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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 11 (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
Abstract Bounded Arithmetic, Proof Complexity and Two Papers of Parikh
"... This article surveys R. Parikh’s work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh’s papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. 1 ..."
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This article surveys R. Parikh’s work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh’s papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. 1
Bounded Arithmetic, Proof Complexity and Two Papers of Parikh
, 2002
"... This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. ..."
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This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs.
Polynomialsize Frege and Resolution Proofs of stConnectivity and Hex Tautologies
, 2005
"... Abstract A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The stconnectivityprinciple 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 thegrid graph un ..."
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Abstract A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The stconnectivityprinciple 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 thegrid graph unless the paths cross somewhere. We prove that the propositional tautologies which encode the stconnectivity principle have polynomial size Frege proofs and polynomial size T C0Frege proofs. For bounded width grid graphs, the stconnectivity tautologies have polynomial size resolution proofs. Akey part of the proof is to show that the group with two generators, both of order two, has word problem in alternating logtime (Alogtime)and even in T C0.Conversely, we show that constant depth Frege proofs of the stconnectivity tautologies require nearexponential size. The proofuses a reduction from the pigeonhole principle, via tautologies that express a &quot;directed single source &quot; principle SINK, which is related toPapadimitriou's search classes PPAD and PPADS (or, PSK). The stconnectivity principle is related to Urquhart's propositionalHex tautologies, and we establish the same upper and lower bounds on proof complexity for the Hex tautologies. In addition, the Hextautology is shown to be equivalent to the SINK tautologies and to the onetoone onto pigeonhole principle. *Supported in part by NSF grants DMS0100589 and DMS0400848.
Polynomialsize Frege and Resolution Proofs of stConnectivity and Hex Tautologies
, 2005
"... Abstract A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The stconnectivityprinciple 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 thegrid graph un ..."
Abstract
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Abstract A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The stconnectivityprinciple 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 thegrid graph unless the paths cross somewhere. We prove that the propositional tautologies which encode the stconnectivity principle have polynomial size Frege proofs and polynomial size T C0Frege proofs. For bounded width grid graphs, the stconnectivity tautologies have polynomial size resolution proofs. Akey part of the proof is to show that the group with two generators, both of order two, has word problem in alternating logtime (Alogtime)and even in T C0.Conversely, we show that constant depth Frege proofs of the stconnectivity tautologies require nearexponential size. The proofuses a reduction from the pigeonhole principle, via tautologies that express a &quot;directed single source &quot; principle SINK, which is related toPapadimitriou's search classes PPAD and PPADS (or, PSK). The stconnectivity principle is related to Urquhart's propositionalHex tautologies, and we establish the same upper and lower bounds on proof complexity for the Hex tautologies. In addition, the Hextautology is shown to be equivalent to the SINK tautologies and to the onetoone onto pigeonhole principle. *Supported in part by NSF grants DMS0100589 and DMS0400848.