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A Proof Procedure for Separation Logic with Inductive Definitions and Data

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Abstract

A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoting structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We also show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii) When the theory is empty, in which case every valid sequent admits a rational proof tree, where the total number of pairwise distinct sequents occurring in the proof tree is doubly exponential w.r.t. the size of the end-sequent.

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Notes

  1. Note that \(\kappa \) is not considered as constant for the complexity analysis in Sect. 10: it is part of the input.

  2. Because we will consider transformations introducing an unbounded number of new predicate symbols, we cannot assume that the predicate atoms have a constant size.

  3. Each symbol s in \({{\mathcal {P}}}_S\cup {{\mathcal {P}}}_{{\mathcal {T}}}\cup {{\mathcal {V}}}\) is counted with a weight equal to its length \(\Vert s\Vert \), and all the logical symbols have weight 1.

  4. We assume that \(\textbf{x}'\) contains no variable in \(\textbf{u}\theta \).

  5. Hence two non-valid sequents with different countermodels are equivalent.

  6. This set is an over-approximation of the set of variables x such that \(x \in fv ^{{{\mathcal {T}}}}_{{{\mathcal {R}}}}(\phi ')\) for some predicate-free formula \(\phi '\) with \(\phi \Leftarrow _{{{\mathcal {R}}}}^* \phi '\), but it is sufficient for our purpose.

  7. This condition is not restrictive since a fresh variable y can always be added both to \(\textbf{u}\) and \( dom (\theta )\), and by letting \(y\theta = x\).

  8. We recall that a rule is invertible if the validity of its conclusion implies the validity of each of its premises.

  9. The key difference is that in our rule the premises are directly written into disjunctive normal form, rather than using universal quantifications over sets of indices and disjunction.

  10. Note that in the proof of Theorem 65 rule is applied when \(\psi \) contains a predicate atom. However, this strategy is not applicable here because it may produce an infinite proof tree.

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Acknowledgements

This work has been partially funded by the the French National Research Agency (ANR-21-CE48-0011). The authors wish to thank Radu Iosif for his comments on the paper and for fruitful discussions.

Funding

This work has been partially funded by the the French National Research Agency (ANR-21-CE48-0011).

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  1. Mnacho Echenim and Nicolas Peltier have contributed equally to this work.

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  1. Department, Univ. Grenoble Alpes, CNRS, LIG, 38000, Grenoble, France

    Mnacho Echenim & Nicolas Peltier

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  1. Mnacho Echenim
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Echenim, M., Peltier, N. A Proof Procedure for Separation Logic with Inductive Definitions and Data. J Autom Reasoning 67, 30 (2023). https://doi.org/10.1007/s10817-023-09680-4

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