Fast Cube Tests for LIA Constraint Solving
We present two tests that solve linear integer arithmetic constraints. These tests are sound and efficiently find solutions for a large number of problems. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width, and we have shown the advantage of our cube tests on these instances by comparing our implementation of the cube test with several state-of-the-art SMT solvers. Our implementation is not only several orders of magnitudes faster, but it also solves all instances, which most SMT solvers do not. Finally, we discovered an additional application for our cube tests: the extraction of equalities implied by a system of linear arithmetic inequalities. This extraction is useful both as a preprocessing step for linear integer constraint solving as well as for the combination of theories by the Nelson-Oppen method.
KeywordsLinear arithmetic SMT Integer arithmetic Constraint solving
- 2.Bobot, F., Conchon, S., Contejean, E., Iguernelala, M., Mahboubi, A., Mebsout, A., Melquiond, G.: A simplex-based extension of Fourier-Motzkin for solving linear integer arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 67–81. Springer, Heidelberg (2012)CrossRefGoogle Scholar
- 8.Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Heidelberg (2014)Google Scholar
- 10.Ehrgott, M.: Scalarization techniques. In: Multicriteria Optimization, pp. 97–126. Springer, Heidelberg (2005)Google Scholar
- 15.Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.): 50 Years of Integer Programming 1958–2008: From the Early Years to the State-of-the-Art. Springer, Heidelberg (2010)Google Scholar