Graph-Based Proof Procedures for Horn Clauses by RAATZ


Preliminaries.- A Semantics for the Hornlog System.- The Hornlog facts Procedure.- Soundness and Completeness effects I.- An Equational Extension.- The He � Refutation Method.- Soundness and Completeness effects II.- Appendix: Implementation concerns.

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Graph-Based Proof Procedures for Horn Clauses

Preliminaries. - A Semantics for the Hornlog procedure. - The Hornlog facts technique. - Soundness and Completeness effects I. - An Equational Extension. - The He � Refutation approach. - Soundness and Completeness effects II. - Appendix: Implementation concerns.

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2) lingle-Iolution protocol: The system returns only one substitution. This assumption has no counterpart in Prolog. The idea is best understood as a simultaneous attempt to explore all possible trials towards unsatisfiability, and to return the first one that succeeds. We will comment more on this protocol after presenting the method. 2 Construction and Ezpansion of H -graphs 39 The two protocols are mutually exclusive. If an H -graph is initially expanded under one protocol, it cannot later at any state be expanded under the other.

And all descendants of 1. whose age is not equal to age(X). 3 This will be clear when we present the algorithm which performs this check on H-graphs in chapter 9. 4 42 THE HORNLOG PROOF PROCEDURE We are at last ready to define the general graph expansion step. 2 (General graph expansion step) Let G be an H-graph obtained at some stage of expansion, node X E G be chosen for expansion, and L = ((Ol,0't},(02,0'2), ... ,(Ok,0'0) be the list of all pairs such that a node X unifies with the head Aj of definite clause OJ with most general unifier O'j (variables have been renamed apart).

G'. Note that in the all-solutions protocol, merging of nodes only occurs within subgraphs whose nodes have the same age. Under the single-solution protocol, the only condition for merging nodes is that their labds are identical, and nodes having different ages can be merged. We mention the following procedural aspect of the method before presenting some examples. e. that no node waits forever for expansion. This corresponds to a breadth-first expansion strategy. To ensure fairness, the &tatu& field of each node is assigned one of the following four values: 1.

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