Related papers: Andrews' Type Theory with Undefinedness

We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces…

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…

This dissertation introduces executable refinement types, which refine structural types by semi-decidable predicates, and establishes their metatheory and accompanying implementation techniques. These results are useful for undecidable type…

In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…

In this paper, we introduce the notion of relation type of analytic and formal algebras and prove that it is well-defined and invariant by describing this notion in terms of the Andr\'e-Quillen homology and using the Jacobi-Zariski long…

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 consider a family U of finite universes. The second order quantifier Q_R, means for each u in U quantifying over a set of n(R)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called…

We develop square zero obstruction theory for modules over $\mathbb{E}_1$-algebras in an arbitrary stable (presentably) monoidal $\infty$-category. We explicitly describe the obstruction element as the homotopy class of a canonically…

We prove "untyping" theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the…

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…

In this paper, we make a preliminary interpretation of Cook's theorem presented in [1]. This interpretation reveals cognitive biases in the proof of Cook's theorem that arise from the attempt of constructing a formula in CNF to represent a…

The aim of this paper is to give a complete classification of irreducible finite dimensional representations of the nonstandard q-deformation U'_q(so(n)) (which does not coincide with the Drinfeld-Jimbo quantum algebra U_q(so(n)) of the…

Uncertainty quantification (UQ) is the process of systematically determining and characterizing the degree of confidence in computational model predictions. In the context of systems biology, especially with dynamic models, UQ is crucial…

Elaboration-based type class resolution, as found in languages like Haskell, Mercury and PureScript, is generally nondeterministic: there can be multiple ways to satisfy a wanted constraint in terms of global instances and locally given…

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…

We propose a new cubical type theory, termed (self-deprecatingly) the naive cubical type theory, and study its semantics using the universe category framework, which is similar to Uemura's categories with representable morphisms. In…

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 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 show that given a rigid C*-tensor category, there is an equivalence of categories between normalized irreducible Q-systems, also known as connected unitary Frobenius algebra objects, and compact connected W*-algebra objects. Although…

There is an increasing need to integrate model-agnostic explanation techniques with concept-based approaches, as the former can explain models across different architectures while the latter makes explanations more faithful and…