Entailment Checking in Separation Logic with Inductive Definitions is 2-EXPTIME hard

Abstract

The entailment between separation logic formulæ with inductive predicates, also known as sym- bolic heaps, has been shown to be decidable for a large class of inductive definitions [7]. Recently, a 2-EXPTIME algorithm was proposed [10, 14] and an EXPTIME-hard bound was established in [8]; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines [5].

Unifying Decidable Entailments in Separation Logic with Inductive Definitions

“…The last line is the main result of the paper and corresponds to the most general (known) decidable class of entailment problems (Definition 8). The following theorem is an easy consequence of previous results [6].…”

“…Together with the results of [10,14,6,8], Theorem 31 draws a clear and complete picture concerning the decidability and complexity of the entailment problem in Separation Logic with inductive definitions. The room for improvement in this direction is probably very limited, since Theorem 31 pushes the frontier quite far.…”

“…The decidability of progressing, connected and left-established entailment problems is an immediate consequence of the result of [10]. Moreover, an analysis of the proof [10] leads to an elementary recursive complexity upper bound, which has been recently tighten down to 2EXPTIME-complete [14,8,6]. In the following, we refer to Table 1 for a recap of the complexity results for the entailment problem.…”

“…Intuitively, the progress (P) condition states that every rule allocates exactly one location, the connectivity (C) condition states that the set of allocated locations has a tree-shaped structure, and the establishment (E) condition states that every existentially quantified variable from a rule defining a predicate is (eventually) allocated in every unfolding of that predicate. A 2EXPTIME algorithm was proposed for testing the validity of PCE entailments [13,14] and a matching 2EXPTIME-hardness lower bound was provided shortly after [6].…”

“…Although the class of data structures that can be described is much larger, we show that this new class of entailments, called safe, is also 2EXPTIME-complete, by a many-one reduction of the validity of safe entailments to the validity of PCE entailments. A second contribution of the paper is the cross-certification of the two independent proofs of the 2EXPTIME upper bounds, for the PCE [6,14,8] and PCR [8] classes of entailments, respectively, by closing the loop. Namely, the reduction given in this paper enables the translation of any of the three entailment problems into an equivalent problem in any other class, while preserving the 2EXPTIME upper bound.…”