Decidability and synthesis of inductive invariants ranging in a given domain play an important role in many software and hardware verification systems. We consider here inductive invariants belonging to an abstract domain $A$ as defined in abstract interpretation, namely, ensuring the existence of the best approximation in $A$ of any system property. In this setting, we study the decidability of the existence of abstract inductive invariants in $A$ of transition systems and their corresponding algorithmic synthesis. Our model relies on some general results which relate the existence of abstract inductive invariants with least fixed points of best correct approximations in $A$ of the transfer functions of transition systems and their completeness properties. This approach allows us to derive decidability and synthesis results for abstract inductive invariants which are applied to the well-known Kildall's constant propagation and Karr's affine equalities abstract domains. Moreover, we show that a recent general algorithm for synthesizing inductive invariants in domains of logical formulae can be systematically derived from our results and generalized to a range of algorithms for computing abstract inductive invariants.

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抽象归纳不变量的可判定性和综合

给定域中归纳不变量的可判定性和综合在许多软件和硬件验证系统中起着重要作用。我们在这里考虑属于抽象域 $A$ 的归纳不变量，如抽象解释中所定义，即确保存在任何系统属性的 $A$ 中的最佳近似。在这种情况下，我们研究了转移系统 $A$ 中抽象归纳不变量存在的可判定性及其相应的算法综合。我们的模型依赖于一些一般结果，这些结果将抽象归纳不变量的存在与转移系统传递函数的 $A$ 及其完整性属性的最佳正确近似的最少不动点相关联。这种方法使我们能够推导出抽象归纳不变量的可判定性和综合结果，这些不变量应用于众所周知的 Kildall 常数传播和 Karr 仿射等式抽象域。此外，我们表明，可以从我们的结果系统地推导出最近用于在逻辑公式域中合成归纳不变量的通用算法，并将其推广到用于计算抽象归纳不变量的一系列算法。