M. Bonet, T. Pitassi, and R. Raz, Lower bounds for cutting planes proofs with small coefficients. In: Proc. 27th ACM STOC (1995), 575--584.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....as a propositional proof system. In particular, they pointed out that the cutting plane proof system is a strengthening of resolution proofs. Since the work of Haken [30] exponential lower bounds are known for the latter. Results of Chvtal, Cook, and Hartmann [14] of Bonet, Pitassi, and Raz [8], of Impagliazzo, Pitassi, and Urquhart [35] and of Pudlk [40] imply exponential lower bounds on the length of cutting plane proofs as well. On the other hand, there is no upper bound on the length of cutting plane proofs in terms of the dimension of the corresponding polyhedron, as the following ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62:708--728, 1997.

....The first proof system working with inequalities was Cutting Planes (CP) Gom63, Chv73, CCT87, CCH89] see also Subsection 2.3. This system uses linear inequalities (with integer coefficients) Exponential lower bounds on proof size were established for CP with polynomially bounded coefficients [BPR95] as well as for the general case [Pud97] Another family of well studied proof systems are so called Lov asz Schrijver calculi (LS) LS91, Lov94] see also [Pud99] and Subsection 2.3 below. In these systems one is allowed to deal with quadratic inequalities. No non trivial complexity lower bounds ....

....a, where A = a 1 x 1 : anxn , x 1 ; xn are (integer) variables, and a 1 ; an ; a are integers. The rounding rule (2.8) transforms into i a i x i a a i x i d a e (where d 2 N; dja 1 ; an ) 5. 3) We define CP with polynomially bounded coefficients (cf. [BPR95]) if the absolute values of a i are bounded by a polynomial in the length of a CP refutation. Theorem 5.2. The following systems polynomially simulate CP with polynomially bounded coefficients: 1. LS ;split . Proof. We fix a CP refutation and simulate it rule by rule. Simulating the rule ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for Cutting Planes proofs with small coefficients. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, STOC'95, pages 575--584. ACM, 1995.

....a natural proof. Combined with the material contained in Section 4 of this paper, this leads to the independence of such lower bounds from these theories (assuming our cryptographic hardness assumption) See also [19, 34] for interpretations of this approach in terms of the propositional calculus, [10, 25] for further results in this direction, and [35] for an informal survey. 1.1. Notation and definitions We denote by F n the set of all Boolean functions in n variables. Most of the time, it will be convenient to think of f n 2 F n as a binary string of length 2 , called the truth table of f n ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proceedings of the 27th ACM Simposium on Theory of Computing, pages 575--584, 1995.

....in the case of instances which lead to successful rebuttals of an argument, there is the issue of the Challenger constructing the best line of attack, i.e. of finding the dispute that minimises dispute complexity. An analogous situation in Proof Complexity was formulated in Bonet et al. [9]: suppose is an unsatisfiable CNF with m clauses and n variables. Letting ( S) denote the size of the shortest proof of : in some Propositional Proof System S, then for a function, q : IN 3 IN, S is said to be q automatizable if there exists a (deterministic) algorithm that produces a ....

Bonet, M., T. Pitassi, and R. Raz: 1997, `Lower bounds for cutting planes proofs with small coefficients'. Journal of Symbolic Logic 62(3), 708--728.

....as a propositional proof system. In particular, they pointed out that the cutting plane proof system is a strengthening of resolution proofs. Since the work of Haken [25] exponential lower bounds are known for the latter. Results of Chvtal, Cook, and Hartmann [13] of Bonet, Pitassi, and Raz [7], of Impagliazzo, Pitassi, and Urquhart [30] and of Pudlk [34] imply exponential lower bounds on the length of cutting plane proofs as well. On the other hand, there is no upper bound on the length of cutting plane proofs in terms of the dimension of the corresponding polyhedron as 2 the ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62:708--728, 1997.

....monotone case since the same technique does not apply. We also consider the formula CLIQUE n k expressing the (n; k) Clique Coclique Principle, used by Bonet, Pitassi and Raz, and for which an exponential lower bound in Cutting Planes with polynomially bounded coefficients (poly CP) was proved [9] (notice the difference with the Clique Principle with common variables introduced by Kraj icek in [23] and used by Pudl ak in [29] to obtain exponential lower bounds for Cutting Planes with unrestricted coefficients. The latter is not a monotone tautology of the form A B) We show that 2 ....

