Related papers: From type theory to setoids and back
The lambda-Pi-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory U, where proofs of several logical systems can be expressed. Moreover, we identify a…
We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a…
As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in…
Conventional bag-of-words approaches for topic modeling, like latent Dirichlet allocation (LDA), struggle with literary text. Literature challenges lexical methods because narrative language focuses on immersive sensory details instead of…
We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version…
We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTW, including Weyl's definition of…
In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory…
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 introduce a generic, compositional and interpretable class of generative world models that supports open-ended learning agents. This is a sparse class of Bayesian networks capable of approximating a broad range of stochastic processes,…
Revisiting a classic result from M. Hofmann's dissertation, we give a direct proof of Morita equivalence, in the sense of V. Isaev, between extensional type theory and intensional type theory extended by the principles of functional…
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 describe a type system for a platform called the General Intensional Programming System (GIPSY), designed to support intensional programming languages built upon intensional logic and their imperative counter-parts for the intensional…
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general…
We propose an automated deduction method which allows us to produce proofs close to the human intuition and practice. This method is based on tableaux, which generate more natural proofs than similar methods relying on clausal forms, and…
We extend intersection types to a computational $\lambda$-calculus with algebraic operations \`a la Plotkin and Power. We achieve this by considering monadic intersections, whereby computational effects appear not only in the operational…
We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that…
We construct a left semi-model structure on the category of intensional type theories (precisely, on $\mathrm{CxlCat_{Id,1,\Sigma(,\Pi_{ext})}}$). This presents an $\infty$-category of such type theories; we show moreover that there is an…
It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming specific instances of non-parametricity. We also address…
We translate properties of the Sigma-type in Martin-L\"of Type Theory (MLTT) to properties of the Grothendieck construction in category theory. Namely, equivalences in MLTT that involve the Sigma-type motivate isomorphisms between…
We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-L\"of's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of…