The Jordan-Scho¨nflies theorem and the classification of surfaces
American Mathematical Monthly
Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
Polynomial-size Frege and resolution proofs of st-connectivity and Hex tautologies
Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
The Complexity of Proving the Discrete Jordan Curve Theorem
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
The complexity of the pigeonhole principle
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Bounded reverse mathematics
Logical Foundations of Proof Complexity
Logical Foundations of Proof Complexity
| Hi-index | 0.01 |
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 V0(2) (corresponding to AC0(2)) proves that any set of edges that form disjoint cycles divides the grid into at least two regions. The theory V0 (corresponding to AC0) 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 AC0(2)-Frege-proofs, which improves results of Buss which only apply to the stronger proof system TC0-Frege.