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Show CHAPTER 1 INTRODUCTION Powerful automated reasoning procedures are key components of "intelligent" machines. Of these procedures, mechanical theorem proving procedures in first-order logic have been a major focus of logicians and computer scientists over the last three decades. The essential problem here is the design of effective and efficient inference systems. A major step forward was the invention of the resolution principle in 1965 by Robinson [99]. Equality is a very important relation, through which many theorems can be conveniently symbolized. For example, elementary number theory can be formalized by employing the equality predicate, the constant function 0, and the successor function [84]. The equality relation has the properties of reflexivity, symmetry, and transitivity. In addition, we can substitute equals for equals and instantiate equations. We will refer to the set of axioms expressing these properties as the classical theory of equality. In resolution procedure(s), we must treat equality as simply one more binary predicate symbol, with a collection of extra axioms describing the properties of the equality relation. This way of treating equality is entirely infeasible in practice (see, for example, [98, 1 00, 118]). Because the special relation of equality commonly occurs in interpretations of first-order theories and Robinson's resolution procedure cannot handle it effectively, special methods are sought when designing mechanical theorem proving procedures. A mechanism superimposed on the resolution procedure(s), named paramodulation, was formalized by G. Robinson and L. Wos in this con- |
