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Parametric completeness for separation theories

Published:08 January 2014Publication History
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Abstract

In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation models is usually specified by a collection of axioms describing the specific model properties that are expected to hold, which we call a separation theory.

Our main contributions are as follows. First, we show that several typical properties of separation theories are not definable in BBI. Second, we show that these properties become definable in a suitable hybrid extension of BBI, obtained by adding a theory of naming to BBI in the same way that hybrid logic extends normal modal logic. The binder-free extension captures most of the properties we consider, and the full extension HyBBI(↓) with the usual ↓ binder of hybrid logic covers all these properties. Third, we present an axiomatic proof system for our hybrid logic whose extension with any set of "pure" axioms is sound and complete with respect to the models satisfying those axioms. As a corollary of this general result, we obtain, in a parametric manner, a sound and complete axiomatic proof system for any separation theory from our considered class. To the best of our knowledge, this class includes all separation theories appearing in the published literature.

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              • Published in

                cover image ACM SIGPLAN Notices
                ACM SIGPLAN Notices  Volume 49, Issue 1
                POPL '14
                January 2014
                661 pages
                ISSN:0362-1340
                EISSN:1558-1160
                DOI:10.1145/2578855
                Issue’s Table of Contents
                • cover image ACM Conferences
                  POPL '14: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
                  January 2014
                  702 pages
                  ISBN:9781450325448
                  DOI:10.1145/2535838

                Copyright © 2014 ACM

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                • Published: 8 January 2014

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