....edge, and no other edge is present in G. A graph G is a k coclique if there is a partition of the nodes of G into k disjoint sets in such a way that any two nodes that belong to different sets are connected by an edge, and no other edges are present in G. The (n; k) clique coclique principle of [9] says that, given a set V of n nodes, if G is a k clique over V and H is a (k Gamma 1) coclique over V , then there is an edge in G that is not present in H. This principle may be stated as a monotone sequent CLIQUE n k as follows. For every l 2 f1; kg and i 2 f1; ng, let x li ....

[Article contains additional citation context not shown here]

M. Bonet, T. Pitassi, R. Raz. Lower Bounds for Cutting Planes Proofs with small Coefficients. Journal of Symbolic Logic 62(3) (1997), 708--728.

....circuits as interpolants: resolution, Cutting Planes are among those which enjoy such property. This fact is used to show that these propositional systems do not have polynomial size proofs for a sequence of tautologies expressing the positive and the negative instances of the k clique problem [4], 7] 8] In this paper, we use the same tautologies to show the opposite result. GCNF permutation has polynomial size proofs for these tautologies, hence it does not enjoy feasible monotone interpolation. At the same time, our results show that Cutting Planes, Hilbert s Nullstellensatz and ....

M. Bonet, T. Pitassi and R. Raz, "Lower bounds for cutting planes proofs with small coefficients", Proc. ACM Symp. Theory of Computing (1995) 575-584.

....the monotone case, as the same technique does not apply. We also consider the formula CLIQUE n k expressing the (n; k) Clique Coclique Principle, used by Bonet, Pitassi and Raz, and for which an exponentialy lower bound in Cutting Planes with polynomially bounded coefficients (poly CP) was proved [8] (notice the difference with the Clique Principle with common variables introduced by J. Kraj icek in [21] and used by Pudl ak in [25] to obtain exponential lower bounds for Cutting Planes with unrestricted coefficients. The latter is not a monotone tautology of the form A B) We show that ....

....edge, and no other edge is present in G. A graph G is a k coclique if there is a partition of the nodes of G into k disjoint sets in such a way that any two nodes that belong to different sets are connected by an edge, and no other edges are present in G. The (n; k) clique coclique principle of [8] says that, given a set V of n nodes, if G is a k clique over V and H is a (k Gamma 1) coclique over V , then there is an edge in G that is not present in H. This principle may be stated as a monotone sequent CLIQUE n k as follows. For every l 2 f1; kg and i 2 f1; ng, let x l;i ....

[Article contains additional citation context not shown here]

M. Bonet, T. Pitassi, R. Raz. Lower Bounds for Cutting Planes Proofs with small Coefficients. Journal of Symbolic Logic, 62 (3), pp. 708-728, 1997. A preliminary version appeared STOC'95. 15

....monotone case, as the same technique does not apply. We also consider the formula CLIQUE n k expressing the (n; k) Clique Coclique Principle, used by Bonet, Pitassi and Raz, and for which an exponentialy lower bound in Cutting Planes with polynomially bounded coefficients (poly CP) was proved [8] (notice the difference with the Clique Principle with common variables introduced by J. Kraj icek in [21] and used by Pudl ak in [25] to obtain exponential lower bounds for Cutting Planes with unrestricted coefficients. The latter is not a monotone tautology of the form A B) We show that ....

....edge, and no other edge is present in G. A graph G is a k coclique if there is a partition of the nodes of G into k disjoint sets in such a way that any two nodes that belong to different sets are connected by an edge, and no other edges are present in G. The (n; k) clique coclique principle of [8] says that, given a set V of n nodes, if G is a k clique over V and H is a (k Gamma 1) coclique over V , then there is an edge in G that is not present in H. This principle may be stated as a monotone sequent CLIQUE n k as follows. For every l 2 f1; kg and i 2 f1; ng, let x li ....

[Article contains additional citation context not shown here]

M. Bonet, T. Pitassi, R. Raz. Lower Bounds for Cutting Planes Proofs with small Coefficients. Journal of Symbolic Logic, 62 (3), pp. 708-728, 1997. A preliminary version appeared in STOC'95.

