The Complexity of Proving the Discrete Jordan Curve Theorem

TL;DR

This paper formalizes and proves the Jordan Curve Theorem within bounded arithmetic theories related to small complexity classes, demonstrating how certain tautologies have polynomial-size proofs in specific proof systems.

Contribution

It establishes formal proofs of the Jordan Curve Theorem in theories of bounded arithmetic corresponding to small complexity classes, improving proof size bounds for related tautologies.

Findings

V^0(2) proves that disjoint cycles divide the grid into at least two regions.

V^0 proves that simple closed curves divide the grid into exactly two regions.

Hex and st-connectivity tautologies have polynomial size AC^0(2)-Frege proofs.

Abstract

The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded arithmetic that correspond to small complexity classes. The theory $V^0(2)$ (corresponding to $AC^0(2)$) proves that any set of edges that form disjoint cycles divides the grid into at least two regions. The theory $V^0$ (corresponding to $AC^0$) proves that any sequence of edges that form a simple closed curve divides the grid into exactly two regions. As a consequence, the Hex tautologies and the st-connectivity tautologies have polynomial size $AC^0(2)$-Frege-proofs, which improves results of Buss which only apply to the stronger proof system $TC^0$-Frege.