Supporting inductive reasoning is an essential component is any framework of use in computer science. To do so, the logical framework must extend that of first-order logic. Transitive closure logic is a known extension of first-order logic that is particularly straightforward to automate. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of a single transitive closure operator has the advantage of uniformly capturing all finitary inductive definitions. To further improve the reasoning techniques for transitive closure logic, we here present an infinitary proof system for it, which is an infinite descent –style counterpart to the existing (explicit induction) proof system for the logic. We show that the infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e., cyclic ) proofs provides the basis for an effective system for automating inductive reasoning.

中文翻译：

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传递闭包逻辑的无根据证明理论

支持归纳推理是计算机科学中任何使用框架的重要组成部分。为此，逻辑框架必须扩展一阶逻辑。传递闭包逻辑是一阶逻辑的已知扩展，特别容易实现自动化。而具有归纳定义的一阶逻辑的其他扩展是先验通过一组归纳定义进行参数化，添加单个传递闭包运算符具有统一捕获所有有限归纳定义的优点。为了进一步改进传递闭包逻辑的推理技术，我们在这里提出了一个无限的它的证明系统，这是一个无限下降– 与现有的（显式归纳）逻辑证明系统相对应的样式。我们证明了无限系统对于标准语义是完整的并且包含显式系统。此外，传递闭包运算符的一致性允许使用简单的句法标准定义语义上有意义的完整限制。因此，对正则无穷大的限制（即，循环的) 证明为自动归纳推理的有效系统提供了基础。