....a natural proof. Combined with the material contained in Section 4 of this paper, this leads to the independence of such lower bounds from these theories (assuming our cryptographic hardness assumption) See also [19, 34] for interpretations of this approach in terms of the propositional calculus, [10, 25] for further results in this direction, and [35] for an informal survey. 1.1. Notation and definitions We denote by F n the set of all Boolean functions in n variables. Most of the time, it will be convenient to think of f n 2 F n as a binary string of length 2 n , called the truth table of f ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proceedings of the 27th ACM Simposium on Theory of Computing, pages 575--584, 1995.

.... fact are known to provide exponentially shorter proofs on some examples (the pigeonhole clauses) An exponential lower bound for cutting plane proofs for clause sets based on a clique coclique distinction, under the restriction that coefficients in the proof are polynomially bounded, was proved in [BPR95], by reducing the problem to lower bounds for monotone Boolean circuits, and applying the Razborov Andreev result mentioned above. Building on this and other work, Pudl ak [Pud95A] showed how the restriction on coefficient size could be eliminated if the monotone circuit lower bound could be ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proc. Twenty-seventh Ann. ACM Symp. Theor. Comput., pages 575--584, 1995.

....propositional proofs must necessarily be at least as hard as complexity lower bounds (and the latter are currently inaccessible) let us at least show that they are just as hard One successful scheme fulfilling this idea for substantially weaker p.p.s. appeared, implicitly and independently, in [32, 9]. More explicit and slightly different treatments of this scheme were later given in [23, 33] and in our presentation we follow an intermediate course between them. Let U; V f0; 1g be two disjoint NP sets, and let us arbitrarily fix their NP representations x 2 U j 9y 2 f0; 1g p(n) An ....

....set L of the complexity prescribed by the interpolation theorem, then the corresponding tautologies :An (p; q) Bn (p; r) do not have short P proofs, and, thus, P is not optimal. Lower bounds based upon this idea independently appeared in [32] for the system R in the uniform framework) and in [9] (for CP ) 27] established the interpolation theorem for cutting planes with arbitrary coefficients, and this is the strongest system for which it is currently known: Theorem 5. There exists a polynomial time algorithm which does the following. Given a CP refutation of some set of clauses ....

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M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proceedings of the 27th ACM STOC, pages 575--584, 1995.

....the propositional calculus. Cutting plane proofs provide a complete refutation system for unsatisfiable sets of propositional clauses. They efficiently simulate resolution proofs, and in fact are known to provide exponentially shorter proofs on some examples (the pigeonhole clauses) Bonet et al. [8] and Pudl ak [19] reduced the problem to lower bounds for circuits with nondecreasing real functions of fanin 2 as gates. Thus, our general lower bound for such circuits (Theorem 2.1) as well as lower bounds for explicit functions, are also lower bounds for the length of cutting plane proofs. 5 ....

M. Bonet, T. Pitassi, and R. Raz, Lower bounds for cutting planes proofs with small coefficients. In: Proc. 27th ACM STOC (1995), 575--584.

....formula of the form f = A(x) B(y) if f has a size s refutation in S, then either A(x) or B(y) has a size p(s) refutation in S. Effective interpolation theorems are very important because they give rise to conditional lower bounds for the corresponding proof system. See the papers [25, 14, 33, 30, 17] for more details. And if a monotone version of the interpolation theorem holds, then using known lower bound for monotone circuits, it is possible to prove an (unconditional) lower bound for the proof systems. For example, this has been the method used to obtain exponential lower bounds for ....

Bonet, M., Pitassi, T., Raz, R. Lower bounds for Cutting Planes proofs with small coefficients. To appear in Journal of Symbolic Logic, 1995.

....and the cutting plane system are among of those which enjoy such property. This fact is used to show that these propositional systems do not have polynomial size proofs for a sequence of tautologies, called k T est(n) which expressing the positive and the negative test for k clique problem [6], 11] 12] In this paper, we use k T est(n) to show the opposite result. GCNF permutation is powerful enough to polynomially prove k T est(n) hence it does not enjoy feasible monotone interpolation. At the same time, k Test(n) witnesses the fact that the cutting plane system does not ....

M. Bonet, T. Pitassi and R. Raz, "Lower bounds for cutting planes proofs with small coefficients", Proc. ACM Symp. Theory of Computing (1995) 575-584.

