Related papers: Path Types in Algebraic Type Theory
Using the theory of extensions of L-infinity algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley-Eilenberg and Harrison…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
Type inference is an application domain that is a natural fit for logic programming (LP). LP systems natively support unification, which serves as a basic building block of typical type inference algorithms. In particular, polymorphic type…
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…
Ten years ago, it was shown that nominal techniques can be used to design coalgebraic data types with variable binding, so that alpha-equivalence classes of infinitary terms are directly endowed with a corecursion principle. We introduce…
We introduce the notion of a "category with path objects", as a slight strengthening of Kenneth Brown's classic notion of a "category of fibrant objects". We develop the basic properties of such a category and its associated homotopy…
Each family $\mathcal{M}$ of means has a natural, partial order (point-wise order), that is $M \le N$ iff $M(x) \le N(x)$ for all admissible $x$. In this setting we can introduce the notion of interval-type set (a subset $\mathcal{I}…
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…
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…
Regular path query languages for data graphs are essentially \emph{untyped}. The lack of type information greatly limits the optimization opportunities for query engines and makes application development more complex. In this paper we…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose iterated-integrals signature…
Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected…
We present an approach to support partiality in type-level computation without compromising expressiveness or type safety. Existing frameworks for type-level computation either require totality or implicitly assume it. For example, type…
Infinite types and formulas are known to have really curious and unsound behaviors. For instance, they allow to type {\Omega}, the auto- autoapplication and they thus do not ensure any form of normalization/productivity. Moreover, in most…
We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of…
Interval identification of parameters such as average treatment effects, average partial effects and welfare is particularly common when using observational data and experimental data with imperfect compliance due to the endogeneity of…
In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…
LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF's…
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…