A Maximal-Literal Unit Strategy for Horn Clauses

A new positive-unit theorem-proving procedure for equational Horn clauses is presented. It uses a term ordering to restrict paxamodulation to potentially maximal sides of equations. Completeness is shown using proof orderings. 1. I n t r o d u c t i o n A conditional equation is a universally-quantified Horn clause in which the only predicate symbol is equality. We write such a clause in the form e l A . . . A e . :=~ 8 N t (n >__ 0), meaning that the equality s ~ t holds whenever all the equations el, called conditions, hold. If n = 0, then the (positive unit) clause, s -~ t, will be called an unconditional equation. Conditional equations are important for specifying abstract data types and expressing logic programs with equations. Our interest here is in procedures for proving validity of equations in all models of a given finite set E of conditional equations. Note that a conditional equation el A . . . A en ~ s ~-t is valid for E iff s -~ t is valid for E U {e l , . . . , en}. Hence, proving validity of conditional equations reduces to proving validity of unconditional ones. The completeness of positive-unit resolution for Horn clauses is weU-known. An advantage of positive-unit resolution is that the number of conditions never grows; it suffers from the disadvantage of being a bottom-up method. Ordinary Horn clauses pl A " 'Apn ::~ Pn+l where the Pi are not equality literals, can be expressed as conditional equations, by turning each literal Pl into a Boolean equation Pl = T, for the truth constant T. Ordered resolution, in wkie~ the literals of each danse are arranged in a linear order >, and only the largest literal may serve as a resolvent, is also complete for Horn clauses (see Boyer, 1971). Positive-unit resolution can be expressed by means of the following inference rule: EU{ q A s ~ T =} u~-T' ~ .................. l~-T J E U l ~T, qa =~ ua ~ T • Thls research supported in part by the National Science Foundation under Grant CGR-9007195.