一个圆柱代数单元的水平构造

IF 0.6 4区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Jasper Nalbach , Erika Ábrahám , Philippe Specht , Christopher W. Brown , James H. Davenport , Matthew England
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引用次数: 0

摘要

可满足模理论(SMT)求解器在不同理论下检验无量词一阶逻辑公式的可满足性。我们考虑非线性实算术理论,其中公式是多项式约束的逻辑组合。这里常用的工具是柱面代数分解(CAD),通过使用投影多项式将实空间分解为约束为真不变的单元。CAD编码的信息比检查满意度所需的要多。解决这个问题的一种方法是将CAD理论重新包装为基于搜索的算法:一种猜测样本点以满足公式,并将猜测冲突约束推广到样本周围的圆柱形单元,这在继续搜索中是避免的。这种方法可以更快地得到令人满意的任务,或者用更少的单元得出不满意的结论。这种方法的一个显著例子是jovanoviki和de Moura的NLSAT算法。由于这些单元是在样本的局部产生的,因此与传统的CAD投影相比,可以使用更少的投影多项式。原始的NLSAT算法对集合进行了一些缩减;而布朗的单细胞结构进一步减少了它。然而,它是一个多项式一个多项式地细化细胞,这意味着产生的细胞的形状和大小取决于考虑多项式的顺序。本文提出了一种逐级构建此类单元的方法,即按变量顺序逐级构建,而不是对所有级别进行多项式逐级构建。我们仍然使用减少数量的投影多项式,但现在可以考虑各种不同的减少,并使用启发式方法来选择投影多项式,以优化正在构建的细胞的形状。因此,新方法可以提高NLSAT算法的性能。我们将支持算法的所有必要理论表述为证明系统:虽然不是该领域工作的常见表示,但它在允许启发式决策与主算法及其正确性证明的优雅解耦方面是有价值的。我们希望符号计算社区也能在其他领域找到它的用途。特别是,该证明系统可能是向非线性实算术的形式化证明迈出的一步。这项工作已经在SMT-RAT求解器中实现,并在SMT-LIB基准库上实验验证了分层构建的好处。我们还比较了几种构建的启发式方法,并观察到每种启发式方法都有优势,为进一步开发新方法提供了潜力。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.

A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.

The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.

This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

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来源期刊
Journal of Symbolic Computation
Journal of Symbolic Computation 工程技术-计算机:理论方法
CiteScore
2.10
自引率
14.30%
发文量
75
审稿时长
142 days
期刊介绍: An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.
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