R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper andlower bounds on tree-like cutting planes proofs. In Proceedings from Logic in Computer Science, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 well known example shows. The Chvtal rank ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting plane proofs. In Proceedings of the IEEE Symposium on Logic in Computer Science, pages 220--228, 1994.

....of LK proofs is polynomial. Thus, using a result of Buss [4] that (all versions of) PHP have polynomial proofs in the sequent calculus, we get al..so polynomial size monotone proofs of the two versions of PHP. Finally, we consider the monotone formulation of the Matching Principle that appears in [10] and get polynomial size monotone proofs as well. Using our technique we derive the following interesting result: MLK is polynomially bounded if and only if LK is. Recall that a system is called polynomially bounded if there is a polynomial bound to the minimal proof of every tautology. Thus, our ....

....every j 6= j gives sequent (2) As regards OPHP n , one simply needs define ij as i 6=i p i ;j where i ranges over f1; n 1g, and reason analogously. tu Let us be given a graph G = V; E) on n = 3m nodes. We consider the following matching principle PM n formulated in [10]. If X is a set of m edges forming a perfect matching in G and Y is an m Gamma 1 subset of V , then there is some edge (u; v) 2 X such that neither u nor v are in V . To encode this principle as a monotone sequent we use variables x i;k for i 2 [m] and k 2 [3m] whose intended meaning is that the ....

[Article contains additional citation context not shown here]

R. Impagliazzo, T. Pitassi, A. Urquhart. Upper and Lower Bounds for Tree-like Cutting Planes Proofs. In Proceedings of Ninth Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 220-228, 1994.

....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 ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In 9th IEEE Symposium on Logic in Computer Science, pages 220--228, 1994.

....resolution can have an exponential speed up over ordered resolution. The Cutting Planes proof system, CP from now on, is a refutation system based on manipulating integer linear inequalities. Exponential lower bounds for the size of CP refutations have already been proven. Impagliazzo et al. [17] proved exponential lower bounds for tree like CP . Bonet et al. 6] proved a lower bound for the subsystem CP # , where the coe#cients appearing in the inequalities are polynomially bounded in the size of the formula being refuted. This is a very important result because all known CP refutations ....

....finding proofs in CP . The separation between tree like and dag like versions of resolution and CP are obtained using the technique of the interpolation method introduced by Krajcek [21] Closely related ideas appeared previously in the mentioned works that gave lower bounds for fragments of CP ([17, 6]) The interpolation method applied on CP , translates proofs of certain formulas to monotone real circuits (a generalization of boolean circuits) The translation has two important features. First, it preserves the size, that is, the size of the circuit is similar to the size of the proof from ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proc. 9th Symposium on Logic in Computer Science, pages 220--228, 1994.

....of LK proofs is polynomial. Thus, using a result of Buss [4] that (all versions of) PHP have polynomial proofs in the sequent calculus, we get al..so polynomial size monotone proofs of the two versions of PHP. Finally, we consider the monotone formulation of the Matching Principle that appears in [8] and get polynomial size monotone proofs as well. 2 Monotone Calculus All our propositional formulas are over the basis f; g. We say that a formula is in De Morgan normal form if all the negations occur in front of the variables. For every formula , let p( be a formula in De Morgan normal ....

....0 6= j gives sequent (2) As regards OPHP n 1 n , one simply needs define ij as W i 0 6=i p i 0 ;j where i 0 ranges over f1; n 1g, and reason analogously. tu Let us be given a graph G = V; E) on n = 3m nodes. We consider the following matching principle PMn formulated in [8]. If X is a set of m edges forming a perfect matching in G and Y is an m Gamma 1 subset of V , then there is some edge (u; v) 2 X such that neither u nor v are in V . To encode this principle as a monotone sequent we use variables x i;k for i 2 [m] and k 2 [3m] whose intended meaning is that the ....

[Article contains additional citation context not shown here]

R. Impagliazzo, T. Pitassi, A. Urquhart. Upper and Lower Bounds for Tree-like Cutting Planes Proofs. Proceedings of Ninth Annual IEEE Symposium on Logic in Computer Science (LICS) pp. 220-228 (1994).

....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 following well known example shows. The Chvtal rank ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bound for tree-like cutting plane proofs. In Proc. Logic in Computer Science, LICS'94, Paris, 1994. 12

....with Gomory s cutting plane method [ it is of practical relevance. On the one hand, it has stimulated to a certain extent the search for problem specific cutting planes which became the basis of an own branch of combinatorial optimization, namely polyhedral combinatorics (see, e.g. [30, 21, 32]) On the other hand, Balas et al. 2] successfully incorporated Gomory s mixed integer cuts within a Branch and Cut framework. Third, since cuttingplane theory implies that certain implications in integer linear programming have cutting plane proofs, it is of particular importance in mathematical ....

