We call a Herbrand model of a set of first-order clauses finite, if each of the predicates in the clauses is interpreted by a finite set of ground terms. We consider first-order clauses with the signature restricted to unary predicate and function symbols and one variable. Deciding the existence of a finite Herbrand model for a set of such clauses is known to be ExpTime-hard even when clauses are restricted to an anti-Horn form. Here we present an ExpTime algorithm to decide if a finite Herbrand model exists in the more general case of arbitrary clauses. Moreover, we describe a way to generate finite Herbrand models, if they exist. Since there can be infinitely many minimal finite Herbrand models, we propose a new notion of acyclic Herbrand models. If there is a finite Herbrand model for a set of restricted clauses, then there are finitely many (at most triple-exponentially many) acyclic Herbrand models. We show how to generate all of them.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:79619 |
| Date | 20 June 2022 |
| Creators | Borgwardt, Stefan, Morawska, Barbara |
| Publisher | Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:report, info:eu-repo/semantics/report, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | urn:nbn:de:bsz:14-qucosa2-785040, qucosa:78504 |
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