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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 ..."
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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
The provable total search problems of bounded arithmetic
, 2007
"... We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument use ..."
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We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a nonlogical question about multiparty communication protocols.
Fragments of Approximate Counting
, 2012
"... We study the longstanding open problem of giving ∀Σ b 1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeˇrábek’s theories for approximate counting and their subtheories. ..."
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We study the longstanding open problem of giving ∀Σ b 1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeˇrábek’s theories for approximate counting and their subtheories. We show that the ∀Σ b 1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP NP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeonhole principle for polynomial time functions.
Improved Witnessing and Local Improvement Principles for SecondOrder Bounded Arithmetic
"... This paper concerns the second order systems U1 2 and V1 2 of bounded arithmetic, which have proof theoretic strengths corresponding to polynomial space and exponential time computation. We formulate improved witnessing theorems for these two theories by using S1 2 as a base theory for proving the c ..."
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This paper concerns the second order systems U1 2 and V1 2 of bounded arithmetic, which have proof theoretic strengths corresponding to polynomial space and exponential time computation. We formulate improved witnessing theorems for these two theories by using S1 2 as a base theory for proving the correctness of the polynomial space or exponential time witnessing functions. We develop the theory of nondeterministic polynomialspace computation, includingSavitch’s theorem,inU 1 2.Kolodziejczyk, Nguyen, and Thapen have introduced local improvement properties to characterize the provably total NP functions of these second order theories. We show that the strengths of their local improvement principles over U1 2 and V1 2 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U1 2 proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically many rounds is complete for the provably total NP search problems of V1 2. Related results are obtained for local improvement principles with one improvement round, and for local improvement over rectangular grids.
On extracting computations from propositional proofs (a survey)
, 2010
"... This paper describes a project that aims at showing that propositional proofs of certain tautologies in weak proof system give upper bounds on the computational complexity of functions associated with the tautologies. Such bounds can potentially be used to prove (conditional or unconditional) lower ..."
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This paper describes a project that aims at showing that propositional proofs of certain tautologies in weak proof system give upper bounds on the computational complexity of functions associated with the tautologies. Such bounds can potentially be used to prove (conditional or unconditional) lower bounds on the lengths of proofs of these tautologies and show separations of some weak proof systems. The prototype are the results showing the feasible interpolation property for resolution. In order to prove similar results for systems stronger than resolution one needs to define suitable generalizations of boolean circuits. We will survey the known results concerning this project and sketch in which direction we want to generalize them. 1