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32
Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic
"... A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We ..."
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Cited by 91 (2 self)
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A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuitsize lower bounds ...
On the Weak Pigeonhole Principle
, 2001
"... We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of Ramsey theorem. In particular, we link the proof complexity of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of ..."
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Cited by 75 (2 self)
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We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of Ramsey theorem. In particular, we link the proof complexity of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) treelike resolution proofs of Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions (the existence of oneway functions). In particular, we show that functions violating WPHP 2n n are oneway and, on the other hand, that oneway permutations give rise to functions violating PHP n+1 n , and that strongly collisionfree families of hash functions give rise to functions violating WPHP 2n n (all in suitable models of bounded arithmetic). Further we formulate few problems and conjectures; in particular, on the structured PHP (introduced here) and on the unrelativised WPHP. The symbol WPHP m n...
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 IN IZVESTIYA OF THE RUSSIAN ACADEMY OF SCIENCE, MATHEMATICS
, 1995
"... 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 smal ..."
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Cited by 54 (6 self)
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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.
A New Proof of the Weak Pigeonhole Principle
, 2000
"... The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with boundeddepth, quasipolynomialsize proofs. Their argument was further re ..."
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Cited by 38 (3 self)
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The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with boundeddepth, quasipolynomialsize proofs. Their argument was further refined by Kraj'icek [9]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomialsize LK proofs where every formula consists of a single AND/OR of polylog fanin. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. 1 Introduction The pigeonhole principle is a fundamental axiom of mathematics, stating that there is no onetoone mapping from m pigeons to n holes when m ? n. It expresses Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136995815, U.S.A. alexis@clarkson.edu. Research supported by NSF grant CCR9877150. y Department of Computer Science, University o...
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Cited by 13 (3 self)
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce
Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
Lifting Independence Results in Bounded Arithmetic
 ARCHIVE FOR MATHEMATICAL LOGIC
, 1999
"... We investigate the problem how to lift the non  8\Sigma b 1 (ff)  conservativity of T 2 2 (ff) over S 2 2 (ff) to the expected non  8\Sigma b i (ff)  conservativity of T i+1 2 (ff) over S i+1 2 (ff), for i ? 1. We give a nontrivial refinement of the "lifting method" develope ..."
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Cited by 13 (1 self)
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We investigate the problem how to lift the non  8\Sigma b 1 (ff)  conservativity of T 2 2 (ff) over S 2 2 (ff) to the expected non  8\Sigma b i (ff)  conservativity of T i+1 2 (ff) over S i+1 2 (ff), for i ? 1. We give a nontrivial refinement of the "lifting method" developed in [4, 8], and we prove a sufficient condition on a 8\Sigma b 1 (f)consequence of T2 (f) to yield the nonconservation result. Further we prove that Ramsey's theorem, a 8\Sigma b 1 (ff)  formula, is not provable in T 1 2 (ff), and that 8\Sigma b j (ff)  conservativity of T i+1 2 (ff) over T i 2 (ff) implies 8\Sigma b j (ff)  conservativity of the whole T2 (ff) over T i 2 (ff), for any j
Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic
, 2005
"... ..."
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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Cited by 9 (4 self)
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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.