Related papers: Stratified Type Theory
Simplicial type theory (STT) was introduced by Riehl and Shulman to leverage homotopy type theory to prove results about $(\infty,1)$-categories. Initial work on simplicial type theory focused on "formal" arguments in higher category theory…
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…
We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types.…
Large knowledge graphs increasingly add value to various applications that require machines to recognize and understand queries and their semantics, as in search or question answering systems. Latent variable models have increasingly gained…
Large knowledge bases typically contain data adhering to various schemas with incomplete and/or noisy type information. This seriously complicates further integration and post-processing efforts, as type information is crucial in correctly…
We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of shape "Gamma |- A : B", our systems just have judgments of shape "A : B". A key feature is that we distinguish free…
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…
We give a new characterization of elementary and deterministic polynomial time computation in linear logic through the proofs-as-programs correspondence. Girard's seminal results, concerning elementary and light linear logic, achieve this…
Erasure enriches type theory with a distinction between runtime relevant and irrelevant data, allowing the compilation step to safely erase the latter. Versions of this feature are implemented by many systems, including Agda, Idris, and…
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…
While the manifold hypothesis is widely adopted in modern machine learning, complex data is often better modeled as stratified spaces -- unions of manifolds (strata) of varying dimensions. Stratified learning is challenging due to varying…
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-L\"of Type Theory to asymmetric hom-types. We articulate…
We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple…
We present Stratified Metric Temporal Logic (SMTL), a novel formalism for specifying and verifying properties of complex cyber-physical systems that exhibit behaviors across multiple temporal and abstraction scales. SMTL extends existing…
Benefits of static type systems are well-known: they offer guarantees that no type error will occur during runtime and, inherently, inferred types serve as documentation on how functions are called. On the other hand, many type systems have…
Multilevel modeling extends traditional modeling techniques with a potentially unlimited number of abstraction levels. Multilevel models can be formally represented by multilevel typed graphs whose manipulation and transformation are…
A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences…
We introduce regular graph constraints and explore their decidability properties. The motivation for regular graph constraints is 1) type checking of changing types of objects in the presence of linked data structures, 2) shape analysis…
When scripts in untyped languages grow into large programs, maintaining them becomes difficult. A lack of explicit type annotations in typical scripting languages forces programmers to must (re)discover critical pieces of design information…
This paper describes a design that can be used for Explainable AI. The lower level is a nested ensemble of patterns created by self-organisation. The upper level is a hierarchical tree, where nodes are linked through individual concepts, so…