A Decidable Fragment of Second Order Logic With Applications to Synthesis.

We propose a fragment of many-sorted second order logic called EQSMT and show that checking satisfiability of sentences in this fragment is decidable. EQSMT formulae have an exists^*forall^* quantifier prefix (over variables, functions and relations) making EQSMT conducive for modeling synthesis problems. Moreover, EQSMT allows reasoning using a combination of background theories provided that they have a decidable satisfiability problem for the exists^*forall^* FO-fragment (e.g., linear arithmetic). Our decision procedure reduces the satisfiability of EQSMT formulae to satisfiability queries of exists^*forall^* formulae of each individual background theory, allowing us to use existing efficient SMT solvers supporting exists^*forall^* reasoning for these theories; hence our procedure can be seen as effectively quantified SMT (EQSMT) reasoning.