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This chapter (3) 3 is the technical heart of the book. It is gives given an epistemic interpretation of CTT (constructive type theory) in terms of knowledge and information . This interpretation is explicated by considering CTT as a framework for the representation of knowledge processes with the operation of retrieving and extending knowledge. This formalization abstracts comon common processes in computer science. The interpretation given here is due to Giuseppe Primiero and is partially based on ideas ofBorghuis, Kamarredine and Nederpelt.
Basic notions in this type theoretical framework are that of context (which are just the contexts familiar to type theorists) and that of environment (which has been presented by Martin-Löf 1991a who attributes this notion to Peter Landin) which is the logical translation of the notion of computer memory.
There are two specific types
subject to the following rules:
saying that contexts can be used as types and that we write contexts in brackets.
where , , and . This rule declares the formation of a type under a environment . It is an extension of a context in some environment.
where the first bracket expects environments and the second- expects contexts.
This rule can be derived from two rules of computation of environments whose genesis is conceptually crucial to this chapter:
To avoid the circularity arising in the formalization of contexts we need some inductive construction called updating. Contexts are hence built by successive updating of the elements in it:
Borghuis, Kammarredine, Nederpelt, Formalizing Belief Revision in Type Theory, BibTex
Peer Martin-Löf, Contexts and environments, unpublished notes by P. Mäenpaä and A. Rata of a series of lectures given at Stockholm University