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Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
A Framework for Using Knowledge in Tableau Proofs
 Proc. International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, PontMousson
, 1997
"... . The problem of automatically reasoning using a knowledge base containing axioms, definitions and theorems from a firstorder theory is recurrent in automated reasoning research. Here we present a sound and complete method for reasoning over an arbitrary firstorder theory using the tableau cal ..."
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. The problem of automatically reasoning using a knowledge base containing axioms, definitions and theorems from a firstorder theory is recurrent in automated reasoning research. Here we present a sound and complete method for reasoning over an arbitrary firstorder theory using the tableau calculus. A natural, wellmotivated and simple restriction (implemented in IPR) to the method provides a powerful framework for the automation of the selection of theorems from a knowledge base for use in theorem proving. The restrictions are related to semantic resolution restrictions and the setofsupport restriction in resolution, and to hypertableaux and the weak connection condition in tableaux. We also present additional tableau rules used by the IPR prover for handling some equality which is not complete but is sufficient for handling the problems in its intended domain of problem solving. 1 Introduction The rules presented in this paper allow an automatic theorem proving pro...
Nonstandard Set Theories and Information Management
"... . The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems. In this paper, a study of various nonstandard treatments of set theory from this perspective is offered. Applications of thes ..."
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. The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems. In this paper, a study of various nonstandard treatments of set theory from this perspective is offered. Applications of these alternative set theories to information or knowledge management are surveyed. Keywords: set theory, knowledge representation, information management, commonsense reasoning, nonwellfounded sets (hypersets) 1. Introduction Set theory is a branch of modern mathematics with a unique place because various other branches can be formally defined within it. For example, Book 1 of the influential works of N. Bourbaki is devoted to the theory of sets and provides the framework for the remaining volumes. Bourbaki said in 1949 (Goldblatt, 1984) 1 : "[A]ll mathematical theories may be regarded as extensions of the general theory of sets : : : [O]n these foundations I can state that I can build up t...
Comprehension and Description in Tableaux
, 1997
"... Various approaches have been invented for enabling an automated theorem proving program to find proofs in set theory. The present approach is completely automatic and quite successful on many problems which are showcased as challenge problems for provers in set theory. In fact, this procedure fi ..."
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Various approaches have been invented for enabling an automated theorem proving program to find proofs in set theory. The present approach is completely automatic and quite successful on many problems which are showcased as challenge problems for provers in set theory. In fact, this procedure finds proofs of several of these examples without search. We implement the comprehension schema by means of tableau reduction and expansion rules. We also discuss the implementation of the definite descriptor in tableaux and special rules for handling equality effectively and in a tractable way in set theory. 1 Introduction An inference rule that "builds in" set theory at the inference level is the objective of Research Problem 8. More precisely, just as the employment of paramodulation permits one to avoid using any equality axioms other than reflexivity, the soughtafter inference rule for set theory would permit one to avoid using a number of the axioms in Godel's approach. L...
A Decidable Tableau Calculus for a Fragment of Set Theory With Iterated Membership
 II. Optimization and Complexity Issues. Journal of Automated Reasoning
, 1997
"... this paper we give a decision procedure and a decidable tableau calculus for the extension of Multilevel Syllogistic ..."
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this paper we give a decision procedure and a decidable tableau calculus for the extension of Multilevel Syllogistic
INDUCTIVE REASONING IN MATHEMATICS
"... We investigate several methods of inductive reasoning in the domain of difference equations, including the method of generalization with beliefs, the method of successive refinement, and temporal methods based on comparisons with previously solved problems. 1. ..."
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We investigate several methods of inductive reasoning in the domain of difference equations, including the method of generalization with beliefs, the method of successive refinement, and temporal methods based on comparisons with previously solved problems. 1.
A Theorem Prover For Meta Theory
 Proc. Fourth Workshop on Automated Deduction
, 1979
"... We describe an automatic theorem prover for 'meta theory which is capable of proving the completeness of quantificational logic from intuitively true assumptions. ..."
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We describe an automatic theorem prover for 'meta theory which is capable of proving the completeness of quantificational logic from intuitively true assumptions.
on Symmetric Logic. Symmetric Logic unlike Sequent Logic Natural Deduction
"... We describe an automatic theorem prover for real algebra which is based on Symmetric Logic. Symmetric Logic unlike Sequent Logic Natural Deduction and Resolution, is based on treating the universal and existential quantifiers analogously, and on treating conjunctions and disjunctions analogously ..."
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We describe an automatic theorem prover for real algebra which is based on Symmetric Logic. Symmetric Logic unlike Sequent Logic Natural Deduction and Resolution, is based on treating the universal and existential quantifiers analogously, and on treating conjunctions and disjunctions analogously Proofs of various theorems of real algebra involving mathematical induction have been obtained. Contents 1.