15 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
The Herbrand Topos
We define a new topos, the Herbrand topos, inspired by the modified
realizability topos and our earlier work on Herbrand realizability. We also
introduce the category of Herbrand assemblies and characterise these as the
double-negation-separated objects in the Herbrand topos. In addition, we show
that the category of sets is included as the category of
double-negation-sheaves and prove that the inclusion functor preserves and
reflects validity of first-order formulas
Herbrandized modified realizability
Realizability notions in mathematical logic have a long history, which can be tracedback to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations ofintuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticatednotions such as Kreisel’s modified realizability and various modern approaches. Inthis context, our work aligns with the lineage of realizability strategies that emphasizethe accumulation, rather than the propagation of precise witnesses. In this paper, weintroduce a new notion of realizability, namely herbrandized modified realizability.This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, whichenables the gathering of witnesses into nonempty finite sets. We also show that theprevious analysis can be extended from logic to (Heyting) arithmetic.The authors are grateful to Fernando Ferreira for interesting discussions on the topic.
They extend their gratitude to the anonymous referee for providing valuable suggestions, which inspired the
addition of Sect. 4.3 to the manuscript. Both authors acknowledge the support of Fundação para a Ciência e
a Tecnologia under the Projects: UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020 and are also
grateful to Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (Universidade de
Lisboa). The first author is also grateful to LASIGE - Computer Science and Engineering Research Centre
Herbrandized modified realizability (Universidade de Lisboa). The second author also benefitted from Fundação para a Ciência e a Tecnologia doctoral Grant 2022.12585.BD.info:eu-repo/semantics/publishedVersio
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
A herbrandized functional interpretation of classical first-order logic
We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio
Functional interpretations and applications
Functional interpretations are maps of formulas from the language of one theory
into the language of another theory, in such a way that provability is preserved.
These interpretations typically replace logical relations by functional relations.
Functional interpretations have many uses, such as relative consistency results, conservation results, and
extraction of computational content from proofs as is the case in the so-called proof mining program.
I will present several recent functional interpretations and some results that come from these interpretations.
I will also give examples of application of functional interpretations, in the spirit of the proof mining program
Stateful Realizers for Nonstandard Analysis
In this paper we propose a new approach to realizability interpretations for
nonstandard arithmetic. We deal with nonstandard analysis in the context of
(semi)intuitionistic realizability, focusing on the Lightstone-Robinson
construction of a model for nonstandard analysis through an ultrapower. In
particular, we consider an extension of the -calculus with a memory
cell, that contains an integer (the state), in order to indicate in which slice
of the ultrapower the computation is being done. We pay
attention to the nonstandard principles (and their computational content)
obtainable in this setting. In particular, we give non-trivial realizers to
Idealization and a non-standard version of the LLPO principle. We then discuss
how to quotient this product to mimic the Lightstone-Robinson construction