Andrej Bauer
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View article: Spreen spaces and the synthetic Kreisel-Lacombe-Shoenfield-Tseitin theorem
Spreen spaces and the synthetic Kreisel-Lacombe-Shoenfield-Tseitin theorem Open
I take a constructive look at Dieter Spreen's treatment of effective topological spaces and the Kreisel-Lacombe-Shoenfield-Tseitin (KLST) continuity theorem. Transferring Spreen's ideas from classical computability theory and numbered sets…
View article: An Imperative Language for Verified Exact Real-Number Computation
An Imperative Language for Verified Exact Real-Number Computation Open
We introduce Clerical, a programming language for exact real-number computation that combines first-order imperative-style programming with a limit operator for computation of real numbers as limits of Cauchy sequences. We address the semi…
View article: The Countable Reals
The Countable Reals Open
We construct a topos in which the Dedekind reals are countable. The topos arises from a new kind of realizability, which we call parameterized realizability, based on partial combinatory algebras whose application depends on a parameter. R…
View article: Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics
Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics Open
Programming languages can be defined from the concrete to the abstract by abstract syntax trees, well-scoped syntax, well-typed (intrinsic) syntax, algebraic syntax (well-typed syntax quotiented by conversion). Another aspect is the repres…
View article: MLFMF: Data Sets for Machine Learning for Mathematical Formalization
MLFMF: Data Sets for Machine Learning for Mathematical Formalization Open
MLFMF MLFMF (Machine Learning for Mathematical Formalization) is a collection of data sets for benchmarking recommendation systems used to support formalization of mathematics with proof assistants. These systems help humans identify which…
View article: MLFMF: Data Sets for Machine Learning for Mathematical Formalization
MLFMF: Data Sets for Machine Learning for Mathematical Formalization Open
MLFMFMLFMF (Machine Learning for Mathematical Formalization) is a collection of data sets for benchmarking recommendation systems used to support formalization of mathematics with proof assistants. These systems help humans identify which …
View article: Finitary Type Theories With and Without Contexts
Finitary Type Theories With and Without Contexts Open
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 prov…
View article: Spreen spaces and the synthetic Kreisel-Lacombe-Shoenfield-Tseitin theorem
Spreen spaces and the synthetic Kreisel-Lacombe-Shoenfield-Tseitin theorem Open
I take a constructive look at Dieter Spreen's treatment of effective topological spaces and the Kreisel-Lacombe-Shoenfield-Tseitin (KLST) continuity theorem. Transferring Spreen's ideas from classical computability theory and numbered sets…
View article: Instance reducibility and Weihrauch degrees
Instance reducibility and Weihrauch degrees Open
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 rever…
View article: A global COVID-19 observatory, monitoring the pandemics through text mining and visualization
A global COVID-19 observatory, monitoring the pandemics through text mining and visualization Open
The global health situation due to the SARS-COV-2 pandemic motivated an unprecedented contribution of science and technology from companies and communities all over the world to fight COVID-19. In this paper, we present the impactful role …
View article: An extensible equality checking algorithm for dependent type theories
An extensible equality checking algorithm for dependent type theories Open
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 computa…
View article: Finitary type theories with and without contexts
Finitary type theories with and without contexts Open
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 prov…
View article: Instance reducibility and Weihrauch degrees
Instance reducibility and Weihrauch degrees Open
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 rever…
View article: Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions
Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions Open
We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably …
View article: Every metric space is separable in function realizability
Every metric space is separable in function realizability Open
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 T0T0-space is separable and every discret…
View article: Every metric space is separable in function realizability
Every metric space is separable in function realizability Open
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 discre…
View article: Formalization of Mathematics in Type Theory (Dagstuhl Seminar 18341)
Formalization of Mathematics in Type Theory (Dagstuhl Seminar 18341) Open
Formalized mathematics is mathematical knowledge (definitions, theorems, and proofs) represented in digital form suitable for computer processing. The central goal of this seminar was to identify the theoretical advances and practical impr…
View article: What is algebraic about algebraic effects and handlers?
What is algebraic about algebraic effects and handlers? Open
This note recapitulates and expands the contents of a tutorial on the mathematical theory of algebraic effects and handlers which I gave at the Dagstuhl seminar 18172 "Algebraic effect handlers go mainstream". It is targeted roughly at the…
View article: Design and Implementation of the Andromeda Proof Assistant
Design and Implementation of the Andromeda Proof Assistant Open
Andromeda is an LCF-style proof assistant where the user builds derivable judgments by writing code in a meta-level programming language AML. The only trusted component of Andromeda is a minimalist nucleus (an implementation of the inferen…
View article: Design and Implementation of the Andromeda Proof Assistant
Design and Implementation of the Andromeda Proof Assistant Open
Andromeda is an LCF-style proof assistant where the user builds derivable judgments by writing code in a meta-level programming language AML. The only trusted component of Andromeda is a minimalist nucleus (an implementation of the inferen…
View article: A Homological Theory of Functions: Nonuniform Boolean Complexity Separation and VC Dimension Bound Via Algebraic Topology, and a Homological Farkas Lemma
A Homological Theory of Functions: Nonuniform Boolean Complexity Separation and VC Dimension Bound Via Algebraic Topology, and a Homological Farkas Lemma Open
In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes A != B especially when A is known to be a subset of B. In this paper we introduc…
View article: The HoTT library: a formalization of homotopy type theory in Coq
The HoTT library: a formalization of homotopy type theory in Coq Open
We present a proof of coherence for monoidal groupoids in homotopy type theory. An important role in the formulation and in the proof of coherence is played by groupoids with a free monoidal structure; these can be represented by 1-truncat…
View article: The HoTT Library: A formalization of homotopy type theory in Coq
The HoTT Library: A formalization of homotopy type theory in Coq Open
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 o…
View article: Five stages of accepting constructive mathematics
Five stages of accepting constructive mathematics Open
On the odd day, a mathematician might wonder what constructive mathematics is all about. They may have heard arguments in favor of constructivism but are not at all convinced by them, and in any case they may care little about philosophy. …
View article: From Theory to Practice of Algebraic Effects and Handlers (Dagstuhl Seminar 16112)
From Theory to Practice of Algebraic Effects and Handlers (Dagstuhl Seminar 16112) Open
Dagstuhl Seminar 16112 was devoted to research in algebraic effects and handlers, a chapter in the principles of programming languages which addresses computational effects (such as I/O, state, exceptions, nondeterminism, and many others).…