A Decidable Fragment of Second Order Logic With Applications to Synthesis (1712.05513v3)

Abstract: 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. Errata: We have modified the transformation step-2 (page 9) to correct for a slight error. Also, the description above Theorem 10 is different from the published version.