Recognizing Unnecessary Inference

Intell igent reasoners sometimes draw conclusions that lack new or relevant information. Similarly, automated reasoning systems can produce formulas that are not necessary for the problem at hand. We concentrate on the problem of unnecessary inference in the context of resolution based systems. In such systems several strategies have been developed that allow for the deletion of clauses wi thout sacrificing completeness. Unfortunately these strategies fai l to recognize other frequently generated unnecessary formulas. We wi l l present a generalized subsumption theorem that can be used to recognize such formulas and to develop new deletion methods which retain completeness. 1 I n t r o d u c t i o n Intell igent reasoners expend much effort deciding what information is necessary for the problem at hand. When a conclusion is drawn we decide if it contains new and relevant information. It is well known that the performance of automated reasoning systems can be enhanced by el iminat ing unnecessary formulas. In such systems conclusions drawn from unnecessary formulas are also unnecessary. Failure to prevent the generation of these formulas can lead to rapid combinatorial explosion [Wos, 1988]. In this paper we concentrate on the problem of recognit ion of unnecessary formulas in the context of resolut ion based systems. In such systems several strategies have been developed that allow for the deletion of clauses wi thout sacrificing completeness. Common strategies include subsumption, tautology el imination, and demodulat ion. The subsumption strategy eliminates clauses that are instances of other clauses. The demodulation strategy pertains to clauses containing the equality predicate. Using this strategy demodulators are used to rewrite clauses that are subsequently deleted. Unfortunately these strategies fail to recognize other frequently generated unnecessary formulas. A less obvious way in which resolution based systems produce unnecessary formulas is related to skolemization. Skolem functions have the effect of automatically creating names for new objects, but sometimes too many names are creThe main result discussed here can be viewed as a generalization of subsumption. This result can be used to develop new deletion methods and to show that these methods do not sacrifice completeness. Using deletion methods we have developed, we were able to prove certain theorems from Hilbert 's axioms for geometry. These theorems present a significant challenge to automated systems due to the constructive nature of the proofs. For a discusion of these theorems see [Benanav, 1988]. In section 2 we discuss an example that illustrates a subtle way in which resolution systems generate unnecessary formulas and present informal arguments as to why these formulas are unnecessary. In later sections we define more precisely what we mean by unnecessary clauses and how to recognize them. 2 The Naming Problem In mathematical arguments one often sees inferences that assert the existence of some object and give it a name. Subsequently, other inferences are made that refer to the object by this name. This process is so natural 366 Automated Deduction