Prenex Separation Logic with One Selector Field

We show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with \(k\ge 1\) selector fields (\(\mathsf {SL}^{\!\scriptstyle {k}}\)). This fact entails the decidability of the finite and infinite satisfiability problems for the class of prenex formulas of \(\mathsf {SL}^{\!\scriptstyle {1}}\), by reduction to the first-order theory of a single unary function symbol and an arbitrary number of unary predicate symbols. We also prove that the complexity of this fragment is not elementary recursive, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Schonfinkel-Ramsey fragment of prenex \(\mathsf {SL}^{\!\scriptstyle {1}}\) formulas with quantifier prefix in the language Open image in new window is PSPACE-complete.