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Polynomialsize Frege and Resolution Proofs of stConnectivity and Hex Tautologies
 Theorectical Computer Science
, 2003
"... A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The stconnectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless ..."
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Cited by 10 (0 self)
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A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The stconnectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere.
Polynomial Local Search in the Polynomial Hierarchy and Witnessing in Fragments of Bounded Arithmetic
, 2008
"... The complexity class of Π p kpolynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p idefinable functions of T k+1 2 are characterized in terms of Π p kPLS problems. These Π p kPLS problems c ..."
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Cited by 8 (3 self)
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The complexity class of Π p kpolynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p idefinable functions of T k+1 2 are characterized in terms of Π p kPLS problems. These Π p kPLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p kPLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new ∀Σb 1(α)principle that is conjectured to separate T k 2 (α) and T k+1 2 (α). 1
Bounded Arithmetic and Constant Depth Frege Proofs
, 2004
"... We discuss the ParisWilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and nonrelativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0depth for PKproofs that makes the translation from boun ..."
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We discuss the ParisWilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and nonrelativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0depth for PKproofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1definable functions of S12are polynomial time computable and that the \Sigma b1definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma