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 unde-cidable 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.
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