Rewriting and reasoning with set-relations II: The non-ground case completeness

  • Valentinas Kriaučiukas
  • Michał Walicki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1130)


We consider reasoning and rewriting with set-relations: inclusion, non-empty intersection and singleton identity, each of which satisfies only two among the three properties of the equivalence relations. The paper presents a complete inference system which is a generalization of ordered paramodulation and superposition calculi. Notions of rewriting proof and confluent rule system are defined for such non-equivalence relations. Together with the notions of forcing and redundancy they are applied in the completeness proof. Ground completeness cannot be lifted to the non-ground case because substitution for variables is restricted to deterministic terms. To overcome the problems of restricted substitutivity and hidden (in relations) existential quantification, unification is defined as a three step process: substitution of determistic terms, introduction of bindings and “on-line” skolemisation. The inference rules based on this unification derive non-ground clauses even from the ground ones, thus making an application of a standard lifting lemma impossible. The completness theorem is proved directly without use of such a lemma.


rewriting theorem proving binary relations nondeterminism completeness ordered superposition 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Valentinas Kriaučiukas
    • 1
  • Michał Walicki
    • 2
  1. 1.Matematikos ir Informatikos InstitutasVilniusLithuania
  2. 2.Institutt for InformatikkUniversitetet i Bergen, HiBBergenNorway

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