Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.651333
Title: Representing logics in type theory
Author: Gardner, Philippa
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1992
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Abstract:
Computer Science today has many examples of logics given by proof systems. Although one intuitively knows how to use these systems and recognise correct derivations, there is no definitive account which captures this intuition. It is therefore natural to seek a framework for representing logics, which unifies the structure common to all logical systems. We introduce such a framework, called ELF+ and based on the Edinburgh Logical Framework (ELF). The major advantage of ELF&43 is that it allows us to give precise definitions of representation. Such definitions are not possible with ELF since information is lost during encoding; the adequacy theorems of ELF representations are only applicable to particular encodings and cannot be generalised. We rectify this deficiency using the extra distinctions between terms provided by the universes of a pure type system which yields a simple presentation of the type theory of ELF+ . To do this, we extend these type systems to include signatures and βeta-equivalence. Using the ideas underlying representation in ELF+ , we give a standard presentation of the logics under consideration, based on Martin-Lof's notion of judgements and Aczel's work on Frege structures. This presentation forms a reference point from which to investigate representations in ELF+ it is not itself a framework since we do not specify a logic using a finite amount of information. Logics which do not fit this pattern are particularly interesting as they are more difficult, if not impossible, to encode. The syntactic definitions of representations have an elegant algebraic formulation which utilises the abstract view of logics as consequence relations. The properties of the ELF+ entailment relation determine the behaviour of the variables and consequence relations of the logics under consideration. Encodings must preserve this common structure. This motivates the presentation of the logics and their corresponding type theories as strict indexed categories (or split fibrations) so that encodings give rise to indexed functors. The syntactic notions of representation now have a simple formulation as indexed isomorphisms.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.651333  DOI: Not available
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