Abstract
Incremental linearization is a conceptually simple, yet effective, technique that we have recently proposed for solving SMT problems over nonlinear real arithmetic constraints. In this paper, we show how the same approach can be applied successfully also to the harder case of nonlinear integer arithmetic problems. We describe in detail our implementation of the basic ideas inside the MathSAT SMT solver, and evaluate its effectiveness with an extensive experimental analysis over all nonlinear integer benchmarks in SMT-LIB. Our results show that MathSAT is very competitive with (and often outperforms) state-of-the-art SMT solvers based on alternative techniques.
This work was funded in part by the H2020-FETOPEN-2016-2017-CSA project SC\(^2\) (712689).
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Notes
- 1.
In the following, we only consider quantifier-free theories, and we abuse the accepted notation by omitting the “QF_” prefix in the names of the theories.
- 2.
In the rest of the paper, for simplifying the presentation we assume that an equality constraint is written as a conjunction of weak inequality constraints, and an inequality constraint is written as a disjunction of strict inequality constraints.
- 3.
This can happen when the tangent lemmas (see Fig. 8) are not used.
- 4.
We are grateful to Andrew Reynolds for fruitful discussions about this.
- 5.
It is important to stress here that we keep the monomials in a normal form by reordering their variables, although this is not shown explicitly in the pseudo-code.
- 6.
In our implementation we use a bounded (dichotomic) search for this. For example, for the ‘UU’ direction we try increasing both a and b until either the tangent plane passing through (a, b) cannot separate the multiplication curve from the bad point c anymore, or we reach a maximum bound on the number of iterations.
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Cimatti, A., Griggio, A., Irfan, A., Roveri, M., Sebastiani, R. (2018). Experimenting on Solving Nonlinear Integer Arithmetic with Incremental Linearization. In: Beyersdorff, O., Wintersteiger, C. (eds) Theory and Applications of Satisfiability Testing – SAT 2018. SAT 2018. Lecture Notes in Computer Science(), vol 10929. Springer, Cham. https://doi.org/10.1007/978-3-319-94144-8_23
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