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Title: Proofs, search and computation in general logic
Author: Pym, David J.
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1990
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The Logical Framework (LF) is a formal system for the representation of logics and formal systems as theories within it, which was developed by Harper, Ilonscll and Plotkin. The LF consists of two components: the All-calculus, the A-calculus with dependent function space (or product) types; and the principle of judgements as types in which judgements, the units of inference in logics and other formal systems, are identified with the types of the All-calculus. This work is strongly influenced by the work of Martin-L6f on judgements in logic. The LF is suitable for mechanical implementation because the AlJ-calculus is decidable. We present a theory of search and logic programming for the Alt-calculus and consequently for logics which are adequately represented in the LF. The presentation of the All-calculus of Harper, Honsell and Plotkin is as a (linearized) natural deduction system. The search space induced by such a system is highly non-deterministic and so the first step is to define a Gcntzenized system, in which the natural deduction style II-elimination rule is replace by a sequent calculus style II-elimination rule, which is sound and complete with respect to the system of Harper, Honsell and Plotkin. The Gentzenized system thus provides the foundation for a metacalculus, possessing an important subformula property, for the inhabitation assertions of the A-calculus which forms a suitable basis for proof-search in the A-11-calculus. By exploiting the structure of a form of hypolhetica-general judgement, we arc able to obtain (complete) calculi based on certain forms of resolution rule, the search spaces of which are properly contained within that of our basic metacalculus. Furthermore, we are able to further constrain proof-search in these calculi by considering a certain uniform form for proofs. We develop a unification algorithm for the All-calculus by extending the work of Huet for the simply-typed A-calculus. We use this unification algorithm to allow us to eliminate non-deterministic choices of terms, when performing proof-search with the (II/) or resolution rules. This is done by the introduction of a class of universal variables which are later instantiated via the unification algorithm. When unification is used to instantiate universal variables well-formedness may not be preserved. However, by extending the work of Bibel and Wallen (for first-order classical, modal and inluitionistic logics) to the All-calculus we are able to obtain a theory which allows to accept as many such instantiations as possible by detecting when the derivation can be reordered in order to yield a well-formed proof. We extend the theory described above to provide a notion of logic programming by admitting universal variables in cndsequents: these are analogous to the logical variables of PROLOG. Finally, we provide a denota-tional semantics for our logic programs by performing a least fixed point construction in a collection of Herbrand interpretations - maps from |C(£)| to (families of) sets of terms in | 7 |, where C(£) is the syntactic category constructed out of the All-calculus with signature E, and 7 is the category of families of sets. This construction is similar to one of Miller, and exploits a Kripke-like satisfaction predicate. We characterize this construction in terms of the model theory of the All-calculus using the Yoneda functor.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available