....it is shown that an integral vector c # Z n is saturated after at most n 2 lg #c## steps of the Gomory Chvatal procedure. Since each 0 1 polytope has a representation Ax # b with A # Z mn , b # Z m such that each absolute value of an entry in A is bounded by n n 2 (see, e.g. [30]) the known bound of O(n 3 lg n) follows. One drawback in this proof is that faces of P which do not contain 0 1 points are taken to have worst case behavior n. The following observation is crucial to derive a better bound. Lemma 4. Let c x # # be valid for P I and c x # # be valid for P , ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bound for tree-like cutting plane proofs. In Proc. Logic in Computer Science, LICS'94, Paris, 1994.

....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 ....

R. Impagliazzo, T. Pitassi and A. Urquhart, Upper and lower bounds for tree-like cutting planes proofs, Proceedings of Logic in Computer Science, (1994).

....resolution can have an exponential speed up over ordered resolution. The Cutting Planes proof system, CP from now on, is a refutation system based on manipulating integer linear inequalities. Exponential lower bounds for the size of CP refutations are already proven. Impagliazzo et al. [22] proved exponential lower bounds for tree like CP . Bonet et al. 9] proved a lower bound for the subsystem CP , where the coecients appearing in the inequalities are polynomially bounded in the size of the formula being refuted. This is a very important result because all known CP refutations ....

....a classical theorem of Mathematical Logic, the Craig s interpolation theorem. Kraj cek [27] reformulated this classical theorem in order to use it to prove lower bounds for proof systems. Closely related ideas appeared previously in the mentioned works that gave lower bounds for fragments of CP ([22, 9]) The interpolation method translates proofs of certain formulas to circuits, preserving sizes. So it is a way to reduce the problem of proving proof complexity lower bounds to circuit complexity lower bounds. This is very important because in some cases there are strong circuit complexity lower ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proc. 9th Symposium on Logic in Computer Science, pages 220-228, 1994.

....lower bound on the size of tree like Cutting Planes proofs. Together with an Supported by DFG grant No. Jo 291 1 1 Preprint submitted to Elsevier Preprint 9 June upper bound from [3] this separates tree like Cutting Planes from their daglike counterparts, answering an open question from [5] . We denote by d R (f) the minimal depth of a monotone real circuit computing f , and by s R (f) the minimal size of a monotone real formula computing f . For a natural number n, n] denotes the set f1; ng. Real Communication Complexity We recall the notion of real games and real ....

....in conjunctive normal form, as shown in [4] note that a clause W i2I x i W j2J :x j is satis able i the inequality P i2I x i P j2J x j 1 jJ j has a f0; 1g solution. It was also shown in [4] that CP can simulate resolutions. For more information on Cutting Planes, see the references [1,5,10]. A CP refutation is called tree like if every line in the refutation is used at most once as a premise to an application of a rule, so that the derivation can be represented as a tree, otherwise it is called dag like. Exponential lower bounds for tree like CP refutations were given in [5] As ....

[Article contains additional citation context not shown here]

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proc. 9th LICS, pages 220-228, 1994.

....is given which directly translates Frege proofs into CP proofs. As the cutting plane system appears to be weaker than Frege systems, it seems the most natural proof system for which one can hope to obtain superpolynomial lower bounds for proof size. Indeed, Impagliazzo, Pitassi and Urquhart [25] have recently obtained exponential lower bounds for tree like cutting plane proofs. A possible candidate for hard CPLE proofs is a suitable propositional logic formulation of the Paris Harrington or Kanamori McAloon combinatorial principle [21, 22, 19] given in the last section. 2 Definitions and ....

T. Pitassi R. Impagliazzo and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proceedings of IEEE 9th Annual Symposium on Logic in Computer Science, 1994. pp. 220--228.

....Planes, see the references cited below. A CP refutation is called tree like if every line in the refutation is used at most once as a premise to an application of a rule, so that the derivation can be represented as a tree. Exponential lower bounds for tree like CP refutations were given in [4]. That paper left open the question whether tree like CP can polynomially simulate arbitrary CP , i.e. whether for some polynomial 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 ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proc. 9th LICS, pages 220--228, 1994.

.... Together with the above mentioned lower bound for tree like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall s theorem (these include tree like cutting planes proofs for which an exponential lower bound was demonstrated by [2]) Various interesting unsatisfiable propositional formulas occurring in lengthof proofs lower bounds can be formulated in the following form. Let U; V f0; 1g be two disjoint NP sets. The formula formalises that the intersection of Un : U f0; 1g n and of Vn : V f0; 1g n is not ....

....satisfying it. This yields a semantic refutation of E i s and F j s. It is easy to see that for every set A occurring in the refutation it holds that MCC R U (A) O(1) The rest follows from Theorem 3.2. q.e.d. 4 Lower bounds for Hall s theorem Impagliazzo, Pitassi and Urquhart [2] proved that a set of clauses related to BPM (similar to Halln below) requires exponential size tree like CP refutations. In this section we derive a mild generalisation of their theorem (with CP like proof systems in place of just CP) as an immediate corollary of the monotone interpolation ....

