Semipositivity in Separation Logic with Two Variables

In a recent work by Demri and Deters (CSL-LICS 2014), first-order separation logic restricted to two variables and separating implication was shown undecidable, where it was shown that even with only two variables, if the use of negations is unrestricted, then they can be nested with separating implication in a complex way to get the undecidability result. In this paper, we revisit the decidability and complexity issues of first-order separation logic with two variables, and proposes semi-positive separation logic with two-variables (SPSL2), where the use of negations is restricted in the sense that negations can only occur in front of atomic formulae. We prove that satisfiability of the fragment of SPSL2 where neither separating conjunction nor septraction (the dual operator of separating implication) occurs in the scope of universal quantifiers, is NEXPTIME-complete. As a byproduct of the proof, we show that the finite satisfiability problem of first-order logic with two variables and a bounded number of function symbols is NEXPTIME-complete (the lower bound holds even with only one function symbol and without unary predicates), which may be of independent interest beyond separation logic community.