....condition of [4] One direction of study of propositional proof systems is to prove lower bounds on the length of proofs in certain restricted proof systems. Exponential lower bounds were obtained for such systems as resolution [12] bounded depth Frege systems [15, 17] cutting planes [6, 18], and Nullstellensatz refutations [3, 8] In 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 ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proc. of the 27th Ann. ACM Symp. on Theory of Computing, pages 575--584. ACM, 1995.

.... p(x) every set of inequalities that has a CP refutation of size s also has a tree like CP refutation of size p(s) The question was answered for the subsystem CP , where every coefficient appearing in a refutation must be bounded by a polynomial in the size of the original inequalitites, by [1]: they showed that CP cannot be simulated by tree like CP . We shall show the same for CP with arbitrary coefficients. Cutting Planes refutations are linked to monotone real circuits by the following interplation theorem due to Pudl ak: Theorem 6 (Pudl ak [8] Let p; q; r be ....

M. L. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proc. 27th STOC, pages 575--584, 1995.

....by a set of small circuit complexity give rise to an implication whose all P proofs must be large. Thus we get in this way lower bounds for proof systems admitting effective interpolation. This idea works for some proof systems, most notably for resolution and cutting planes proof systems (see [27, 2, 10, 4, 25, 11, 26, 12]) In this paper we study limitations to this method. In particular, we shall show that Extended Frege proof system does not admit effective interpolation unless the RSA cryptosystem is not secure. The assumption about RSA is stronger than P=poly 6= NP , so it will be very difficult to prove. ....

Bonet, M. L., Pitassi, T., and Raz, R. (1994) Lower bounds for cutting planes proofs with small coefficients, preprint.

....to split Partially supported by the US Czechoslovak Science and Technology Program grant # 93025, and by grant #A1019602 of the AV CR. repeatedly) the proof into two parts, depending on whether a CP inequality is or is not satisfied (see Example 3 in Section 2 for a formalization due to [4]) A natural question arises to extend the lower bounds for CP to these presumably stronger proof systems. It appeared to us that a convenient formalization of CP with the deduction rule is resolution working with clauses formed by CP inequalities (see Section 3) In particular, it allows us to ....

....proof systems. It appeared to us that a convenient formalization of CP with the deduction rule is resolution working with clauses formed by CP inequalities (see Section 3) In particular, it allows us to discuss also proofs in which the deduction formulas are not organized in a tree (as it is in [4]) We noticed that an interpolation theorem for such a proof system, and, in fact, for its first order extension LK(CP) is an immediate consequence of a universal interpolation theorem for semantic derivations [16, Thm. 5.1] the monotone interpolation for R(CP) needs an extra argument) Let us ....

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Bonet, M. L., Pitassi, T., and Raz, R. (1994) Lower bounds for cutting planes proofs with small coefficients, preprint.

....the propositional calculus. Cutting plane proofs provide a complete refutation system for unsatisfiable sets of propositional clauses. They efficiently simulate resolution proofs, and in fact are known to provide exponentially shorter proofs on some examples (the pigeonhole clauses) Bonet et al. [6] and Pudl ak [22] reduced the problem to lower bounds for circuits with nondecreasing real functions of fanin 2 as gates. Thus, our general lower bound for such circuits (Theorem 2) as well as lower bounds for explicit functions, are also lower bounds for the length of cutting plane proofs. 2. ....

M. Bonet, T. Pitassi, and R. Raz, Lower bounds for cutting planes proofs with small coefficients. In: Proc. Twenty-seventh Ann. ACM Symp. Theor. Comput. , (1995), 575--584.

....depth Frege systems corresponded above to AC 0 . As we already mentioned, no superpolynomial lower bounds on the length of proofs function for unrestricted Frege systems are known. Such bounds, however, are known for resolution (Tseitin 1968, Haken 1985, Urquhart 1987, Chvatal Szemeredi 1988, Bonet et al. 1995, Kraj icek 1994b) bounded depth Frege, with and without counting axioms modulo m (Ajtai 1988, Bellantoni et al. 1992, Kraj icek 1994a, Kraj icek et al. 1995, Pitassi et al. 1993, Ajtai 1990, Beame Pitassi 1993, Riis 1993, Ajtai 1994, Beame et al. 1994, Riis 1994) and for cutting planes ....

