[论文解读] Truth Predicate of Inductive Definitions and Logical Complexity of Infinite-Descent Proofs

Sohei Ito, Makoto Tatsuta|arXiv (Cornell University)|Mar 4, 2026

Logic, programming, and type systems被引用 0

一句话总结

该论文证明 LKID-omega 的可证明性（针对带自顶式定义谓词的一阶逻辑的无穷降证明系统）是 Pi-1-1 完全的，并将 omega-语言真命题扩展到自顶式定义。

ABSTRACT

Formal reasoning about inductively defined relations and structures is widely recognized not only for its mathematical interest but also for its importance in computer science, and has applications in verifying properties of programs and algorithms. Recently, several proof systems of inductively defined predicates based on sequent calculus including the cyclic proof system CLKID-omega and the infinite-descent proof system LKID-omega have attracted much attention. Although the relation among their provabilities has been clarified so far, the logical complexity of these systems has not been much studied. The infinite-descent proof system LKID-omega is an infinite proof system for inductive definitions and allows infinite paths in proof figures. It serves as a basis for the cyclic proof system. This paper shows that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete. To show this, first it is shown that the validity for inductive definitions in standard models is equivalent to the validity for inductive definitions in standard term models. Next, using this equivalence, this paper extends the truth predicate of omega-languages, as given in Girard's textbook, to inductive definitions by employing arithmetical coding of inductive definitions. This shows that the validity of inductive definitions in standard models is a (Pi-1-1) relation. Then, using the completeness of LKID-omega for standard models, it is shown that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete.

研究动机与目标

• 对自顶定义谓词进行形式推理的动机及其在编程语言验证中的相关性.
• 确立扩展为自顶定义的符号集下，标准模型与标准项模型之间的有效性等价性。
• 为 FOL_ID 构建一个 Pi-1-1 真命题，分析自顶定义的逻辑复杂性。
• 通过从 Pi-1-1 困难问题的归约，证明 LKID_omega 的可证明性为 Pi-1-1 完全。

提出的方法

• 给出带自顶定义谓词的第一阶语言 FOL_ID，并给出自顶定义谓词的产生规则。
• 引入标准模型和项模型，并证明在这些模型之间的有效性等价性。
• 利用对自顶定义的算术编码，将 Girard 的 omega-语言真命题扩展到自顶定义。
• 为 FOL_ID 构造一个真命题 I(f)，以表明自顶定义的有效性是 Pi^1_1 关系。
• 定义无穷降证明系统 LKID_omega，并通过归约确立其 Pi^1_1-完备性。

实验结果

研究问题

• RQ1 自顶定义在标准模型中的有效性在逻辑复杂性上是什么？
• RQ2 标准模型中的有效性是否等价于扩展符号集并带新常量的标准项模型中的有效性？
• RQ3 是否可以利用自顶定义的阿里映射将 omega-语言的真命题扩展？
• RQ4 LKID_omega 的可证明性是否是 Pi^1_1 完全，以及有哪些证据支持这一点？
• RQ5 向下 Skolem-Löwenheim 定理如何帮助理解在此情境中的模型等价性？

主要发现

• 在标准模型中自顶定义的有效性是一个 Pi-1-1 关系。
• 在扩展符号集并引入新常量的标准模型中，标准模型的有效性等价于标准项模型的有效性。
• 通过将自顶定义的阿里映射编码扩展到 FOL_ID，构造了一个 Pi-1-1 的真命题。
• LKID_omega 的可证明性是 Pi-1-1 完全的，通过 Pi-1-1 成员性和 Pi-1-1 硬性归约确立。
• 该方法为高阶语言（超越 omega-语言）的真定义提供了基础。