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**Published in:** LIPIcs, Volume 279, 34th International Conference on Concurrency Theory (CONCUR 2023)

In this paper we study a Separation Logic of Relations (SLR) and compare its expressiveness to (Monadic) Second Order Logic [(M)SO]. SLR is based on the well-known Symbolic Heap fragment of Separation Logic, whose formulæare composed of points-to assertions, inductively defined predicates, with the separating conjunction as the only logical connective. SLR generalizes the Symbolic Heap fragment by supporting general relational atoms, instead of only points-to assertions. In this paper, we restrict ourselves to finite relational structures, and hence only consider Weak (M)SO, where quantification ranges over finite sets. Our main results are that SLR and MSO are incomparable on structures of unbounded treewidth, while SLR can be embedded in SO in general. Furthermore, MSO becomes a strict subset of SLR, when the treewidth of the models is bounded by a parameter and all vertices attached to some hyperedge belong to the interpretation of a fixed unary relation symbol. We also discuss the problem of identifying a fragment of SLR that is equivalent to MSO over models of bounded treewidth.

Radu Iosif and Florian Zuleger. Expressiveness Results for an Inductive Logic of Separated Relations. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{iosif_et_al:LIPIcs.CONCUR.2023.20, author = {Iosif, Radu and Zuleger, Florian}, title = {{Expressiveness Results for an Inductive Logic of Separated Relations}}, booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-299-0}, ISSN = {1868-8969}, year = {2023}, volume = {279}, editor = {P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.20}, URN = {urn:nbn:de:0030-drops-190146}, doi = {10.4230/LIPIcs.CONCUR.2023.20}, annote = {Keywords: Separation Logic, Model Theory, Monadic Second Order Logic, Treewidth} }

Document

**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider concurrent systems consisting of replicated finite-state processes that synchronize via joint interactions in a network with user-defined topology. The system is specified using a resource logic with a multiplicative connective and inductively defined predicates, reminiscent of Separation Logic [John C. Reynolds, 2002]. The problem we consider is if a given formula in this logic defines an invariant, namely whether any model of the formula, following an arbitrary firing sequence of interactions, is transformed into another model of the same formula. This property, called havoc invariance, is quintessential in proving the correctness of reconfiguration programs that change the structure of the network at runtime. We show that the havoc invariance problem is many-one reducible to the entailment problem ϕ ⊧ ψ, asking if any model of ϕ is also a model of ψ. Although, in general, havoc invariance is found to be undecidable, this reduction allows to prove that havoc invariance is in 2EXP, for a general fragment of the logic, with a 2EXP entailment problem.

Marius Bozga, Lucas Bueri, and Radu Iosif. On an Invariance Problem for Parameterized Concurrent Systems. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bozga_et_al:LIPIcs.CONCUR.2022.24, author = {Bozga, Marius and Bueri, Lucas and Iosif, Radu}, title = {{On an Invariance Problem for Parameterized Concurrent Systems}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {24:1--24:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.24}, URN = {urn:nbn:de:0030-drops-170874}, doi = {10.4230/LIPIcs.CONCUR.2022.24}, annote = {Keywords: parameterized verification, invariant checking, resource logics, reconfigurable systems, tree automata} }

Document

**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

We define a class of Separation Logic [Ishtiaq and O'Hearn, 2001; J.C. Reynolds, 2002] formulae, whose entailment problem given formulae ϕ, ψ₁, …, ψ_n, is every model of ϕ a model of some ψ_i? is 2-EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work [Iosif et al., 2013; Jens Katelaan et al., 2019; Jens Pagel and Florian Zuleger, 2020] consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated location, i.e. from the domain of the heap. In particular, this guarantees that each structure has treewidth bounded by the size of the largest rule in the set. In contrast, here we show that establishment, although sufficient for decidability (alongside two other natural conditions), is not necessary, by providing a condition, called equational restrictedness, which applies syntactically to (dis-)equalities. The entailment problem is more general in this case, because equationally restricted rules define richer classes of structures, of unbounded treewidth. In this paper we show that
(1) every established set of rules can be converted into an equationally restricted one and
(2) the entailment problem is 2-EXPTIME-complete in the latter case, thus matching the complexity of entailments for established sets of rules [Jens Katelaan et al., 2019; Jens Pagel and Florian Zuleger, 2020].

Mnacho Echenim, Radu Iosif, and Nicolas Peltier. Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Establishment. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{echenim_et_al:LIPIcs.CSL.2021.20, author = {Echenim, Mnacho and Iosif, Radu and Peltier, Nicolas}, title = {{Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Establishment}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.20}, URN = {urn:nbn:de:0030-drops-134546}, doi = {10.4230/LIPIcs.CSL.2021.20}, annote = {Keywords: Separation logic, Induction definitions, Inductive theorem proving, Entailments, Complexity} }

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