By RAATZ

ISBN-10: 0817635300

ISBN-13: 9780817635305

ISBN-10: 1489935568

ISBN-13: 9781489935564

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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**Extra resources for Graph-Based Proof Procedures for Horn Clauses**

**Example text**

6. 5 with substitution [a/V). 1.. 4 Note that a merging of nodes labeled u( a, f( a)) has occurred, but has not resulted in any redirection of arcs from the two nodes. 2 Construction and E:l:pansion of H -graphs is equivalent to marking the second node as "already seen", as done in various methods that use AND/OR trees. 5, however, the two methods are not equivalent. 7. 1.. 7 Intermediate expansion stage The node labeled u(Y, J( a)) has incoming arcs from all the nodes labeled {mI,"" mk}, and outgoing arcs to all the nodes labeled {mk+l,'''' mk+i}.

Proof: First, we prove that if D U {N1 , ••• ,N1o } is unsatisfiable, then D contains some atomic formula and the set {N1, ... ,N1o} is nonempty. If D does not contain any atomic formulae, then every formula in D U {Nl , ... , N 1o} contains some negative literal. Then, D U {NIt ... , N1o} is satisfied in the one-point structure such that every predicate symbol is interpreted as the constant function false. If {Nl' ... ,N1o} = 0, then every formula in D contains some (positive) atomic formula. Then, D is satisfied in the one-point structure such that, for every atomic formula of the form P(tl,"" t,,) in D, p is interpreted as the constant function true.

Q} by the method of P. We will also be able to make precise the relationship between the substitutions 8 and the n-tuples they represent. It is clear that we would like that the relationships Mp,Q ~ Op,Q and Op,Q ~ hold in order to establish that the operational and declarative semantics are the same. However, a relation between Mp,Q and Op,Q other than that of equality between sets is necessary, since for n-tuples {t 1, ... • V Q( t ih ) for a proper subset {i 1 , ... ,ih} of {I, ... , the disjunction Q(t1)V ...

### Graph-Based Proof Procedures for Horn Clauses by RAATZ

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