Representations of stream processors using nested fixed points

We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finite-path) trees of a certain kind. The underlying idea can be traced back to Brouwer's justification of bar-induction, or to Kreisel and Troelstra's elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity

Quotienting the delay monad by weak bisimilarity

The delay datatype was introduced by Capretta as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. It is a monad and it constitutes a constructive alternative to the maybe monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay monad quotiented by weak bisimilarity is still a monad. In this paper, we consider Hofmann's alternative approach of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. We have fully formalized our results in the Agda dependently typed programming language

Internal Effectful Forcing in System T

The effectful forcing technique allows one to show that the denotation of a closed System T term of type (ι ⇒ ι) ⇒ ι in the set-theoretical model is a continuous function (ℕ → ℕ) → ℕ. For this purpose, an alternative dialogue-tree semantics is defined and related to the set-theoretical semantics by a logical relation. In this paper, we apply effectful forcing to show that the dialogue tree of a System T term is itself System T-definable, using the Church encoding of trees

From Partial to Monadic: Combinatory Algebra with Effects

Partial Combinatory Algebras (PCAs) provide a foundational model of the untyped λ-calculus and serve as the basis for many notions of computability, such as realizability theory. However, PCAs support a very limited notion of computation by only incorporating non-termination as a computational effect. To provide a framework that better internalizes a wide range of computational effects, this paper puts forward the notion of Monadic Combinatory Algebras (MCAs). MCAs generalize the notion of PCAs by structuring the combinatory algebra over an underlying computational effect, embodied by a monad. We show that MCAs can support various side effects through the underlying monad, such as non-determinism, stateful computation and continuations. We further obtain a categorical characterization of MCAs within Freyd Categories, following a similar connection for PCAs. Moreover, we explore the application of MCAs in realizability theory, presenting constructions of effectful realizability triposes and assemblies derived through evidenced frames, thereby generalizing traditional PCA-based realizability semantics. The monadic generalization of the foundational notion of PCAs provides a comprehensive and powerful framework for internally reasoning about effectful computations, paving the path to a more encompassing study of computation and its relationship with realizability models and programming languages

Inductive Continuity via Brouwer Trees

Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this "inductive" continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Russian Constructivism in a Prefascist Theory

International audienc

THE EUROPEAN EXTERNAL ACTION SERVICE AND GENDER PROMOTION - A comparative text analysis of how the EEAS adapts its gender promotion messages to audiences in the west and east

A qualitative text analysis on how the European External Action Service, EEAS, promotes gender. It examines how the former High Representative Federica Mogherini adjust her language towards a “western” and “eastern” region. The aim is to see if there are any differences in how the EEAS choose to approach different audience when specking about gender equality. Previous research shows that there is a lot of research on gender and the EU, and its institutions including the EEAS. Though, there is research gap on how the EEAS represent gender towards different audiences. Theories that will be used to analyse the result is social constructivism and the concept of representation. The concept of representation is useful to se how actors frame and present their interest towards different audiences. A comparative text analysis of speeches made by Federica Mogherini, representing the EEAS, will be made to outsource possible differences in how she concepts gender equality. The analysis will be made through Bacchi’s framework “What is the problem represented to be?”. The results show that in general the aim of why gender equality is important does not differ but there is a difference within what topic on gender equality is discussed and how it is presented

Realizing Continuity Using Stateful Computations

The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. This paper presents a class of intuitionistic theories that features stateful computations, such as reference cells, and shows that these theories can be extended with continuity axioms. The modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled in the theory