Incomplete SMT techniques for solving non-linear formulas over the integers
We present new methods for solving the Satisfiability Modulo Theories problem over the theory of QuantifierFree Non-linear Integer Arithmetic, SMT(QF-NIA), which consists of deciding the satisfiability of ground formulas with integer polynomial constraints. Following previous work, we propose to sol...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/177600 |
| Acceso en línea: | https://hdl.handle.net/2117/177600 https://dx.doi.org/10.1145/3340923 |
| Access Level: | acceso abierto |
| Palabra clave: | Integer programming Non-linear arithmetic Satisfiability modulo theories Programació en nombres enters Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica |
| Sumario: | We present new methods for solving the Satisfiability Modulo Theories problem over the theory of QuantifierFree Non-linear Integer Arithmetic, SMT(QF-NIA), which consists of deciding the satisfiability of ground formulas with integer polynomial constraints. Following previous work, we propose to solve SMT(QF-NIA) instances by reducing them to linear arithmetic: non-linear monomials are linearized by abstracting them with fresh variables and by performing case splitting on integer variables with finite domain. For variables that do not have a finite domain, we can artificially introduce one by imposing a lower and an upper bound and iteratively enlarge it until a solution is found (or the procedure times out). The key for the success of the approach is to determine, at each iteration, which domains have to be enlarged. Previously, unsatisfiable cores were used to identify the domains to be changed, but no clue was obtained as to how large the new domains should be. Here, we explain two novel ways to guide this process by analyzing solutions to optimization problems: (i) to minimize the number of violated artificial domain bounds, solved via a Max-SMT solver, and (ii) to minimize the distance with respect to the artificial domains, solved via an Optimization Modulo Theories (OMT) solver. Using this SMT-based optimization technology allows smoothly extending the method to also solve Max-SMT problems over non-linear integer arithmetic. Finally, we leverage the resulting Max-SMT(QF-NIA) techniques to solve ∃∀ formulas in a fragment of quantified non-linear arithmetic that appears commonly in verification and synthesis applications. |
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