On Spatial Conjunction as Second-Order Logic

Spatial conjunction is a powerful construct for reasoning about dynamically al- located data structures, as well as concurrent, distribute d and mobile computation. While researchers have identified many uses of spatial conjunctio n, its precise expressive power compared to traditional logical constructs was not previously known. In this paper we establish the expressive power of spatial conjunction. We construct an embedding from first-order logic with spatial conjunction i nto second-order logic, and more surprisingly, an embedding from full second order logic into first-order logic with spatial conjunction. These embeddings show that the satisfiability of formulas in first-order logic with spatial conjunction is equivalent to the satisfiabilit y of formulas in second-order logic. These results explain the great expressive power of spatial conjunction and can be used to show that adding unrestricted spatial conjunction to a decidable logic leads to an un- decidable logic. As one example, we show that adding unrestricted spatial conjunction to two-variable logic leads to undecidability. On the side of decidability, the embedding into second-order logic immediately implies the decidability of first-order logic with a form of spatial conj unction over trees. The embedding into spatial conjunction also has useful consequences: because a restricted form of spatial conjunction in two-variable logic preserves decidability, we obtain that a correspondingly restricted form of second-order quantification in two-vari able logic is decidable. The result- ing language generalizes the first-order theory of boolean a lgebra over sets and is useful in reasoning about the contents of data structures in object-o riented languages.