The Complexity of Prenex Separation Logic with One Selector

Abstract

We first show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with $k\geq1$ selector fields (\seplogk{k}). Second, we show that this entails the decidability of the finite and infinite satisfiability problem for the class of prenex formulas of \seplogk{1}, by reduction to the first-order theory of one unary function symbol and unary predicate symbols. We also prove that the complexity is not elementary, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey fragment of prenex \seplogk{1} formulae with quantifier prefix in the language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete (hierarchical) classification of the complexity of prenex \seplogk{1}, according to the quantifier alternation depth is left as an open problem