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A Ring-Theoretic Basis for Logic Programming

Investigations into the semantics of logic programming have largely restricted themselves to the case when the set of truth values being considered is a complete lattice. While a few theorems have been obtained which make do with weaker structures, to our knowledge there is no semantical characterization of logic programming which does not require that the set of truth values be partially ordered. We derive here semantical results on logic programming over a space of truth values that forms a commutative pseudo-ring (an algebraic structure a bit weaker than a ring) with identity. This permits us to study the semantics of multi-valued logic programming having a (possibly) non-partially ordered set of truth values.