Abstract. Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite which were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8. 1.

Abstract. A computer implementation of Gödel’s algorithm for class formation in Mathematica TM was used to formulate definitions and theorems about iteration in Gödel’s class theory. The intent is to use this approach for automated reasoning about a variety of applications of iteration using Mc-Cune’s automated reasoning program Otter. The applications include the theory of transitive closures of relations, the arithmetic of natural numbers, construction of invariant subsets, and the Schröder-Bernstein theorem. 1

Abstract. The reflection theorem has been proved using Isabelle/ZF. This theorem cannot be expressed in ZF, and its proof requires reasoning at the meta-level. There is a particularly elegant proof that reduces the meta-level reasoning to a single induction over formulas. Each case of the induction has been proved with Isabelle/ZF, whose built-in tools can prove specific instances of the reflection theorem upon demand. 1

As a part of ongoing research on automated reasoning in set theory, we focus here on an example of a computer proof that involves a recursive definition.

Abstract. We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite first-order axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework. 1

Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of first-order (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of first-order logic; recent

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