Related papers: Nominal Type Theory by Nullary Internal Parametric…
Conventional closed-world information extraction (IE) approaches rely on human ontologies to define the scope for extraction. As a result, such approaches fall short when applied to new domains. This calls for systems that can automatically…
We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated…
We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying…
We introduce $\infty$-type theories as an $\infty$-categorical generalization of the categorical definition of type theories introduced by the second named author. We establish analogous results to the previous work including 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…
We study nominal recursors from the literature on syntax with bindings and compare them with respect to expressiveness. The term "nominal" refers to the fact that these recursors operate on a syntax representation where the names of bound…
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…
We use type-theoretic techniques to present an algebraic theory of $\infty$-categories with strict units. Starting with a known type-theoretic presentation of fully weak $\infty$-categories, in which terms denote valid operations, we extend…
The contribution of this paper is the development of the syntax and semantics of multi-sorted nominal abstract binding trees (abts), an extension of second order universal algebra to support symbol-indexed families of operators. Nominal…
There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and…
Induction is the process by which we obtain predictive laws or theories or models of the world. We consider the structural aspect of induction. We answer the question as to whether we can find a finite and minmalistic set of operations on…
We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not…
We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic/linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a…
Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have…
Permissive-Nominal Logic (PNL) is an extension of first-order predicate logic in which term-formers can bind names in their arguments. This allows for direct axiomatisations with binders, such as of the lambda-binder of the lambda-calculus…
Many definitions of weak and strict $\infty$-categories have been proposed. In this paper we present a definition for $\infty$-categories with strict associators, but which is otherwise fully weak. Our approach is based on the existing type…
Initial Semantics aims at interpreting the syntax associated to a signature as the initial object of some category of 'models', yielding induction and recursion principles for abstract syntax. Zsid\'o proves an initiality result for…
Type theory plays an important role in foundations of mathematics as a framework for formalizing mathematics and a base for proof assistants providing semi-automatic proof checking and construction. Derivation of each theorem in type theory…
In this paper we examine a number of term rewriting system for integer number representations, building further upon the datatype defining systems described in [2]. In particular, we look at automated methods for proving confluence and…
We present a type system that combines, in a controlled way, first-order polymorphism with intersectiontypes, union types, and subtyping, and prove its safety. We then define a type reconstruction algorithm that issound and terminating.…