.... 1995, Kraj icek 1994b) bounded depth Frege, with and without counting axioms modulo m (Ajtai 1988, Bellantoni et al. 1992, Kraj icek 1994a, Kraj icek et al. 1995, Pitassi et al. 1993, Ajtai 1990, Beame Pitassi 1993, Riis 1993, Ajtai 1994, Beame et al. 1994, Riis 1994) and for cutting planes (Bonet et al. 1995, Kraj icek 1994b, Pudlak 1995b) The survey Urquhart (1995) discusses many of these superpolynomial lower bounds. The second hierarchy comprises algebraic proof systems. Let be a fixed (commutative) ring, usually Zm (integers modulo m) for some integer m. Instead of adding modular reasoning to ....

M. Bonet, T. Pitassi, and R. Raz, Lower bounds for cutting planes proofs with small coefficients. In Proceedings of the 27th ACM STOC, 1995, 575--584.

....cutting plane proof system. The idea of the lower bound proof goes back to Kraj icek s paper [16] There he proposed to use interpolation theorem to reduce lower bounds on the lengths of propositional proofs to lower bounds on circuits. Such an idea was implicitly used by Bonet, Pitassi and Raz in [4]. Razborov used interpolation in the context of bounded arithmetic [23] In [18] Kraj icek stated explicitly and proved interpolation theorems for certain proof systems using which it is possible to reduce the problem of proving lower bounds on the length of propositional proofs to lower bounds on ....

.... with a restriction on the size of coefficients (the restriction is that the absolute value of the coefficients in the inequalities used in the proof is bounded by a polynomial in the size of the proof) For such cutting plane proofs this generalizes the earlier result of Bonet, Pitassi and Raz [4]. In both cases, resolution and cutting planes, the reduction is to lower bounds on monotone boolean circuits. In this paper we give different proofs of the interpolation theorems of Kraj icek. Our proofs provide more information about the connection between the proofs and the interpolating ....

[Article contains additional citation context not shown here]

Bonet, M., Pitassi, T. and Raz, R., Lower bounds for Cutting Planes proofs with small coefficients, Proc. 27-th STOC, 1995, 575-584.

....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 circuit size lower bounds in the bounded arithmetic theory S 2 2 (ff) 1991 Mathematics Subject Classification. Primary 03F20, 03B05, 03F30; Secondary 68Q25. Partially supported by the US Czechoslovak ....

....done while I was visiting the Department of Mathematics of the University of California at San Diego in April 1994. I replaced the original statements about the linear equational calculus over Q by present Corollaries 6.4 and 7.3(2. after learning about the proof system CP and the result of [4] from a lecture by M. L. Bonet at the meeting Logic and Computational Complexity (Indianapolis, October 1994) I thank A.A.Razborov for explaining to me the remarks on one way functions made in [43, Sect. 8] but not described there. I thank P. Pudl ak and J. Sgall for helpful comments on the ....

Bonet, M. L., Pitassi, T., and Raz, R. (1994) Lower bounds for cutting planes proofs with small coefficients, preprint.

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M. L. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, 1997.

No context found.

M. L. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, 1997. A preliminary version appeared in STOC'95.

.... of F require size 2 Omega (n 1=3 = log(n) 5 Separation between Res(2) and Resolution In this section we prove that Resolution cannot polynomially simulate Res(2) More precisely, we prove that a certain Clique Coclique principle, as defined by Bonet, Pitassi and Raz in [6], has polynomial size Res(2) refutations, but every Resolution refutation requires quasipolynomial size. The Clique Coclique principle that we use, CLIQUE 0 , is: x i;1 Delta Delta Delta x i;n 1 l k x l;i x l;j 1 l k; 1 i; j n; i 6= j x l;i x l 0 ;i 1 l; l k; 1 i n; l ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic, 62(3):708--728, Sept. 1997.

....system S, there will not be an algorithm that will produce S proofs of a tautology in time polynomial in the size of the tautology. This is because in some cases we might require exponential time just to write down the proof. Considering this limitation of proof systems, Bonet, Pitassi and Raz [12] proposed the following definition. A propositional proof system S is automatizable if there exists an algorithm that, given a tautology, it produces an S proof of it in time polynomial in the size of the smallest S proof of the tautology. The idea behind this definition is that if short ....

