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arXiv · 1712.05513

A Decidable Fragment of Second Order Logic With Applications to Synthesis

P. Madhusudan·Umang Mathur·Shambwaditya Saha·Mahesh Viswanathan
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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.

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P. Madhusudan, Umang Mathur, Shambwaditya Saha, Mahesh Viswanathan. 2017-12-15. A Decidable Fragment of Second Order Logic With Applications to Synthesis. https://doi.org/10.4230/lipics.csl.2018.31

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