Use of unit clauses and clause splitting in automatic deduction ...

Use of Unit Clauses and Clause Splitting in Automatic Deduction * David A. Plaisted Department of Computer Science University of North Carolina Chapel Hill, NC 27599-3175, U.S.A.

Shie-Jue Lee Department of Electrical Engineering National Sun Yat-Sen University Kaohsiung, Taiwan 80424, R.O.C.

unified with be unit clauses. Small proof checking requires that literals to be unified with be unit clauses or UR resolvents. All these four techniques take advantage of unit clauses. Making the value of n small is done by clause split*ting. New predicate symbols are introduced and used to separate a long clause C into shorter clauses. In this way, we replace the time complexity N ” for processing C to ELl N l n nwhere m is the number of shorter clauses obtained from C, N1 is the new average number of literals to be considered for unifying any literal, and ni is the number of literals contained in each shorter clause. Normally, we don’t need to add too many new predicate symbols, so N is not much different from N I . Notice that we want all ni, 1 i m to NI”’ N n . be smaller than n. In many cases, In the sequel, we start with a brief introduction to the hyper-linking theorem prover. Then we discuss how to use unit clauses to reduce the search space. We then present the clause splitting technique.

Abstract A mechanical theorem prover usually has t o perform unification which is a very time-consuming operation. Therefore, it is necessary t o reduce the number of unification operations t o obtain speedups. There are two ways t o do this. One i s t o restrict the literals t o be unified with the underlying literal, and the other is t o make clauses small. Four techniques: unit simplification, ground unit clause generation, UR simulation, and small proof checking, can restrict the literals t o be unified with. All these techniques take advantage of unit clauses. T h e clause splitting technique is able t o reduce the length of a clause b y splitting the clause into shorter clauses. These techniques improve dramatically the e f i c i e n c y of a theorem prover, using the hyper-linking method, i n m a n y cases.

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