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Thorsten Altenkirch

Publications and source records attributed to Thorsten Altenkirch.

16 recordsLinked to original sources
arXiv2026

Logics and Type Theory: essays dedicated to Stefano Berardi on the occasion of his 1000000th birthday

Proof Theory and Type Theory are two branches of mathematical logic and theoretical computer science that explore the structure of mathematical proofs and the foundations of computation. Both are crucial for understanding formal systems, programming languages, and constructive mathematics. Stefano Berardi is an influential researcher in these fields, particularly known for his work on constructive logic, dependent types, and more recently, cyclic proofs. These proceedings intend to collect papers by researchers who are active in the field and often have been coauthors of Stefano, aimed to illustrate the achievements and the perspectives of this research area.

cs.LO↗
arXiv2025

Substitution Without Copy and Paste

Defining substitution for a language with binders like the simply typed $λ$-calculus requires repetition, defining substitution and renaming separately. To verify the categorical properties of this calculus, we must repeat the same argument many times. We present a lightweight method that avoids repetition and that gives rise to a simply typed category with families (CwF) isomorphic to the initial simply typed CwF. Our paper is a literate Agda script.

cs.LO↗
arXiv2025

The Groupoid-Syntax of Type Theory is a Set

Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of Homotopy Type Theory (HoTT), one of the limitations of the traditional notion of CwFs is the requirement to set-truncate types, which excludes models based on univalent categories, such as the standard set model. To address this limitation, we introduce the concept of a Groupoid Category with Families (GCwF). This framework truncates types at the groupoid level and incorporates coherence equations, providing a natural extension of the CwF framework when starting from a 1-category. We demonstrate that the initial GCwF for a type theory with a base family of sets and Pi-types (groupoid-syntax) is set-truncated. Consequently, this allows us to utilize the conventional intrinsic syntax of type theory while enabling interpretations in semantically richer and more natural models. All constructions in this paper were formalised in Cubical Agda.

cs.LO↗
arXiv2025

Formalising Inductive and Coinductive Containers

Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of locally cartesian closed categories (LCCCs) with disjoint coproducts and W-types, and uniqueness of identity proofs (UIP) is implicitly assumed throughout. Although it is claimed that these developments can also be interpreted in extensional Martin-Löf type theory, this interpretation is not made explicit. In this paper, we present a formalisation of the results that 'containers preserve least and greatest fixed points' in Cubical Agda, thereby giving a formulation in intensional type theory. Our proofs do not make use of UIP and thereby generalise the original results from talking about container functors on Set to container functors on the wild category of types. Our main incentive for using Cubical Agda is that its path type restores the equivalence between bisimulation and coinductive equality. Thus, besides developing container theory in a more general setting, we also demonstrate the usefulness of Cubical Agda's path type to coinductive proofs.

cs.LO↗
arXiv2024

Synthetic 1-Categories in Directed Type Theory

The field of directed type theory seeks to design type theories capable of reasoning synthetically about (higher) categories, by generalizing the symmetric identity types of Martin-Löf Type Theory to asymmetric hom-types. We articulate the directed type theory of the category model, with appropriate modalities for keeping track of variances and a powerful directed-J rule capable of proving results about arbitrary terms of hom-types; we put this rule to use in making several constructions in synthetic 1-category theory. Because this theory is expressed entirely in terms of generalized algebraic theories, we know automatically that this directed type theory admits a syntax model and is the first step towards directed higher observational type theory.

math.CT↗
arXiv2023

Internal parametricity, without an interval

Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the appearance of higher dimensional cubes. In previous theories with internal parametricity, either an explicit syntax for higher cubes is present or the theory is extended with a new sort for the interval. In this paper we present a type theory with internal parametricity which is a simple extension of Martin-Löf type theory: there are a few new type formers, term formers and equations. Geometry is not explicit in this syntax, but emergent: the new operations and equations only refer to objects up to dimension 3. We show that this theory is modelled by presheaves over the BCH cube category. Fibrancy conditions are not needed because we use span-based rather than relational parametricity. We define a gluing model for this theory implying that external parametricity and canonicity hold. The theory can be seen as a special case of a new kind of modal type theory, and it is the simplest setting in which the computational properties of higher observational type theory can be demonstrated.

cs.LO↗
arXiv2020

The Integers as a Higher Inductive Type

We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

cs.LO↗
arXiv2020

Free Higher Groups in Homotopy Type Theory

Given a type A in homotopy type theory (HoTT), we can define the free infinity-group on A as the loop space of the suspension of A+1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit : F(A), cons : A -> F(A) -> F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that the 1-truncation of F(A) is a set.

cs.LO↗
arXiv2017

Quotient inductive-inductive types

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules.

cs.LO↗
arXiv2017

Normalisation by Evaluation for Type Theory, in Type Theory

We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no preterms or realizers in our construction, and every construction respects the conversion relation. NBE for simple types uses a logical relation between the syntax and the presheaf interpretation. In our construction, we merge the presheaf interpretation and the logical relation into a proof-relevant logical predicate. We prove normalisation, completeness, stability and decidability of definitional equality. Most of the constructions were formalized in Agda.

cs.LO↗
arXiv2017

Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type

Capretta's delay monad can be used to model partial computations, but it has the "wrong" notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the "right" notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory without assuming (instances of) the axiom of countable choice. Using an idea from homotopy type theory - a higher inductive-inductive type - we construct a partiality monad without relying on countable choice. We prove that, in the presence of countable choice, our partiality monad is equivalent to the delay monad quotiented by weak bisimilarity. Furthermore we outline several applications.

cs.LO↗
arXiv2017

Notions of Anonymous Existence in Martin-Löf Type Theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-Löf type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.

cs.LO↗
arXiv2016

Extending Homotopy Type Theory with Strict Equality

In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an "outer" one, containing a strict equality type former, and an "inner" one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of "infinite structures" which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

cs.LO↗
arXiv2005

An Algebra of Pure Quantum Programming

We develop a sound and complete equational theory for the functional quantum programming language QML. The soundness and completeness of the theory are with respect to the previously-developed denotational semantics of QML. The completeness proof also gives rise to a normalisation algorithm following the normalisation by evaluation approach. The current work focuses on the pure fragment of QML omitting measurements.

quant-ph↗
arXiv2005

A functional quantum programming language

We introduce the language QML, a functional language for quantum computations on finite types. Its design is guided by its categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive semantics of irreversible quantum computations realisable as quantum gates. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings explicit. Strict programs are free from decoherence and hence preserve superpositions and entanglement - which is essential for quantum parallelism.

quant-ph↗