Every metric space is separable in function realizability

We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every discrete space is countable. It follows that intuitionistic logic does not show the existence of a non-separable metric space, or an uncountable set with decidable equality, even if we assume principles that are validated by function realizability, such as Dependent and Function choice, Markov's principle, and Brouwer's continuity and fan principles

An Effect System for Algebraic Effects and Handlers

We present an effect system for core Eff, a simplified variant of Eff, which is an ML-style programming language with first-class algebraic effects and handlers. We define an expressive effect system and prove safety of operational semantics with respect to it. Then we give a domain-theoretic denotational semantics of core Eff, using Pitts's theory of minimal invariant relations, and prove it adequate. We use this fact to develop tools for finding useful contextual equivalences, including an induction principle. To demonstrate their usefulness, we use these tools to derive the usual equations for mutable state, including a general commutativity law for computations using non-interfering references. We have formalized the effect system, the operational semantics, and the safety theorem in Twelf

First Steps in Synthetic Computability Theory

AbstractComputability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. Consequently, there have been a number of presentations of computability theory that aimed to present the subject in an abstract and conceptually pleasing way. We build on two such approaches, Hyland's effective topos and Richman's formulation in Bishop-style constructive mathematics, and develop basic computability theory, starting from a few simple axioms. Because we want a theory that resembles ordinary mathematics as much as possible, we never speak of Turing machines and Gödel encodings, but rather use familiar concepts from set theory and topology

Instance reducibility and Weihrauch degrees

We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in reverse constructive mathematics (formal Church's thesis, Brouwer's Continuity principle and Fan theorem, Excluded middle, Limited principle, Function choice, Markov's principle, etc.). We show that the instance degrees form a frame, i.e., a complete lattice in which finite infima distribute over set-indexed suprema. They turn out to be equivalent to the frame of upper sets of truth values, ordered by the reverse Smyth partial order. We study the overall structure of the lattice: the subobject classifier embeds into the lattice in two different ways, one monotone and the other antimonotone, and the $\lnot\lnot$-dense degrees coincide with those that are reducible to the degree of Excluded middle. We give an explicit formulation of instance degrees in a relative realizability topos, and call these extended Weihrauch degrees, because in Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees correspond precisely to Weihrauch degrees. The extended degrees improve the structure of Weihrauch degrees by equipping them with computable infima and suprema, an implication, the ability to control access to parameters and computation of results, and by generally widening the scope of Weihrauch reducibility

Locally Non-compact Spaces and Continuity Rinciples

We give a constructive proof that Baire space embeds in any inhabited locally non-compact complete separable metric space, X, in such a way that every sequentially continuous function from Baire space to Z extends to a function from X to R. As an application, we show that, in the presence of certain choice and continuity principles, the statement “all functions from X to R is continuous ” is false. This generalizes a result previously obtained by Escardó and Streicher, in the context of “domain realizability”, for the special case X = C[0, 1].

An extensible equality checking algorithm for dependent type theories

We present a general and user-extensible equality checking algorithm that is applicable to a large class of type theories. The algorithm has a type-directed phase for applying extensionality rules and a normalization phase based on computation rules, where both kinds of rules are defined using the type-theoretic concept of object-invertible rules. We also give sufficient syntactic criteria for recognizing such rules, as well as a simple pattern-matching algorithm for applying them. A third component of the algorithm is a suitable notion of principal arguments, which determines a notion of normal form. By varying these, we obtain known notions, such as weak head-normal and strong normal forms. We prove that our algorithm is sound. We implemented it in the Andromeda~2 proof assistant, which supports user-definable type theories. The user need only provide the equality rules they wish to use, which the algorithm automatically classifies as computation or extensionality rules, and select appropriate principal arguments

Runners in action

Runners of algebraic effects, also known as comodels, provide a mathematical model of resource management. We show that they also give rise to a programming concept that models top-level external resources, as well as allows programmers to modularly define their own intermediate "virtual machines". We capture the core ideas of programming with runners in an equational calculus $\lambda_{\mathsf{coop}}$, which we equip with a sound and coherent denotational semantics that guarantees the linear use of resources and execution of finalisation code. We accompany $\lambda_{\mathsf{coop}}$ with examples of runners in action, provide a prototype language implementation in OCaml, as well as a Haskell library based on $\lambda_{\mathsf{coop}}$.Comment: ESOP 2020 final version + online appendi

On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos

In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs