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 ....

[Article contains additional citation context not shown here]

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.

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.

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.

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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