Friedman-Simpson’s original program of reverse mathematics, as is also the case for most of standard mathematics, has been developed in classical subsystems of second-order arithmetic. As such, (classical) reverse mathematics presents various limitations from a constructive point of view, since for instance it is unable to distinguish between a statement and its contrapositive (e.g. dependent choice and the bar induction principles). The case of (Weak) Kőnig’s Lemma (WKL) and Fan Theorem (FT) is particularly interesting in that regard: while WKL is well-known to imply FT, and if constructivists like Brouwer rejected the former while admitting the latter, the converse implication has not been much studied for years. It is only recently that a growing enthusiasm for constructive reverse mathematics pushed towards a finer-grained analysis of the connection between such principles. In addition to intuitionistic reverse mathematics, the realizability approach to logical principles adds a computational meaning to purely logical statements. We follow this path to investigate the computational meaning of Brouwer’s Fan Theorem: building on recent work by Lubarsky and Rathjen, we first construct a realizability interpretation of higher-order logic validating FT while refuting WKL. This interpretation relies on a λ-calculus extended with oracles while preserving a notion of continuity for realizers. We then push this approach a step further to show the robustness of this realizability interpretation by identifying, in the abstract and general setting of evidenced frames, sufficient computational conditions entailing FT.

Realizability interprets propositions as specifications for computational entities in programming languages. Specifically, syntactic realizability is a powerful machinery that handles realizability as a syntactic translation of propositions into new propositions that describe what it means to realize the input proposition. This paper introduces EffHOL (Effectful Higher-Order Logic), a novel framework that expands syntactic realizability to uniformly support modern programming paradigms with side effects. EffHOL combines higher-kinded polymorphism, enabling typing of realizers for higher-order propositions, with a computational term language that uses monads to represent and reason about effectful computations. We craft a syntactic realizability translation from (intuitionistic) higher-order logic (HOL) to EffHOL, ensuring the extraction of computable realizers through a constructive soundness proof. EffHOL’s parameterization by monads allows for the synthesis of effectful realizers for propositions unprovable in pure HOL, bridging the gap between traditional and effectful computational paradigms. Examples, including continuations and memoization, showcase EffHOL’s capability to unify diverse computational models, with traditional ones as special cases. For a semantic connection, we show that any instance of EffHOL induces an evidenced frame, which, in turn, yields a tripos and a realizability topos.

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 $\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\cal{M}^{\mathbb{N}}$ 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.

In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of 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 λ-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower ℳ^{ℕ} the computation is being done. We shall pay attention to the nonstandard principles (and their computational content) obtainable in this setting. We then discuss how this product could be quotiented to mimic the Lightstone-Robinson construction.

Constructive foundations have for decades been built upon realizability models for higher-order logic and type theory. However, traditional realizability models have a rather limited notion of computation, which only supports non-termination and avoids many other commonly used effects. Work to address these limitations has typically overlaid structure on top of existing models, such as by using powersets to represent non-determinism, but kept the realizers themselves deterministic. This paper alternatively addresses these limitations by making the structure underlying realizability models more flexible. To this end, we introduce evidenced frames: a general-purpose framework for building realizability models that support diverse effectful computations. We demonstrate that this flexibility permits models wherein the realizers themselves can be effectful, such as λ-terms that can manipulate state, reduce non-deterministically, or fail entirely. Beyond the broader notions of computation, we demonstrate that evidenced frames form a unifying framework for (realizability) models of higher-order dependent predicate logic. In particular, we prove that evidenced frames are complete with respect to these models, and that the existing completeness construction for implicative algebras-another foundational framework for realizability-factors through our simpler construction. As such, we conclude that evidenced frames offer an ideal domain for unifying and broadening realizability models.

In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability subsumes Cohen's forcing, and even more, gives rise to unexpected models of set theories. Pursuing the algebraic analysis of these models that was first undertaken by Streicher, Miquel recently proposed to lay the algebraic foundation of classical realizability and forcing within new structures which he called implicative algebras. These structures are a generalization of Boolean algebras based on an internal law representing the implication. Notably, implicative algebras allow for the adequate interpretation of both programs (i.e. proofs) and their types (i.e. formulas) in the same structure. The very definition of implicative algebras takes position on a presentation of logic through universal quantification and the implication and, computationally, relies on the call-by-name $λ$-calculus. In this paper, we investigate the relevance of this choice, by introducing two similar structures. On the one hand, we define disjunctive algebras, which rely on internal laws for the negation and the disjunction and which we show to be particular cases of implicative algebras. On the other hand, we introduce conjunctive algebras, which rather put the focus on conjunctions and on the call-by-value evaluation strategy. We finally show how disjunctive and conjunctive algebras algebraically reflect the well-known duality of computation between call-by-name and call-by-value.

The call-by-need evaluation strategy for the λ-calculus is an evaluation strategy that lazily evaluates arguments only if needed, and if so, shares computations across all places where it is needed. To implement this evaluation strategy, abstract machines require some form of global environment. While abstract machines usually lead to a better understanding of the flow of control during the execution, facilitating in particular the definition of continuation-passing style translations, the case of machines with global environments turns out to be much more subtle.

Dependent types are a key feature of the proof assistants based on the Curry-Howard isomorphism. It is well known that this correspondence can be extended to classical logic by enriching the language of proofs with control operators. However, they are known to misbehave in the presence of dependent types, unless dependencies are restricted to values. Moreover, while sequent calculi naturally support continuation-passing-style interpretations, there is no such presentation of a language with dependent types. The main achievement of this article is to give a sequent calculus presentation of a call-by-value language with a control operator and dependent types, and to justify its soundness through a continuation-passing-style translation. We start from the call-by-value version of the λμ˜μ -calculus. We design a minimal language with a value restriction and a type system that includes a list of explicit dependencies to maintain type safety. We then show how to relax the value restriction and introduce delimited continuations to directly prove the consistency by means of a continuation-passing-style translation. Finally, we relate our calculus to a similar system by Lepigre and present a methodology to transfer properties from this system to our own.

In a recent paper [11], Herbelin developed dPAω, a calculus in which constructive proofs for the axioms of countable and dependent choices could be derived via the encoding of a proof of countable universal quantification as a stream of it components. However, the property of normalization (and therefore the one of soundness) was only conjectured. The difficulty for the proof of normalization is due to the simultaneous presence of dependent types (for the constructive part of the choice), of control operators (for classical logic), of coinductive objects (to encode functions of type N→A into streams (a0, a1, ...)) and of lazy evaluation with sharing (for these coinductive objects).

In this paper, we treat the specification problem in Krivine classical realizability (Krivine 2009Panoramas et synthèses27), in the case of arithmetical formulæ. In the continuity of previous works from Miquel and the first author (Guillermo 2008Jeux de réalisabilité en arithmétique classique, Ph.D. thesis, Université Paris 7; Guillermo and Miquel 2014Mathematical Structures in Computer Science, Epub ahead of print), we characterize the universal realizers of a formula as being the winning strategies for a game (defined according to the formula). In the first sections, we recall the definition of classical realizability, as well as a few technical results. In Section 5, we introduce in more details the specification problem and the intuition of the game-theoretic point of view we adopt later. We first present a game 1, that we prove to be adequate and complete if the language contains no instructions ‘quote’ (Krivine 2003Theoretical Computer Science308259–276), using interaction constants to do substitution over execution threads. We then show that as soon as the language contain ‘Quote,’ the game is no more complete, and present a second game 2that is both adequate and complete in the general case. In the last Section, we draw attention to a model-theoretic point of view and use our specification result to show that arithmetical formulæ are absolute for realizability models.