A Theory of Modules for Logic Programming

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A Theory of Modules for Logic Programming (1986)

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by Dale Miller

Venue:	In Symp. Logic Programming
Citations:	45 - 6 self

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@INPROCEEDINGS{Miller86atheory,
    author = {Dale Miller},
    title = {A Theory of Modules for Logic Programming},
    booktitle = {In Symp. Logic Programming},
    year = {1986},
    pages = {106--114}
}

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Abstract

Abstract: We present a logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses. The operational meaning of a goal which is an implication is given by the deduction theorem. That is, a goal D ⊃ G is satisfied by a program P if the goal G is satisfied by the larger program P ∪ {D}. If the formula D is the conjunction of a collection of universally quantified clauses, we interpret the goal D ⊃ G as a request to load the code in D prior to attempting G, and then unload that code after G succeeds or fails. This extended use of implication provides a logical explanation of parametric modules, some uses of Prolog’s assert predicate, and certain kinds of abstract datatypes. Both a model-theory and proof-theory are presented for this logical language. We show how to build a possible-worlds (Kripke) model for programs by a fixed point construction and show that the operational meaning of implication mentioned above is sound and complete for intuitionistic, but not classical, logic. 1. Implications as Goals Let A be a syntactic variable which ranges over atomic formulas of first-order logic. Let G range over a class of formulas, called goal formulas, to be specified shortly. We shall assume, however, that this class always contains ⊤ (true) and all atomic formulas. The formulas represented by A and G may contain free variables. Given these two classes, we define definite clauses, denoted by the syntactic variable D, as follows: D: = G ⊃ A | ∀x D | D1 ∧ D2 A program is defined to be a finite set of closed definite clauses. P will be a syntactic variable for programs. A clause of the form ⊤ ⊃ A will often be written as simply A. Let P be a program. Define [P] to be the smallest set of formulas satisfying the following recursive definitions. (i) P ⊆ [P].

Keyphrases

syntactic variable logic programming operational meaning atomic formula logical language parametric module assert predicate free variable d1 d2 program quantified clause logical explanation closed definite clause definite clause certain kind abstract datatypes deduction theorem goal formula extended use first-order logic finite set positive horn clause fixed point construction following recursive definition

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