Daniel Gratzer
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View article: The Yoneda embedding in simplicial type theory
The Yoneda embedding in simplicial type theory Open
Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical, man…
View article: A Modal Deconstruction of Löb Induction
A Modal Deconstruction of Löb Induction Open
We present a novel analysis of the fundamental Löb induction principle from guarded recursion. Taking advantage of recent work in modal type theory and univalent foundations, we derive Löb induction from a simpler and more conceptual set o…
View article: Unifying cubical and multimodal type theory
Unifying cubical and multimodal type theory Open
In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory (C…
View article: The category of iterative sets in homotopy type theory and univalent foundations
The category of iterative sets in homotopy type theory and univalent foundations Open
When working in homotopy type theory and univalent foundations, the traditional role of the category of sets, $\mathcal{Set}$ , is replaced by the category $\mathcal{hSet}$ of homotopy sets (h-sets); types with h-propositional identity typ…
View article: Directed univalence in simplicial homotopy type theory
Directed univalence in simplicial homotopy type theory Open
Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic…
View article: The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations
The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations Open
When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with h-propositional identity types. Many of the propertie…
View article: Towards Univalent Reference Types: The Impact of Univalence on Denotational Semantics
Towards Univalent Reference Types: The Impact of Univalence on Denotational Semantics Open
We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky’s univalent foundations. We observe for the first ti…
View article: Towards univalent reference types
Towards univalent reference types Open
We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky's univalent foundations. We observe for the first ti…
View article: UNDER LOCK AND KEY: A PROOF SYSTEM FOR A MULTIMODAL LOGIC
UNDER LOCK AND KEY: A PROOF SYSTEM FOR A MULTIMODAL LOGIC Open
We present a proof system for a multimode and multimodal logic, which is based on our previous work on modal Martin-Löf type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e., a s…
View article: Normalization for multimodal type theory
Normalization for multimodal type theory Open
We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding…
View article: {mitten}: A Flexible Multimodal Proof Assistant
{mitten}: A Flexible Multimodal Proof Assistant Open
Recently, there has been a growing interest in type theories which include modalities, unary type constructors which need not commute with substitution. Here we focus on MTT [Daniel Gratzer et al., 2021], a general modal type theory which …
View article: Under Lock and Key: A Proof System for a Multimodal Logic
Under Lock and Key: A Proof System for a Multimodal Logic Open
We present a proof system for a multimodal logic, based on our previous work on a multimodal Martin-Loef type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e. a small 2-category.…
View article: Controlling unfolding in type theory
Controlling unfolding in type theory Open
We present a new way to control the unfolding of definitions in dependent type theory. Traditionally, proof assistants require users to fix whether each definition will or will not be unfolded in the remainder of a development; unfolding d…
View article: Denotational semantics of general store and polymorphism
Denotational semantics of general store and polymorphism Open
We contribute the first denotational semantics of polymorphic dependent type theory extended by an equational theory for general (higher-order) reference types and recursive types, based on a combination of guarded recursion and impredicat…
View article: A Cubical Language for Bishop Sets
A Cubical Language for Bishop Sets Open
We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets \`a la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of …
View article: Unifying cubical and multimodal type theory
Unifying cubical and multimodal type theory Open
In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory (C…
View article: The directed plump ordering
The directed plump ordering Open
Based on Taylor's hereditarily directed plump ordinals, we define the directed plump ordering on W-types in Martin-Löf type theory. This ordering is similar to the plump ordering but comes equipped with non-empty finite joins in addition t…
View article: An inductive-recursive universe generic for small families
An inductive-recursive universe generic for small families Open
We show that it is possible to construct a universe in all Grothendieck topoi with injective codes a la Pujet and Tabareau which is nonetheless generic for small families. As a trivial consequence, we show that their observational type the…
View article: A Stratified Approach to Löb Induction
A Stratified Approach to Löb Induction Open
Guarded type theory extends type theory with a handful of modalities and constants to encode productive recursion. While these theories have seen widespread use, the metatheory of guarded type theories, particularly guarded dependent type …
View article: A Stratified Approach to Löb Induction
A Stratified Approach to Löb Induction Open
Guarded type theory extends type theory with a handful of modalities and constants to encode productive recursion. While these theories have seen widespread use, the metatheory of guarded type theories, particularly guarded dependent type …
View article: Multimodal Dependent Type Theory
Multimodal Dependent Type Theory Open
We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode …
View article: Transfinite Iris: resolving an existential dilemma of step-indexed separation logic
Transfinite Iris: resolving an existential dilemma of step-indexed separation logic Open
\n Contains fulltext :\n 236635.pdf (Publisher’s version ) (Open Access)\n \n Contains fulltext :\n 236635.pdf (Author’s version preprint ) (Open Access)\n
View article: Normalization for multimodal type theory
Normalization for multimodal type theory Open
We consider the conversion problem for multimodal type theory (MTT) by characterizing the normal forms of the type theory and proving normalization. Normalization follows from a novel adaptation of Sterling's Synthetic Tait Computability w…
View article: Syntactic categories for dependent type theory: sketching and adequacy
Syntactic categories for dependent type theory: sketching and adequacy Open
We argue that locally Cartesian closed categories form a suitable doctrine for defining dependent type theories, including non-extensional ones. Using the theory of sketches, one may define syntactic categories for type theories in a style…
View article: Multimodal Dependent Type Theory
Multimodal Dependent Type Theory Open
sponsorship: Alex Kavvos was supported in part by a research grant (12386, Guarded Homotopy Type Theory) from the VILLUM Foundation. Andreas Nuyts holds a PhD Fellowship from the Research Foundation -Flanders (FWO). This work was supported…
View article: Implementing a modal dependent type theory
Implementing a modal dependent type theory Open
Modalities are everywhere in programming and mathematics! Despite this, however, there are still significant technical challenges in formulating a core dependent type theory with modalities. We present a dependent type theory MLTT 🔒 suppor…
View article: Cubical Syntax for Reflection-Free Extensional Equality
Cubical Syntax for Reflection-Free Extensional Equality Open
We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-Löf's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of ident…
View article: Iron: managing obligations in higher-order concurrent separation logic
Iron: managing obligations in higher-order concurrent separation logic Open
Precise management of resources and the obligations they impose, such as the need to dispose of memory, close locks, and release file handles, is hard---especially in the presence of concurrency, when some resources are shared, and differe…
View article: Cubical Syntax for Reflection-Free Extensional Equality
Cubical Syntax for Reflection-Free Extensional Equality Open
We contribute XTT, a cubical reconstruction of Observational Type Theory [Altenkirch et al., 2007] which extends Martin-Löf’s intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental versi…