....Resolution Res(2) or Res(k) for k constant) has feasible interpolation. This notion will be defined in Section 4. Let us say for the moment, that Resolution, Cutting Planes, Relativized Bounded Arithmetic, Polynomial Calculus, Lov asz Schrijver and Nullstellensatz have feasible interpolation (see [20, 12,26, 15, 22, 30, 29, 27]) On the other hand, the stronger system Frege, and any system that simulates AC Frege do not have feasible interpolation under a cryptographic conjecture. To obtain this characterization we show that Res(2) has polynomial size proofs of the reflection principle of Resolution, which is a form ....

[Article contains additional citation context not shown here]

M. L. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, 1997. A preliminary version appeared in STOC'95.

....of the smallest resolution refutation (resp. DLL refutation) of F . A fundamental problem is to find effective algorithms for constructing resolution refutations and DLL refutations whose size is close to optimal. This is the automatizibility problem for proof systems, which was formalized in [BPR97]. DEFINITION 1.1: Let S be an arbitrary propositional proof system. For the unsatisfiable formula F , let s(F) denote the size of the smallest refutation of F in S . Then S is said to be automatizable if there exists a deterministic algorithm that takes as input an unsatisfiable formula F on n ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, September 1997.

.... = 1 1 jKj=jF j = 1 1 2 b Gamma 1 = 1 2 b tu (of claim 4) 5 Separation between Res(2) and Resolution In this section we prove that Resolution cannot polynomially simulate Res(2) More precisely, we prove that a certain Clique Coclique principle, as defined by Bonet, Pitassi and Raz in [6], has polynomial size Res(2) refutations, but every Res refutation requires quasipolynomial size. The Clique Coclique principle that we use, CLIQUE n k;k 0 , is the conjunction of the following set of clauses: x i;1 Delta Delta Delta x i;n 1 l k (12) x l;i x l;j 1 l k; 1 i; j n; i ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic, 62(3):708--728, Sept. 1997.

.... exponential separation between Res(2) and Res(log) 5 Superpolynomial Separation between Res(2) and Resolution In this section we prove that Resolution cannot polynomially simulate Res(2) More precisely, we prove that a certain Clique Coclique principle, as defined by Bonet, Pitassi and Raz in [5], has polynomial size Res(2) refutations, but every Res refutation requires quasipolynomial size. The Clique Coclique principle that we use, CLIQUE n k;k 0 , is the conjunction of the following set of clauses: x i;1 Delta Delta Delta x i;n 1 l k (5) x l;i x l;j 1 l k; 1 i; j n; i ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic, 62(3):708--728, Sept. 1997.

....in its own right. Indeed, determining the monotone size (or depth) of a function is a very natural combinatorial problem, and monotone complexity may be relevant for several other complexity issues. One important example is propositional proof theory, where following 3 [Ra94] and Bonet et. al [BoPiRa95], reductions to monotone complexity were extensively used. In particular, using techniques developed in the sequence of papers [ImPiUr94, BoPiRa95, Kr95] Pudlak [Pu95] used monotone complexity to obtain an impressive exponential lower bound for the length of cutting planes proofs (see also, ....

....complexity may be relevant for several other complexity issues. One important example is propositional proof theory, where following 3 [Ra94] and Bonet et. al [BoPiRa95] reductions to monotone complexity were extensively used. In particular, using techniques developed in the sequence of papers [ImPiUr94, BoPiRa95, Kr95], Pudlak [Pu95] used monotone complexity to obtain an impressive exponential lower bound for the length of cutting planes proofs (see also, CoHa95, Fu96] Other applications of monotone complexity are also known. 1.2 Methods and Other Results We use Karchmer and Wigderson s communication ....

M. Bonet, T. Pitassi and R. Raz, Lower bounds for cutting planes proofs with small coefficients, Proc. of the 27th ACM Symp. on the Theory of Computing (1995), pp. 575--584. Full version to appear in: Journal of Symbolic Logic.

....of the smallest resolution refutation (resp. DLL refutation) of F . A fundamental problem is to find effective algorithms for constructing resolution refutations and DLL refutations whose size is close to optimal. This is the automatizibility problem for proof systems, which was formalized in [BPR97]. DEFINITION 1.1: Let S be an arbitrary propositional proof system. 1 For the unsatisfiable formula F , let s(F) denote the size of the smallest refutation of F in S . Then S is said to be automatizable if there exists a deterministic algorithm that takes as input an unsatisfiable formula F on ....

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, September 1997.

