Descriptive Complexity in Lean: Completeness by First-Order Reductions

2026-09-16

AI Summary

本文通过在Lean中构建一个基于描述复杂性的库，利用一阶归约等方法，证明了73个关于计算复杂性结果的完备性定理。

Abstract

We show that descriptive complexity can serve as a foundation for formalizing computational complexity results in a proof assistant, by constructing a Lean library centered around the following concepts: decision problems are isomorphism-invariant predicates on finite structures; complexity classes are defined by their logical characterization; membership is shown by definability witnesses; hardness is shown by first-order reductions from a known hard problem. We also establish bridges to traditional machine models such as (non)deterministic Turing machines. The library proves 73 completeness results, on 68 problems or problem families, over 14 different classes; relations between the classes established inside the logic and not by machine simulation, among them NL = coNL and the Abiteboul-Vianu theorem; and unconditional lower bounds, among them $\mathrm{FO}(\leq) \subsetneq \mathrm{FO}(\leq, \mathrm{TC})$ and the failure of order-free FO(IFP) to capture PTIME.

Problem

• descriptive complexity
• computational complexity
• proof assistant
• complexity classes
• first-order reductions

Innovation

• descriptive complexity
• proof assistant Lean
• first-order reductions
• isomorphism-invariant predicates
• logical characterization