Related papers: A Classical Realizability Model arising from a Sta…
The lambda-PRK-calculus is a typed lambda-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend lambda-PRK to…
In a recent paper, a realizability technique has been used to give a semantics of a quantum lambda calculus. Such a technique gives rise to an infinite number of valid typing rules, without giving preference to any subset of those. In this…
We show how to extract a monotonic learning algorithm from a classical proof of a geometric statement by interpreting the proof by means of interactive realizability, a realizability sematics for classical logic. The statement is about the…
The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three…
In this paper we continue with the algebraic study of Krivine's realizability, refining some of the authors' previous constructions by introducing two categories, with objects the abstract Krivine structures and the implicative algebras…
Besides recalling the basic definitions of Realizability Lattices, Abstract Krivine Structures, Ordered Combinatory Algebras and Tripos and reviewing its relationships, we propose a new foundational framework for realizability. Motivated by…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…
A longstanding open problem is whether there exists a non syntactical model of the untyped lambda-calculus whose theory is exactly the least lambda-theory (l-beta). In this paper we investigate the more general question of whether the…
In Hayashi and Leigh (2024), the authors formulate classical number realisability for first-order arithmetic and a corresponding axiomatic system based on Krivine's classical realisability interpretation. This paper presents a…
The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in physics,…
We define an extension of lambda-calculus with dependents types that enables us to encode transparent and opaque probabilistic programs and prove a strong normalisation result for it by a reducibility technique. While transparent…
Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene's initial realizability laid the ground…
The classic model of computable randomness considers martingales that take real or rational values. Recent work by Bienvenu et al. (2012) and Teutsch (2014) shows that fundamental features of the classic model change when the martingales…
Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically…
Accattoli, Dal Lago, and Vanoni have recently proved that the space used by the Space KAM, a variant of the Krivine abstract machine, is a reasonable space cost model for the lambda-calculus accounting for logarithmic space, solving a…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
Classical programming languages cannot model essential elements of complex systems such as true random number generation. This paper develops a formal programming language called the lambda-q calculus that addresses the fundamental…
The denotational semantics of the untyped lambda-calculus is a well developed field built around the concept of solvable terms, which are elegantly characterized in many different ways. In particular, unsolvable terms provide a consistent…
In this paper, we introduce a semantics of realisability for the classical propositional natural deduction and we prove a correctness theorem. This allows to characterize the operational behaviour of some typed terms.
The present paper constructs three new systems of clarithmetic (arithmetic based on computability logic --- see http://www.cis.upenn.edu/~giorgi/cl.html): CLA8, CLA9 and CLA10. System CLA8 is shown to be sound and extensionally complete…