Beyond Shapes: Lists with Ordered Data

Abstract

Abstract. Standard analysis on recursive data structures restrict their attention to shape properties (for instance, a program that manipulates a list returns a list), excluding properties that deal with the actual content of these structures. For instance, these analysis would not establish that the result of merging two ordered lists is an ordered list. Separation logic, one of the prominent framework for these kind of analysis, proposed a heap model that could represent data, but, to our knowledge, no predicate dealing with data has ever been integrated to the logic while preserving decidability. We establish decidability for (first-order) separation logic with a predicate that allows to compare two successive data in a list. We then consider the extension where two data in arbitrary positions may be compared, and establish the undecidability in general. We define a guarded fragment that turns out to be both decidable and sufficiently expressive to prove the preservation of the loop invariant of a standard program merging ordered lists. We finally consider the extension with the magic-wand and prove that, by constrast with the data-free case, even a very restricted use of the magic wand already introduces undecidability.

Two-Variable Separation Logic and Its Inner Circle

“…This natural encoding generalizes the encoding of finite words by heaps in Brochenin et al [2012, Section 3] (see also Bansal et al [2009]) while providing a much more concise representation. Note also that the encoding by itself is of no use since it is essential to be able to operate on it with the logical language at hand.…”

“…Additionally, we provide evidence that an undecidable logic on data words from Bojańczyk et al [2011] can be reduced to two-variable first-order separation logic (Section 4.6). Logics on data words have already been used to get undecidability results for separation logic with data fields (see, e.g., Bansal et al [2009]). Herein, we use a simple version of first-order separation logic without program variables and without data fields, and apart from equality, there is only one binary relation, and it is functional and finite.…”

Reasoning about sequences of memory states

“…This permutation satisfies the hypotheses of Lemma 11, and thus may be applied to (s, ∅), which then still satisfies A. We apply this type of permutation until there is no k such that s( Decidable fragments of first-order SL can be found in [25,34,35].…”

Separation logics and modalities: a survey

“…• are without data values (by contrast, see e.g. [BDES09,BBL09,MPQ11]), • use concrete models (by contrast to abstract models considered in [COY07,BK10,LWG10,BV14]), • are not multi-dimensional extensions of non-classical logics (by contrast, see e.g. [YRSW03,BDL09,CG13]), • do not provide general inductive predicates (lists, trees, etc.)…”