article

Combining Sets with Cardinals

Author:

Calogero G. ZarbaAuthors Info & Claims

Journal of Automated Reasoning, Volume 34, Issue 1

Pages 1 - 29

https://doi.org/10.1007/s10817-005-3075-8

Published: 01 January 2005 Publication History

Abstract

We introduce a quantifier-free set-theoretic language for combining sets with elements in the presence of the cardinality operator. We prove that the language is decidable by providing a combination method specifically tailored to the combination domain of sets, cardinal numbers, and elements. Our method uses as black boxes a decision procedure for the elements and a decision procedure for cardinal numbers. To be correct, our method requires that the theory of elements be stably infinite. However, we show that if we restrict set variables to range over finite sets only, then one can modify our method so that it works even when the theory of the elements is not stably infinite.

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Cited By

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Chocron PFontaine PRingeissen C(2020)Politeness and Combination Methods for Theories with Bridging FunctionsJournal of Automated Reasoning10.1007/s10817-019-09512-464:1(97-134)Online publication date: 1-Jan-2020
https://dl.acm.org/doi/10.1007/s10817-019-09512-4
Barthe GBelaïd SDupressoir FFouque PGrégoire BStrub PZucchini RWeippl EKatzenbeisser SKruegel CMyers AHalevi S(2016)Strong Non-Interference and Type-Directed Higher-Order MaskingProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security10.1145/2976749.2978427(116-129)Online publication date: 24-Oct-2016
https://dl.acm.org/doi/10.1145/2976749.2978427
Kuncak VMayer MPiskac RSuter P(2013)Functional synthesis for linear arithmetic and setsInternational Journal on Software Tools for Technology Transfer (STTT)10.1007/s10009-011-0217-715:5-6(455-474)Online publication date: 1-Oct-2013
https://dl.acm.org/doi/10.1007/s10009-011-0217-7
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Combining Sets with Cardinals

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Reviews

Reviewer: George Hacken

The introduction to this excellent, instructive, highly theoretical paper is also a clear statement of its general motivation: to enable (decidable) reasoning about "logical formulas involving sets of elements of a given nature (such as integers, reals, or lists)," and about "the cardinalities of these sets" within the context of program verification (safety and termination proofs). Sections 2 and 5 subsequently provide a specific motivational example of the paper's results. The author's two-level syllogistic with cardinality (2LSC) quantifier-free set-theoretic language with equality, which he introduces, comprises elements, cardinal numbers, and sets as sorts, and expresses constraints via first-order signatures. Readers already familiar with more-than-naive set theory, predicate calculus, decision procedures, and such notions as Hoare triple, signatures, sorts, Herbrand universes, polynomial time (P) and nondeterministic polynomial time (NP), and the Lowenheim-Skolem theorem are the target audience. (Zarba recalls the definition of stably infinite, and shows that his decision procedure works for computing applications where set variables necessarily range over finite sets, irrespective of this stability.) I am a down-in-the-trenches user of the set theory intensive B-method, and pose the somewhat philosophical question, "Is set theory really necessary__?__" to myself almost daily (and I have since my university days). The paper, while not purporting to address this ill-posed question, displays exemplary rigor and meticulousness in addressing decidability issues regarding sets, elements, and cardinality. It has the salutary side effect of showing how useful set theory can be to program verification, if its applications are guided by such experts as the paper's author. Online Computing Reviews Service

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cover image Journal of Automated Reasoning

Journal of Automated Reasoning Volume 34, Issue 1

January 2005

99 pages

ISSN:0168-7433

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Berlin, Heidelberg

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Published: 01 January 2005

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Chocron PFontaine PRingeissen C(2020)Politeness and Combination Methods for Theories with Bridging FunctionsJournal of Automated Reasoning10.1007/s10817-019-09512-464:1(97-134)Online publication date: 1-Jan-2020
https://dl.acm.org/doi/10.1007/s10817-019-09512-4
Barthe GBelaïd SDupressoir FFouque PGrégoire BStrub PZucchini RWeippl EKatzenbeisser SKruegel CMyers AHalevi S(2016)Strong Non-Interference and Type-Directed Higher-Order MaskingProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security10.1145/2976749.2978427(116-129)Online publication date: 24-Oct-2016
https://dl.acm.org/doi/10.1145/2976749.2978427
Kuncak VMayer MPiskac RSuter P(2013)Functional synthesis for linear arithmetic and setsInternational Journal on Software Tools for Technology Transfer (STTT)10.1007/s10009-011-0217-715:5-6(455-474)Online publication date: 1-Oct-2013
https://dl.acm.org/doi/10.1007/s10009-011-0217-7
Formisano AOmodeo E(2010)Theory-specific automated reasoningA 25-year perspective on logic programming10.5555/1880413.1880417(37-63)Online publication date: 1-Jan-2010
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Kuncak VMayer MPiskac RSuter P(2010)Complete functional synthesisACM SIGPLAN Notices10.1145/1809028.180663245:6(316-329)Online publication date: 5-Jun-2010
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Kuncak VMayer MPiskac RSuter PZorn BAiken A(2010)Complete functional synthesisProceedings of the 31st ACM SIGPLAN Conference on Programming Language Design and Implementation10.1145/1806596.1806632(316-329)Online publication date: 5-Jun-2010
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Marnette BKuncak VRinard M(2007)Polynomial constraints for sets with cardinality boundsProceedings of the 10th international conference on Foundations of software science and computational structures10.5555/1760037.1760058(258-273)Online publication date: 24-Mar-2007
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Kuncak VRinard M(2007)Towards Efficient Satisfiability Checking for Boolean Algebra with Presburger ArithmeticProceedings of the 21st international conference on Automated Deduction: Automated Deduction10.1007/978-3-540-73595-3_15(215-230)Online publication date: 17-Jul-2007
https://dl.acm.org/doi/10.1007/978-3-540-73595-3_15

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