The power of modal separation logics

Abstract

We introduce a modal separation logic MSL whose models are memory states from separation logic and the logical connectives include modal operators as well as separating conjunction and implication from separation logic. With such a combination of operators, some fragments of MSL can be seen as genuine modal logics whereas some others capture standard separation logics, leading to an original language to speak about memory states. We analyse the decidability status and the computational complexity of several fragments of MSL, obtaining surprising results by design of proof methods that take into account the modal and separation features of MSL. For example, the satisfiability problem for the fragment of MSL with $\Diamond $, the difference modality $\langle \neq \rangle $ and separating conjunction $\ast $ is shown Tower-complete whereas the restriction either to $\Diamond $ and $\ast $ or to $\langle \neq \rangle $ and $\ast $ is only NP-complete. We establish that the full logic MSL admits an undecidable satisfiability problem. Furthermore, we investigate variants of MSL with alternative semantics and we build bridges with interval temporal logics and with logics equipped with sabotage operators.

Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

“…This work can be continued in several directions, for instance to characterise the expressiveness of QCTL t X along the lines of [14] or to analyse whether our results can be lifted to modal separation logics where separating conjunction can encoded by propositional quantification, see e.g. [54], [55].…”

An Auxiliary Logic on Trees: on the Tower-Hardness of Logics Featuring Reachability and Submodel Reasoning

“…We now display the usefulness of ALT as a tool for proving the TOWER-hardness of logics interpreted on tree-like structures. In particular, we provide semantically faithful reductions from SAT(ALT) to the satisfiability problem of four logics that were independently found to be TOWER-complete: first-order separation logic [9], quantified CTL on trees [28], modal logic of heaps [17] and modal separation logic [18]. Our reduction only use strict fragments of these formalisms, allowing us to draw some new results on these logics.…”

“…In [17] and later in [18] two families of logics are presented, respectively called modal logic of heaps (MLH) and modal separation logic (MSL). At their core, both logics can be seen as modal logics extended with separating connectives, hence mixing separation logic (Section 4.1) with temporal aspects as in quantified CTL (Section 4.2).…”

“…By looking at the diagram above, compared to the work in [18], ALT allows us to show that propositional symbols and the elsewhere modality can be removed from MSL without changing the complexity status of its satisfiability problem. Similarly, ALT allows us to refine the analysis on the complexity of SAT(MLH) by showing that the ◊ −1 modality is not needed in order to achieve non-elementary complexities.…”

“…T , as well as modal logic of heaps (MLH) and modal separation logics (MSL), two other logics introduced in [17] and [18], respectively. In this context, beside exposing that all these logics are TOWER-hard because of the way they reason about reachability and submodels, we discover interesting sublogics that are still TOWER-complete: -T restricted to ( ) modalities, where , are Boolean combinations of atomic propositions, or to the modality, which can be nested at most once.…”