Abstract
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 2EXPTIMEcomplete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of storeheap 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 2EXPTIMEcomplete 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].
BibTeX  Entry
@InProceedings{echenim_et_al:LIPIcs:2021:13454,
author = {Mnacho Echenim and Radu Iosif and Nicolas Peltier},
title = {{Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Establishment}},
booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
pages = {20:120:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771757},
ISSN = {18688969},
year = {2021},
volume = {183},
editor = {Christel Baier and Jean GoubaultLarrecq},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13454},
URN = {urn:nbn:de:0030drops134546},
doi = {10.4230/LIPIcs.CSL.2021.20},
annote = {Keywords: Separation logic, Induction definitions, Inductive theorem proving, Entailments, Complexity}
}
Keywords: 

Separation logic, Induction definitions, Inductive theorem proving, Entailments, Complexity 
Collection: 

29th EACSL Annual Conference on Computer Science Logic (CSL 2021) 
Issue Date: 

2021 
Date of publication: 

13.01.2021 