Abstract
We discuss an algebra of open formulas, SOAF. The algebra is equipped with the operator mgi-o , which computes suprema of sets of open formulas, ordered according to a weak form of instance ordering. Rules to compute such points are provided. These rules supply a computation procedure to unify open formulas. Next, we consider an algebra of closed formulas, SCAF. It has an operator mgi-c , which computes suprema of set of formulas ordered according to the instance preordering. We give rules to compute such points. These rules supply a computation procedure to unify closed formulas. Relations between these operators and unification in open and closed formulas, and weak unification in closed formulas, are addressed. Also, we consider clausal formulas and apply these considerations to the resolution process. This provides a rule alternative to resolution and shows that resolvents can be expressed and effectively obtained as suprema of closed formulas. The new rule need not resort to renaming, unification. and instantiation as first class operations. Finally, applications to resolution in Horn clause logic and logic programming are discussed. This leads to a more compact theory of logic programming, in which, for instance, the domain of substitutions is eliminated.
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