An extended Herbrand theorem for first-order theories with equality interpreted in partial algebras

  • Uwe Petermann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 379)


Two derivation operations are defined which are sound and complete for interpretations of first-order theories with equality in classes of partial algebraic structures. Łoś' theorem on ultraproducts has been generalized for such interpretations.

The following generalized form of Herbrand's theorem is proved: For an arbitrary first-order theory & and an arbitrary formula α may be constructed an enumerable set of open formulas Hα such that & ↣ α holds iff there exists β ε Hα with & ↣ β. whereby ↣ denotes either the usual or one of the mentioned above derivation operators. The enumeration of Hα is determined by a procedure which is closely related to the connection method proof procedure.

Key words

Theorem Proving Herbrand Disjunctions Connection Method Partial Algebras Built in Theories 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Uwe Petermann
    • 1
  1. 1.Department of MathematicsKarl Marx University Karl-Marx-PlatzLeipzigGDR

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