Introduction to higher order categorical logic
Introduction to higher order categorical logic
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Category theory for computing science
Category theory for computing science
Logical frameworks
A framework for defining logics
Journal of the ACM (JACM)
Polymorphism is Set Theoretic, Constructively
Category Theory and Computer Science
Independence Results for Calculi of Dependent Types
Category Theory and Computer Science
A Note on Categorical Datatypes
Category Theory and Computer Science
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In this paper we define a logical framework, called &;lambda;TT, that is well suited for semantic analysis. We introduce the notion of a fibration ℒ1 : ℱ1 ⟹ 𝒞1 being internally definable in a fibration ℒ2 : ℱ2 ⟹ 𝒞2. This notion amounts to distinguishing an internal category L in ℒ2 and relating ℒ1 to the externalization of L through a pullback. When both ℒ1 and ℒ2 are term models of typed calculi ℒ1 and ℒ2, respectively, we say that ℒ1 is an internal typed calculus definable in the frame language ℒ2. We will show by examples that if an object language is adequately represented in λTT, then it is an internal typed calculus definable in the frame language λTT. These examples also show a general phenomenon: if the term model of an object language has categorical structure S, then an adequate encoding of the language in λTT imposes an explicit internal categorical structure S in the term model of λTT and the two structures are related via internal definability. Our categorical investigation of logical frameworks indicates a sensible model theory of encodings.