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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 ..."
Abstract

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
Cut Elimination In Situ Comments appreciated.
, 2012
"... We present methods for removing toplevel cuts from a sequent calculus or Taitstyle proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from treelike to daglike form, but it most doubles the number of lines in the ..."
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We present methods for removing toplevel cuts from a sequent calculus or Taitstyle proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from treelike to daglike form, but it most doubles the number of lines in the proof. For firstorder logic, the proof size can grow exponentially, but the proof has a succinct description and is polynomialtimeuniform. We usedirect, globalconstructionsthat give polynomial time methods for removing all toplevel cuts from proofs. Byexploitingprenexrepresentations,this extendsto removingall cuts, with final proof size bounded superexponentially in the alternation of quantifiers in cut formulas. 1