Related papers: Type Universes as Allocation Effects
Following the types-as-sets paradigm, we present a mechanized embedding of dependent function types with a hierarchy of universes into schematic first-order logic with equality, with axiom schemas of Tarski-Grothendieck set theory. We carry…
Heap data is potentially unbounded and seemingly arbitrary. As a consequence, unlike stack and static memory, heap memory cannot be abstracted directly in terms of a fixed set of source variable names appearing in the program being…
Awodey, later with Newstead, showed how polynomial functors with extra structure (termed ``natural models'') hold within them the categorical semantics for dependent type theory. Their work presented these ideas clearly but ultimately led…
We present a soundness theorem for a dependent type theory with context constants with respect to an indexed category of (finite, abstract) simplical complexes. The point of interest for computer science is that this category can be seen to…
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in…
Simple type theory is suited as framework for combining classical and non-classical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be…
We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…
We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories…
We give a model of dependent type theory with one univalent universe and propositional truncation interpreting a type as a stack, generalising the groupoid model of type theory. As an application, we show that countable choice cannot be…
In this paper we describe how to leverage higher-order unification to type check a dependently typed language with meta-variables. The literature usually presents the unification algorithm as a standalone component, however the need to…
Abstract algebra provides a large hierarchy of properties that a collection of objects can satisfy, such as forming an abelian group or a semiring. These classifications can arranged into a broad and typically acyclic directed graph. This…
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…
This work extends the present author's computational game semantics of Martin-L\"{o}f type theory to the cumulative hierarchy of universes. This extension completes game semantics of all standard types of Martin-L\"{o}f type theory for the…
By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In…
The dependently-typed lambda calculus LF is often used as a vehicle for formalizing rule-based descriptions of object systems. Proving properties of object systems encoded in this fashion requires reasoning about formulas over LF typing…
As originally proposed, type classes provide overloading and ad-hoc definition, but can still be understood (and implemented) in terms of strictly parametric calculi. This is not true of subsequent extensions of type classes. Functional…
Type-free systems of logic are designed to consistently handle significant instances of self-reference. Some consistent type-free systems also have the feature of allowing the sort of general abstraction or comprehension principle that…
Type-preserving translations are effective rigorous tools in the study of core programming calculi. In this paper, we develop a new typed translation that connects sequential and concurrent calculi; it is governed by type systems that…
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…
Humans can generate reasonable answers to novel queries (Schulz, 2012): if I asked you what kind of food you want to eat for lunch, you would respond with a food, not a time. The thought that one would respond "After 4pm" to "What would you…