The Complexity of Proving the Discrete Jordan Curve Theorem
نویسندگان
چکیده
منابع مشابه
A 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 V0(2) (corresponding to AC(2)) proves that any set of edges that form disjoint cycles divi...
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Let F be a Jordan curve in the plane, i.e. the image of the unit circle C = {(x,y);x + y = 1} under an injective continuous mapping y into R. The Jordan curve theorem [1] says that / ? 2 \ F is disconnected and consists of two components. (We shall use the original definition whereby two points are in the same component if and only if they can be joined by a continuous path (image of [0,1]).) A...
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This article defends Jordan’s original proof of the Jordan curve theorem. The celebrated theorem of Jordan states that every simple closed curve in the plane separates the complement into two connected nonempty sets: an interior region and an exterior. In 1905, O. Veblen declared that this theorem is “justly regarded as a most important step in the direction of a perfectly rigorous mathematics”...
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There has been recent interest in combinatorial versions of classical theorems in topology. In particular, Stahl [S] and Little [3] have proved discrete versions of the Jordan Curve Theorem. The classical theorem states that a simple closed curve y separates the 2-sphere into two connected components of which y is their common boundary. The statements and proofs of the combinatorial versions in...
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ژورنال
عنوان ژورنال: ACM Transactions on Computational Logic
سال: 2012
ISSN: 1529-3785,1557-945X
DOI: 10.1145/2071368.2071377