A Self-Modifying Theorem Prover

Theorem provers can be viewed as containing declarative knowledge (in the form of axioms and lemmas) and procedural knowledge (in the form of an algorithm for proving theorems). Sometimes, as in the case of commutative laws in a Knuth-Bendix prover, it is appropriate or necessary to transfer knowledge from one category to the other. We describe a theorem proving system that independently recognizes opportunities for such transfers and performs them dynamically. Theorem proving algorithms Theorem provers can be divided into two general classes: those that operate without human intervention to prove straightforward consequences of a set of axioms, and those that serve as a mathematician's assistant in the search for a proof of a mathematically significant theorem. The first type of prover is needed for program verification and artificial intelligence applications, where the necessity of human intervention would severely disrupt the intended application. The second type of theorem prover ususally contains one or more of the first type. It is the first type of prover that we are concerned with. Our model of a theorem prover is thus an algorithm that establishes the truth of a statement by showing that it is a logical consequence of a given set of axioms. In the process of establishing that truth, the theorem prover may obtain intermediate results that play the role of lemmas. The power and efficiency of the theorem prover depend on the algorithm that is employed (and there is often a trade-off between these two characteristics of the system). Two major classes of theorem proving algorithms are the resolution-based methods [ROB65,OVE75,BOY71] and Knuth-Bendix type methods [KNU70,HUE80,HUE80b,JEA8O,MUS8O,PET81, STI81]. There are several ambitious theorem provers of the second type that incorporate one or the other of these methods; for example, the Affirm system [TH079] includes a Knuth-Bendix prover, and the ITP theorem prover [MCC76,OVE75,LUS84] uses hyperresolution, an efficient form of resolution, along with other techniques. Both the Knuth-Bendix algorithm and resolution methods can also be used as the basis of an algorithmic theorem prover. The Knuth-Bendix method is usually more efficient in the cases where it applies, but resolution is much more general. Declarative versus procedural knowledge in