A Decidable Fragment of Second Order Logic With Applications to Synthesis
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We propose a fragment of many-sorted second order logic ESMT and show that checking satisfiability of sentences in this fragment is decidable. This logic has an ∃^*∀^* quantifier prefix that is conducive to modeling synthesis problems. Moreover, it allows reasoning using a combination of background theories provided that they have a decidable satisfiability problem for the ∃^*∀^* FO-fragment (e.g., linear arithmetic). Our decision procedure reduces the satisfiability of ESMT formulae to satisfiability queries of the background theories, allowing us to use existing efficient SMT solvers for these theories; hence our procedure can be seen as effectively SMT (ESMT) reasoning.
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