Abstract
Separation Logic (\(\mathsf {SL}\)) is a well-known assertion language used in Hoare-style modular proof systems for programs with dynamically allocated data structures. In this paper we investigate the fragment of first-order \(\mathsf {SL}\) restricted to the Bernays-Schönfinkel-Ramsey quantifier prefix \(\exists ^*\forall ^*\), where the quantified variables range over the set of memory locations. When this set is uninterpreted (has no associated theory) the fragment is PSPACE-complete, which matches the complexity of the quantifier-free fragment [7]. However, \(\mathsf {SL}\) becomes undecidable when the quantifier prefix belongs to \(\exists ^*\forall ^*\exists ^*\) instead, or when the memory locations are interpreted as integers with linear arithmetic constraints, thus setting a sharp boundary for decidability within \(\mathsf {SL}\). We have implemented a decision procedure for the decidable fragment of \(\exists ^*\forall ^*\mathsf {SL}\) as a specialized solver inside a DPLL(T) architecture, within the CVC4 SMT solver. The evaluation of our implementation was carried out using two sets of verification conditions, produced by (i) unfolding inductive predicates, and (ii) a weakest precondition-based verification condition generator. Experimental data shows that automated quantifier instantiation has little overhead, compared to manual model-based instantiation.
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Notes
- 1.
Strictly speaking, the Bernays-Schönfinkel-Ramsey class refers to the \(\exists ^*\forall ^*\) fragment of first-order logic with equality and predicate symbols, but no function symbols [17].
- 2.
By writing \(\mathcal {I}[\sigma \leftarrow S]\) we ensure that all variables of sort \(\sigma \) are mapped by \(\mathcal {I}\) to elements of S.
- 3.
If \(\mathsf {x}_1 \mapsto (\mathsf {x}_2, \ldots , \mathsf {x}_n)\) and \(\mathsf {x}_1 \mapsto (\mathsf {x}'_2, \ldots , \mathsf {x}'_n)\) hold, this forces \(\mathsf {x}_i=\mathsf {x}'_i\), for all \(i=2,\ldots ,n\).
- 4.
Extending the interpretation of \(\mathsf {Loc}\) to include negative integers does not make any difference for the undecidability result.
- 5.
The procedure is incorporated into the master branch of the SMT solver CVC4 (https://github.com/CVC4), and can be enabled by command line parameter - -quant-epr.
- 6.
Available at http://cvc4.cs.nyu.edu/web/.
- 7.
The CVC4 binary and examples used in these experiments are available at http://cs.uiowa.edu/~ajreynol/VMCAI2017-seplog-epr.
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Reynolds, A., Iosif, R., Serban, C. (2017). Reasoning in the Bernays-Schönfinkel-Ramsey Fragment of Separation Logic. In: Bouajjani, A., Monniaux, D. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2017. Lecture Notes in Computer Science(), vol 10145. Springer, Cham. https://doi.org/10.1007/978-3-319-52234-0_25
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