In type theory, an oracle may be specified abstractly by a predicate whose domain is the type of queries asked of the oracle, and whose proofs are the oracle answers. Such a specification induces an oracle modality that captures a computational intuition about oracles: at each step of reasoning we either know the result, or we ask the oracle a query and proceed upon receiving an answer. We characterize an oracle modality as the least one forcing the given predicate. We establish an adjoint retraction between modalities and propositional containers, from which it follows that every modality is an oracle modality. The left adjoint maps sums to suprema, which makes suprema of modalities easy to compute when they are given in terms of oracle modalities. We also study sheaves for oracle modalities. We describe sheafification in terms of a quotient-inductive type of computation trees, and describe sheaves as algebras for the corresponding monad. We also introduce equifoliate trees, an intensional notion of oracle computation given by a (non-propositional) container. Equifoliate trees descend to sheaves, and modally cover them. As an application, we give a concrete description of all Lawvere-Tierney topologies in a realizability topos, closely related to a game-theoretic characterization by Takayuki Kihara.

We introduce MLFMF, a collection of data sets for benchmarking recommendation systems used to support formalization of mathematics with proof assistants. These systems help humans identify which previous entries (theorems, constructions, datatypes, and postulates) are relevant in proving a new theorem or carrying out a new construction. Each data set is derived from a library of formalized mathematics written in proof assistants Agda or Lean. The collection includes the largest Lean 4 library Mathlib, and some of the largest Agda libraries: the standard library, the library of univalent mathematics Agda-unimath, and the TypeTopology library. Each data set represents the corresponding library in two ways: as a heterogeneous network, and as a list of s-expressions representing the syntax trees of all the entries in the library. The network contains the (modular) structure of the library and the references between entries, while the s-expressions give complete and easily parsed information about every entry.We report baseline results using standard graph and word embeddings, tree ensembles, and instance-based learning algorithms. The MLFMF data sets provide solid benchmarking support for further investigation of the numerous machine learning approaches to formalized mathematics. The methodology used to extract the networks and the s-expressions readily applies to other libraries, and is applicable to other proof assistants. With more than $250\,000$ entries in total, this is currently the largest collection of formalized mathematical knowledge in machine learnable format.

Abstract We give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin–Löf type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We prove several general meta-theorems about finitary type theories: weakening, admissibility of substitution and instantiation of metavariables, derivability of presuppositions, uniqueness of typing, and inversion principles. We then give a second formulation of finitary type theories in which there are no explicit contexts. Instead, free variables are explicitly annotated with their types. We provide translations between finitary type theories with and without contexts, thereby showing that they have the same expressive power. The context-free type theory is implemented in the nucleus of the Andromeda 2 proof assistant.

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.

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.

Abstract 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 $$\lambda _{\mathsf {coop}}$$ λ 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 $$\lambda _{\mathsf {coop}}$$ λ coop with examples of runners in action, provide a prototype language implementation in OCaml, as well as a Haskell library based on $$\lambda _{\mathsf {coop}}$$ λ coop .

We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of synthetic homotopy theory, as well as category theory and modalities. The library has been used as a basis for several independent developments. We discuss the decisions that led to the design of the library, and we comment on the interaction of homotopy type theory with recently introduced features of Coq, such as universe polymorphism and private inductive types.

We provide a realizability model based on infinite time Turing machines in which there is an injection from the internal Baire space, the object of infinite sequences of numbers, to the object of natural numbers.

We consider similarity-based relational databases that allow to retrieve approximate data, find data within a given range of distance or similarity, and support imprecise queries. We focus on the recently introduced relational algebra with similarities on [Formula: see text]-relations, which are annotated with multi-dimensional similarity values with each dimension referring to a single attribute. The codomains of the annotated relations are De Morgan frames, and the annotations express the relevance of the tuples as answers to a similarity-based query. In this paper, we study Datalog programs on [Formula: see text]-relations, with and without negation. We describe the least-fixpoint algorithm for safe and rectified Datalog programs on [Formula: see text]-relations with finite support but without negative literals in the body. We further describe the perfect-minimal-fixpoint algorithm of a Datalog program on [Formula: see text]-relations with finite support and negative literals in the body when rules are safe, rectified and stratified. We introduce the idea of controlling the calculation of the annotations such that the tuples that enter an IDB relation last will be announced less desirable than those that enter first. For this we define a damping function that augments/diminishes the individual annotations that contribute to the final annotations of tuples. With a damping function, for instance, long chains of inferences may be made significantly less desirable or even totally undesirable.

Realizability theory is not just a fundamental tool in logic and computability. It also has direct application to the design and implementation of programs, since it can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between the worlds of constructive mathematics and programming. By using the realizability interpretation of constructive mathematics, RZ translates specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools.

Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambda calculus, of which we provide a self-contained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using two-sided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore's, because we work constructively, and we also consider back-to-front (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis.

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally Cartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic.