The automation of syllogistic. II. optimization and complexity issues
Journal of Automated Reasoning
Computable set theory
First-order logic and automated theorem proving (2nd ed.)
First-order logic and automated theorem proving (2nd ed.)
Set theory for computing: from decision procedures to declarative programming with sets
Set theory for computing: from decision procedures to declarative programming with sets
A Fast Saturation Strategy for Set-Theoretic Tableaux
TABLEAUX '97 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
Towards Tableau-Based Decision Procedures for Non-Well-Founded Fragments of Set Theory
TABLEAUX '00 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
A Tableau Calculus for Integrating First-Order and Elementary Set Theory Reasoning
TABLEAUX '00 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
Combining type theory and untyped set theory
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
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MLSS is a decidable fragment of set theory involving the predicates membership and set equality and the operators union, intersection, set difference, and singleton. In this paper we extend MLSS with the iterated membership predicate, that is, with a predicate denoting the transitive closure of the membership relation. We call the resulting language MLSS+. We prove that MLSS+ is decidable by providing a decision procedure for it based on Smullyan semantic tableaux. As an application of our results, we show how our decision procedure can be used as a black box in order to allow an interactive theorem prover to verify some basic properties of the ordinal numbers.