Systems of fixpoint equations over complete lattices, consisting of (mixed) least and greatest fixpoint equations, allow one to express a number of verification tasks such as model-checking of various kinds of specification logics or the check of coinductive behavioural equivalences. In this paper we develop a theory of approximation for systems of fixpoint equations in the style of abstract interpretation: a system over some concrete domain is abstracted to a system in a suitable abstract domain, with conditions ensuring that the abstract solution represents a sound/complete overapproximation of the concrete solution. Interestingly, up-to techniques, a classical approach used in coinductive settings to obtain easier or feasible proofs, can be interpreted as abstractions in a way that they naturally fit in our framework and extend to systems of equations. Additionally, relying on the approximation theory, we can provide a characterisation of the solution of systems of fixpoint equations over complete lattices in terms of a suitable parity game, generalising some recent work that was restricted to continuous lattices. The game view opens the way to the development of on-the-fly algorithms for characterising the solution of such equation systems.

中文翻译：

---

定点方程组的抽象、最新技术和游戏

完整格上的定点方程系统，由（混合）最小和最大定点方程组成，允许表达一系列验证任务，例如各种规范逻辑的模型检查或共归纳行为等价的检查。在本文中，我们以抽象解释的方式开发了不动点方程组的近似理论：将某个具体域上的系统抽象为合适的抽象域中的系统，条件确保抽象解代表一个健全的/完整的对具体解的过度逼近。有趣的是，最新的技术，一种在共归纳设置中使用的经典方法，以获得更容易或可行的证明，可以被解释为抽象，它们自然适合我们的框架并扩展到方程组。此外，依靠近似理论，我们可以根据合适的奇偶博弈对完整格上的不动点方程组的解进行表征，概括一些近期仅限于连续格的工作。游戏视图为开发用于表征此类方程系统解的动态算法开辟了道路。