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Context logic as modal logic: completeness and parametric inexpressivity

Published:17 January 2007Publication History
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Abstract

Separation Logic, Ambient Logic and Context Logic are based on a similar style of reasoning about structured data. They each consist of a structural (separating) composition for reasoning about disjoint subdata, and corresponding structural adjoint(s) for reasoning hypothetically about data. We show how to interpret these structural connectives as modalities in Modal Logic and prove completeness results. The structural connectives are essential for describing properties of the underlying data, such as weakest preconditions for Hoare reasoning for Separation and Context Logic, and security properties for Ambient Logic. In fact, we introduced Context Logic to reason about tree update, precisely because the structural connectives of the Ambient Logic did not have enough expressive power. Despite these connectives being essential, first Lozes then Dawar, Gardner and Ghelli proved elimination results for Separation Logic and Ambient Logic (without quantifiers). In this paper, we solve this apparent contradiction. We study parametric inexpressivity results, which demonstrate that the structural connectives are indeed fundamental for this style of reasoning.

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  1. Context logic as modal logic: completeness and parametric inexpressivity

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                  cover image ACM SIGPLAN Notices
                  ACM SIGPLAN Notices  Volume 42, Issue 1
                  Proceedings of the 2007 POPL Conference
                  January 2007
                  379 pages
                  ISSN:0362-1340
                  EISSN:1558-1160
                  DOI:10.1145/1190215
                  Issue’s Table of Contents
                  • cover image ACM Conferences
                    POPL '07: Proceedings of the 34th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
                    January 2007
                    400 pages
                    ISBN:1595935754
                    DOI:10.1145/1190216

                  Copyright © 2007 ACM

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                  • Published: 17 January 2007

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