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Two Results on Separation Logic With Theory Reasoning

Two results are presented concerning the entailment problem in Separation Logic with inductively defined predicate symbols and theory reasoning. First, we show that the entailment problem is undecidable for rules with bounded tree-width, if theory reasoning is considered. The result holds for a wide class of theories, even with a very low expressive power. For instance it applies to the natural numbers with the successor function, or with the usual order. Second, we show that every entailment problem can be reduced to an entailment problem containing no equality (neither in the formulas nor in the recursive rules defining the semantics of the predicate symbols).