or alternatively has an
interpretation for untyped terms but need an extra condition to deduce strong
normalisation. As a main application, we show that Martin-L\"{o}f dependent
type theory extended with a program for Spector double negation shift.Comment: 16 page
- …