Related papers: Foundations of Substructural Dependent Type Theory
We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured…
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a…
Neural networks excel at pattern recognition but struggle with reliable logical reasoning, often violating basic logical principles during inference. We address this limitation by developing a categorical framework that systematically…
In this article we show that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby…
We introduce the Delta-framework, LF-Delta, a dependent type theory based on the Edinburgh Logical Framework LF, extended with the strong proof-functional connectives, i.e. strong intersection, minimal relevant implication and strong union.…
Higher-order logic HOL offers a very simple syntax and semantics for representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a…
In dependent type theory, being able to refer to a type universe as a term itself increases its expressive power, but requires mechanisms in place to prevent Girard's paradox from introducing logical inconsistency in the presence of…
This manuscript presents a novel framework that integrates higher-order symmetries and category theory into machine learning. We introduce new mathematical constructs, including hyper-symmetry categories and functorial representations, to…
Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws,…
This thesis is concerned with type-logical grammars and their practical applicability as tools of reasoning about sentence syntax and semantics. The focal point is narrowed to Dutch, a language exhibiting a large degree of word order…
We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it.…
Derivations provide a way of transporting ideas from the calculus of manifolds to algebraic settings where there is no sensible notion of limit. In this paper, we consider derivations in certain monoidal categories, called codifferential…
Dependent pattern matching is a key feature in dependently typed programming. However, there is a theory-practice disconnect: while many proof assistants implement pattern matching as primitive, theoretical presentations give semantics to…
This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized…
Existing methods for differentiable structure learning in discrete data typically assume that the data are generated from specific structural equation models. However, these assumptions may not align with the true data-generating process,…
Graded Type Theory provides a mechanism to track and reason about resource usage in type systems. In this paper, we develop GraD, a novel version of such a graded dependent type system that includes functions, tensor products, additive…
We introduce layers to modal type theories, which subsequently enables type theories for pattern matching on code in meta-programming and clean and straightforward semantics.
Type analyses of logic programs which aim at inferring the types of the program being analyzed are presented in a unified abstract interpretation-based framework. This covers most classical abstract interpretation-based type analyzers for…
We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes…
Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been…