....feasible interpolation, in both (monotone and non monotone) cases: In the monotone case, it was proved that a (sufficiently strong) proof system S , that admits monotone feasible interpolation, cannot have polynomial size proofs for all tautologies. This was presented by the sequence of papers [IPU, BPR, K1], and was first used in [BPR] to prove lower bounds for propositional proof systems. The idea is also implicit in [Razb2] In short, the statement F that is used is the Clique interpolation formula, A 0 (g; x) A 1 (g; y) where A 0 states that g is a graph containing a clique of size k ....

....and non monotone) cases: In the monotone case, it was proved that a (sufficiently strong) proof system S , that admits monotone feasible interpolation, cannot have polynomial size proofs for all tautologies. This was presented by the sequence of papers [IPU, BPR, K1] and was first used in [BPR] to prove lower bounds for propositional proof systems. The idea is also implicit in [Razb2] In short, the statement F that is used is the Clique interpolation formula, A 0 (g; x) A 1 (g; y) where A 0 states that g is a graph containing a clique of size k (where the clique is described by ....

[Article contains additional citation context not shown here]

Bonet, M., Pitassi, T., and Raz, R. "Lower bounds for Cutting Planes proofs with small coefficients," Proceedings of the ACM Symposium on the Theory of Computing, 1995, pp. 575-584. Also to appear in Journal of Symbolic Logic.

....and thus does not have polynomial size refutations in any proof system that has the monotone interpolation property. In the last few years, the interpolation method has been used to prove many lower bounds. In particular, lower bounds have been shown for all of the following systems: Resolution [2], Cutting Planes [6, 2, 13, 4] generalizations of Cutting Planes [2, 8, 7] relativized bounded arithmetic [15] Hilbert s Nullstellensatz [14] the polynomial calculus [14] and the Lovasz Schriver proof system [12] On the other hand, in a separate sequence of papers beginning with a key idea ....

....have polynomial size refutations in any proof system that has the monotone interpolation property. In the last few years, the interpolation method has been used to prove many lower bounds. In particular, lower bounds have been shown for all of the following systems: Resolution [2] Cutting Planes [6, 2, 13, 4], generalizations of Cutting Planes [2, 8, 7] relativized bounded arithmetic [15] Hilbert s Nullstellensatz [14] the polynomial calculus [14] and the Lovasz Schriver proof system [12] On the other hand, in a separate sequence of papers beginning with a key idea due to Krajcek and Pudlak [9, ....

[Article contains additional citation context not shown here]

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, 1997.

....feasible interpolation, in both (monotone and non monotone) cases: In the monotone case, it was proved that a (sufficiently strong) proof system S , that admits monotone feasible interpolation, cannot have polynomial size proofs for all tautologies. This was presented by the sequence of papers [IPU, BPR, K1], and was first used in [BPR] to prove lower bounds for propositional proof systems. The idea is also implicit in [Razb2] In short, the statement F that is used is the Clique interpolation formula, A 0 (g; x)A 1 (g; y) where A 0 states that g is a graph containing a clique of size k (where ....

....and non monotone) cases: In the monotone case, it was proved that a (sufficiently strong) proof system S , that admits monotone feasible interpolation, cannot have polynomial size proofs for all tautologies. This was presented by the sequence of papers [IPU, BPR, K1] and was first used in [BPR] to prove lower bounds for propositional proof systems. The idea is also implicit in [Razb2] In short, the statement F that is used is the Clique interpolation formula, A 0 (g; x)A 1 (g; y) where A 0 states that g is a graph containing a clique of size k (where the clique is described by the ....

[Article contains additional citation context not shown here]

Bonet, M., Pitassi, T., and Raz, R. "Lower bounds for Cutting Planes proofs with small coefficients," Proceedings of the ACM Symposium on the Theory of Computing, 1995, pp. 575-584. Also to appear in Journal of Symbolic Logic.

No context found.

M. Bonet, T. Pitassi, and R. Raz, Lower bounds for cutting planes proofs with small coefficients. In: Proc. 27th ACM STOC (1995), 575--584.

No context found.

M. Bonet, T. Pitassi, and R. Raz, Lower bounds for cutting planes proofs with small coefficients. In: Proc. Twenty-seventh Ann. ACM Symp. Theor. Comput. , (1995), 575--584.

No context found.

M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62(3):708--728, 1997.

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