Some fragments of second-order logic over the reals for which satisability and equivalence are (un)decidable

Abstract

We consider the \Sigma_0^1-fragment of second-order logic over the vocabulary ⟨+, ×, 0, 1, <, S_1, ..., S_k⟩, interpreted over the reals, where the predicate symbols S_i are interpreted as semi- algebraic sets. We show that, in this context, satisfiability of formulas is decidable for the first-order \exists^\ast-quantifier fragment and undecidable for the \exists^\ast\forall- and \forall^\ast-fragments. We also show that for these three fragments the same (un)decidability results hold for containment and equivalence of formulas.