Journal of Automated Reasoning

, Volume 34, Issue 1, pp 49–72 | Cite as

A Tableau-Based Decision Procedure for a Fragment of Set Theory with Iterated Membership

  • Domenico Cantone
  • Calogero G. Zarba
  • Rosa Ruggeri Cannata
Article
Accepted:

Abstract

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.

Keywords

set theory decision procedure semantic tableau 
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Copyright information

© Springer 2005

Authors and Affiliations

  • Domenico Cantone
    • 1
  • Calogero G. Zarba
    • 2
  • Rosa Ruggeri Cannata
    • 3
  1. 1.Dipartimento di Matematica e InformaticaUniversità di CataniaCataniaItaly
  2. 2.LORIA and INRIA-LorraineVillers-lès-Nancy CedexFrance
  3. 3.European Commission ESTAT.C.6, Economic indicators of the Euro zoneBruxellesBelgium

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