Results 1  10
of
22
Witnessing Functions in Bounded Arithmetic and Search Problems
, 1994
"... We investigate the possibility to characterize (multi)functions that are \Sigma b i definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomialtime structures. We obtain the following results: 1. A reformulation of known ..."
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Cited by 35 (4 self)
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We investigate the possibility to characterize (multi)functions that are \Sigma b i definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomialtime structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are \Sigma b 1  and \Sigma b 2 definable in the theories S 1 2 and T 1 2 . 2. New characterizations of (multi)functions that are \Sigma b 2  and \Sigma b 3  definable in the theory T 2 2 . 3. A new nonconservation result: the theory T 2 2 (ff) is not 8\Sigma b 1 (ff) conservative over the theory S 2 2 (ff). To prove that the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)conservative over the theory S 2 2 (ff), we present two examples of a \Sigma b 1 (ff)principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 ; f; g) formalizing that no function f is a bijection between a 2 and a with the inverse...
How to Lie Without Being (easily) Convicted and the Lengths of Proofs in Propositional Calculus
"... We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the sec ..."
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Cited by 15 (1 self)
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We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.
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 8 (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.
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
Propositional PSPACE reasoning with Boolean programs versus quantified Boolean formulas
 In ICALP
, 2004
"... Abstract. We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new sys ..."
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Cited by 5 (4 self)
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Abstract. We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new system, BPLK, is polynomially equivalent to the system G, which is based on the familiar but very different quantified Boolean formula (QBF) characterization of PSPACE due to Stockmeyer and Meyer. This equivalence is proved by way of two translations, one of which uses an idea reminiscent of the ɛterms of Hilbert and Bernays. 1
Separating daglike and treelike proof systems
 Accepted in LICS
, 2007
"... We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on ..."
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Cited by 4 (1 self)
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We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on a class of cutformulas that satisfies some plausible hardness assumption. This gives (in some cases, conditional) negative answers to several questions from a recent work of Maciel and Pitassi (LICS 2006). Our technique is inspired by the technique from Maciel and Pitassi. While the sequents used in earlier work are derived from the Pigeonhole principle, here we generalize Statman’s sequents. This gives the desired separation, and at the same time provides stronger results in some cases. 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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Cited by 3 (0 self)
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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
Complexity of the Intuitionistic Sequent Calculus
, 1998
"... . We initiate the study of proof complexity for intuitionistic propositional proof systems: It is known that the set of intuitionistic tautologies is PSPACEcomplete (as opposed to coNPcomplete in the classical case). We show that formulas derived from the "clique tautologies " used in classical pr ..."
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Cited by 1 (0 self)
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. We initiate the study of proof complexity for intuitionistic propositional proof systems: It is known that the set of intuitionistic tautologies is PSPACEcomplete (as opposed to coNPcomplete in the classical case). We show that formulas derived from the "clique tautologies " used in classical proof complexity have only exponentialsize proofs in intuitionistic sequent calculi or Frege systems (without substitution). This is in contrast to the classical case where the complexity of Frege systems is still open. Introduction The theory of classical propositional proof systems is motivated by the conjecture NP 6= coNP. Moreover, recently there is increased interest in the impact of this theory on questions of a more algorithmic nature,for example [Bo et al.97],[Be et al. 97]. As the set of classical tautologies is coNP complete and propositional proof systems are just nondeterministic algorithms having at least 1 accepting computation for each tautology, there should be tautologies...