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Categorical properties of logical frameworks

Published online by Cambridge University Press:  01 February 1997

YUXI FU
Affiliation:
Present address: Department of Computer Science, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, People's Republic of China. Department of Computer Science, University of Manchester, Manchester M13 9PL, England

Abstract

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 [Lscr ]1 : [Fscr ]1 [xrarr ] [Cscr ] 1 being internally definableThe definability as used in this paper should not be confused with Bénabou's ‘definability’ (Bénabou 1985). in a fibration [Lscr ]2 : [Fscr ]2 [xrarr ] [Cscr ] 2. This notion amounts to distinguishing an internal category L in [Lscr ]2 and relating [Lscr ]1 to the externalization of L through a pullback. When both [Lscr ]1 and [Lscr ]2 are term models of typed calculi [Lscr ]1 and [Lscr ]2, respectively, we say that [Lscr ]1 is an internal typed calculus definable in the frame language [Lscr ]2. 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.

Type
Research Article
Copyright
© 1997 Cambridge University Press

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