We investigate decidability, complexity and expressive power issues for (first-order) separation logic with one record field (herein called SL) and its fragments. SL can specify properties about the memory heap of programs with singly-linked lists. Separation logic with two record fields is known to be undecidable by reduction of finite satisfiability for classical predicate logic with one binary relation. Surprisingly, we show that second-order logic is as expressive as SL and as a by-product we get undecidability of SL. This is refined by showing that SL without the separating conjunction is as expressive as SL, whence undecidable too. As a consequence of this deep result, in SL the magic wand can simulate the separating conjunction. By contrast, we establish that SL without the magic wand is decidable with non-elementary complexity by reduction from satisfiability for the first-order theory over finite words. Equivalence between second-order logic and separation logic extends to the case with more than one selector.

We investigate decidability, complexity and expressive power issues for (first-order) separation logic with one record field (herein called SL) and its fragments. SL can specify properties about the memory heap of programs with singly-linked lists. Separation logic with two record fields is known to be undecidable by reduction of finite satisfiability for classical predicate logic with one binary relation. Surprisingly, we show that second-order logic is as expressive as SL and as a by-product we get undecidability of St.. This is refined by showing that SL without the separating conjunction is as expressive as SL, whence undecidable too. As a consequence, in SL the separating implication (also known as the magic wand) can simulate the separating conjunction. By contrast, we establish that SL without the magic wand is decidable, and we prove a non-elementary complexity by reduction from satisfiability for the first-order theory over finite words. This result is extended with a bounded use of the magic wand that appears in Hoare-style rules. As a generalization, it is shown that kSL, the separation logic over heaps with k >= 1 record fields, is equivalent to kSO, the second-order logic over heaps with k record fields. (C) 2012 Elsevier Inc. All rights reserved

Motivated by the verification of programs with pointer variables, we introduce a temporal logic LTL mem whose underlying assertion language is the quantifier-free fragment of separation logic and the temporal logic on the top of it is the standard linear-time temporal logic LTL. We analyze the complexity of various model-checking and satisfiability problems for LTL mem , considering various fragments of separation logic (including pointer arithmetic), various classes of models (with or without constant heap), and the influence of fixing the initial memory state. We provide a complete picture based on these criteria. Our main decidability result is pspace-completeness of the satisfiability problems on the record fragment and on a classical fragment allowing pointer arithmetic. Σ 0 1-completeness or Σ 1 1-completeness results are established for various problems by reducing standard problems for Minsky machines, and underline the tightness of our decidability results.

Motivated by the verification of programs with pointer variables, we introduce a temporal logic LTL mem whose underlying assertion language is the quantifier-free fragment of separation logic and the temporal logic on the top of it is the standard linear-time temporal logic LTL. We analyze the complexity of various model-checking and satisfiability problems for LTL mem , considering various fragments of separation logic (including pointer arithmetic), various classes of models (with or without constant heap), and the influence of fixing the initial memory state. We provide a complete picture based on these criteria. Our main decidability result is pspace-completeness of the satisfiability problems on the record fragment and on a classical fragment allowing pointer arithmetic. Σ 0 1 -completeness or Σ 1 1 -completeness results are established for various problems by reducing standard problems for Minsky machines, and underline the tightness of our decidability results.

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.

Answer set programming is a declarative programming paradigm oriented towards difficult combinatorial search problems. A fundamental task in answer set programming is to compute stable models, i.e., solutions of logic programs. Answer set solvers are the programs that perform this task. The problem of deciding whether a disjunctive program has a stable model is Σ P 2 -complete. The high complexity of reasoning within disjunctive logic programming is responsible for few solvers capable of dealing with such programs, namely dlv, gnt, cmodels, clasp and wasp. In this paper we show that transition systems introduced by Nieuwenhuis, Oliveras, and Tinelli to model and analyze satisfiability solvers can be adapted for disjunctive answer set solvers. Transition systems give a unifying perspective and bring clarity in the description and comparison of solvers. They can be effectively used for analyzing, comparing and proving correctness of search algorithms as well as inspiring new ideas in the design of disjunctive answer set solvers. In this light, we introduce a general template, which accounts for major techniques implemented in disjunctive solvers. We then illustrate how this general template captures solvers dlv, gnt, and cmodels. We also show how this framework provides a convenient tool for designing new solving algorithms by means of combinations of techniques employed in different solvers.

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