Results 1  10
of
32
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 in Izvestiya of the Russian Academy of Science, mathematics
, 1995
"... To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators ..."
Abstract

Cited by 54 (6 self)
 Add to MetaCart
To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constantdepth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolationlike theorems for certain “split versions ” of classical systems of Bounded Arithmetic introduced in this paper.
Lower Bounds For The Polynomial Calculus
, 1998
"... We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumpt ..."
Abstract

Cited by 49 (5 self)
 Add to MetaCart
We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of PHP n , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function f n in n variables, every polynomial calculus proof of the statement "f n cannot be computed by any circuit of size t," must have degree t=n). Loosely speaking, this means that low degree polynomial calculus proofs do not prove NP 6 P=poly.
On the Automatizability of Resolution and Related Propositional Proof Systems
, 2002
"... We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove
On reducibility and symmetry of disjoint NPpairs
, 2001
"... . We consider some problems about pairs of disjoint NP sets. ..."
Abstract

Cited by 29 (0 self)
 Add to MetaCart
. We consider some problems about pairs of disjoint NP sets.
Disjoint NPPairs
, 2003
"... We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is N ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is NPhard. We show under reasonable hypotheses that nonsymmetric disjoint NPpairs exist, which provides additional evidence for the existence of Pinseparable disjoint NPpairs. We construct
Reductions between Disjoint NPPairs
 Information and Computation
, 2004
"... We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for one ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for onetoone, invertible reductions; the class of all disjoint NPpairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NPpairs (A, B) and (C, D) is a Turing reduction with the additional property that if D. We prove under the reasonable assumption UP coUP has a Pbiimmune set that there exist disjoint NPpairs (A, B) and (C, D) such that (A, B) is truthtable reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NPpairs. We exhibit an oracle relative to which DisjNP has a truthtablecomplete disjoint NPpair, but has no manyonecomplete disjoint NPpair.
Survey of disjoint NPpairs and relations to propositional proof systems
 Essays in Theoretical Computer Science in Memory of Shimon Even
, 2006
"... ..."
Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
Properties of NPcomplete sets
 In Proceedings of the 19th IEEE Conference on Computational Complexity
, 2004
"... We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreo ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p mhard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences for longstanding open questions about whether NPcomplete sets are immune. For example, assuming that pseudorandom generators and secure oneway permutations exist, it follows easily that NPcomplete sets are not pimmune. Assuming only that secure oneway permutations exist, we prove that no NPcomplete set is DTIME(2nɛ)immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomialclose to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for NP are not closed under union. Our hypothesis asserts existence of a UPmachine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ timebounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)biimmune sets, then this hypothesis is true.