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21
On Interpolation and Automatization for Frege Systems
, 2000
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
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Cited by 52 (7 self)
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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexitytheoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 Frege. As a corollary, we obtain that TC 0 Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...
On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems
, 2000
"... An exponential lower bound for the size of treelike Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the treelike versions and the daglike versions of resolution and Cutting Planes. ..."
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Cited by 39 (9 self)
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An exponential lower bound for the size of treelike Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the treelike versions and the daglike versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known [29, 18, 8]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [25] are extended to monotone real circuits. An exponential separation is also proved between treelike resolution and several refinements of resolution: negative resolution and regular resolution. Actually this last separation also provides a separation between treelike resolution and ordered resolution, thus the corresponding superpolynomial separation of [29] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [13].
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 ..."
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Cited by 37 (6 self)
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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. ..."
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Cited by 31 (0 self)
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. We consider some problems about pairs of disjoint NP sets.
Nonautomatizability of boundeddepth Frege proofs
, 1999
"... In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that boundeddepth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the DiffieHellman function is sufficiently hard. It follows as a corollary that boundedde ..."
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Cited by 29 (10 self)
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In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that boundeddepth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the DiffieHellman function is sufficiently hard. It follows as a corollary that boundeddepth Frege is not automatizable; in other words, there is no deterministic polynomialtime algorithm that will output a short proof if one exists. A notable feature of our argument is its simplicity.
Exponential Separations between Restricted Resolution and Cutting Planes Proof Systems
, 1998
"... We prove an exponential lower bound for treelike Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between treelike and daglike proofs for both CuttingPlanes and resolution; in both cases only superpolynomial se ..."
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Cited by 28 (5 self)
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We prove an exponential lower bound for treelike Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between treelike and daglike proofs for both CuttingPlanes and resolution; in both cases only superpolynomial separations were known before [30, 20, 10]. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie [26] to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of treelike resolution from (daglike) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a DavisPutnam resolution proof. This extends the corresponding superpolynomial separation of [30]. Finally, we prove an exponential separation between DavisPutnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously...
Algebraic Models of Computation and Interpolation for Algebraic Proof Systems
, 1998
"... this paper we are interested in systems that use uses polynomials instead of boolean formulas. From the previous list this includes the Nullstellensatz refutations. Recently a stronger system using polynomials was proposed, the polynomial calculus, also called the Groebner calculus [9]. The proof sy ..."
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Cited by 23 (2 self)
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this paper we are interested in systems that use uses polynomials instead of boolean formulas. From the previous list this includes the Nullstellensatz refutations. Recently a stronger system using polynomials was proposed, the polynomial calculus, also called the Groebner calculus [9]. The proof systems form a similar hierarchy as the complexity classes or classes of circuits in the computational complexity, but there is no direct relation between the two hierarchies. Recently a new method was found which makes it possible to prove lower bounds on the length of proofs for some propositional proof systems using lower bounds on circuit complexity. This method is based on proving computationally efficient versions of Craig's interpolation theorem for the proof system in question [14, 18]. For appropriate tautologies the interpolation theorem
Discretely Ordered Modules as a FirstOrder Extension of the Cutting Planes Proof System
"... We define a firstorder extension LK(CP) of the cutting planes proof system CP as the firstorder sequent calculus LK whose atomic formulas are CPinequalities P i a i \Delta x i b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary ..."
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Cited by 14 (0 self)
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We define a firstorder extension LK(CP) of the cutting planes proof system CP as the firstorder sequent calculus LK whose atomic formulas are CPinequalities P i a i \Delta x i b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CPinequalities, we prove a monotone interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CPinequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Thm. 5.1]. LK(CP) can be viewed as a twosorted firstorder theory of Z considered itself as a discretely ordered Zmodule. One sort of variables are module ele...
Symmetric Approximation Arguments for Monotone Lower Bounds without Sunflowers
 Comput. Complexity
, 1997
"... We propose a symmetric version of Razborov's method of approximation to prove lower bounds for monotone circuit complexity. Traditionally, only DNF formulas have been used as approximators, whereas we use both CNF and DNF formulas. As a consequence we no longer need the Sunflower lemma that has been ..."
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Cited by 11 (0 self)
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We propose a symmetric version of Razborov's method of approximation to prove lower bounds for monotone circuit complexity. Traditionally, only DNF formulas have been used as approximators, whereas we use both CNF and DNF formulas. As a consequence we no longer need the Sunflower lemma that has been essential for the method of approximation. The new approximation...