
Ticking clocks as dependent right adjoints: Denotational semantics for clocked type theory
Clocked Type Theory (CloTT) is a type theory for guarded recursion usefu...
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Guarded Computational Type Theory
Nakano's later modality can be used to specify and define recursive func...
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Nonaccessible localizations
In a 2005 paper, Casacuberta, Scevenels and Smith construct a homotopy i...
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The clocks they are adjunctions:Denotational semantics for Clocked Type Theory
Clocked Type Theory (CloTT) is a type theory for guarded recursion usefu...
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Construction of Convex Sets on Quadrilateral Ordered Tiles or Graphs with Propagation Neighborhood Operations. Dales, Concavity Structures. Application to Gray Image Analysis o
An effort has been made to show mathematicians some new ideas applied to...
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Simple Type Theory is not too Simple: Grothendieck's Schemes without Dependent Types
We report on a formalization of schemes in the proof assistant Isabelle/...
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Bisimulation as path type for guarded recursive types
In type theory, coinductive types are used to represent processes, and a...
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Denotational semantics for guarded dependent type theory
We present a new model of Guarded Dependent Type Theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with, and reason about coinductive types. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion, and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves orthogonal to the object of clocks. In the case of dependent types, this translates to a unique lifting condition similar to the one found in homotopy theoretic models of type theory. Since the universes defined by the standard HofmannStreicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. A large and technical part of the paper is devoted to showing that these can be constructed in such a way that inclusions between universes induced by inclusions of clock variable contexts commute on the nose with type operations on the universes.
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