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The Complexity of Proving the Discrete Jordan Curve Theorem

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.