Impagliazzo, R., Pitassi, T., and Urquhart, A. (1994) Upper and lower bounds for tree-like cutting planes proofs, in: Proc. of the 9th Annual IEEE Symposium on Logic in Computer Science. Piscataway, NJ, IEEE Computer Science Press. pp.220-228.

....showing that using the Davis Putnam restriction is not, in general, a good strategy for finding resolution proofs. The Cutting Planes proof system (CP ) is a refutation system based on manipulating integer linear inequalities for which the task of findinghard to provetautologies is solved. [18] were the first to show such a result in the restricted case of CP proofs whose underlying graph is a tree. Pudlak [25] and Cook and Haken [11] give general circuit complexity results from which a exponential lower bounds for CP follow. Nothing is known about automatization of CP proofs. Since ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper andlower bounds for tree-like cutting planes proofs. In Proc. 9th LICS, pages 220--228, 1994.

No context found.

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper andlower bounds on tree-like cutting planes proofs. In Proceedings from Logic in Computer Science, 1994.

....complexity, and proof systems having feasible interpolation, in both (monotone and non monotone) cases: In the monotone case, superpolynomial lower bounds can be proven for a (sufficiently strong) proof system that admits feasible interpolation. 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 ....

....by polynomial sized circuits. Many researchers have used these ideas to prove lower bounds for propositional proof systems. In particular, in the last five years, lower bounds have been shown for all of the following systems using the interpolation method: Resolution [BPR] Cutting Planes [IPU, BPR, Pud, CH], generalizations of Cutting Planes [BPR, K1, K3] relativized bounded arithmetic [Razb2] Hilbert s Nullstellensatz [PS] the polynomial calculus [PS] and the LovaszSchriver proof system [Pud3] 1.1 Automatizability and k provability As explained in the previous paragraphs, the existence of ....

Impagliazzo, R., Pitassi, T., Urquhart, A., "Upper and lower bounds for tree-like Cutting Planes proofs," Proceedings from the IEEE Symposium on Logic in Computer Science, 1994.

....lower bound, we show how to extract a small monotone circuit computing clique on many inputs, from a small CP proof. The lower bound for CP then follows using known monotone lower bounds for the clique function [Razb1] AB] This lower bound method can be viewed as an extension of the method in [IPU]. 1 We also show how our lower bound method can be applied to obtain exponential lower bounds for several generalizations of CP . In particular, our method works for any propositional proof system consisting of a sound family of inference rules, each of which takes a constant number of formulas ....

....to a single formula (in the above definition, we have fixed the constant to be 2) 8 3 Methods and Results In this paper, we will use the well known lower bounds for monotone complexity [Razb1, AB] to prove lower bounds for the length of CP proofs. Our result is inspired by the result of [IPU], who prove an exponential lower bound for the length of tree like CP proofs, for some tautology. Below we give the main ideas of their proof. Given a monotone boolean function, f , a minterm x of f , and a maxterm y of f , there must be at least one coordinate i, such that x i = 1, and y i = 0. ....

[Article contains additional citation context not shown here]

Impagliazzo, R., Pitassi, T., Urquhart, A., "Upper and lower bounds for treelike Cutting Planes proofs," Proceedings from Logic in Computer Science, 1994. 28

....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 ....

....exist pseudorandom number generators) Many researchers have used these ideas to prove lower bounds for propositional proof systems. In particular, in the last five years, lower bounds have been shown for all of the following systems using the interpolation method: Resolution [BPR] Cutting Planes [IPU, BPR, Pud, CH], generalizations of Cutting Planes [BPR, K1, K2] relativized bounded arithmetic [Razb2] Hilbert s Nullstellensatz [PS] the polynomial calculus [PS] and the Lovasz Schriver proof system [Pud3] One of the most important questions in propositional proof complexity is to show that there is a ....

Impagliazzo, R., Pitassi, T., Urquhart, A., "Upper and lower bounds for tree-like Cutting Planes proofs," Proceedings from the IEEE Symposium on Logic in Computer Science, 1994.

....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, ....

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proceedings of the IEEE Symposium on Logic in Computer Science, 1994.

....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 ....

....exist pseudorandom number generators) Many researchers have used these ideas to prove lower bounds for propositional proof systems. In particular, in the last five years, lower bounds have been shown for all of the following systems using the interpolation method: Resolution [BPR] Cutting Planes [IPU, BPR, Pud, CH], generalizations of Cutting Planes [BPR, K1, K2] relativized bounded arithmetic [Razb2] Hilbert s Nullstellensatz [PS] the polynomial calculus [PS] and the LovaszSchriver proof system [Pud3] One of the most important questions in propositional proof complexity is to show that there is a ....

Impagliazzo, R., Pitassi, T., Urquhart, A., "Upper and lower bounds for tree-like Cutting Planes proofs," Proceedings from the IEEE Symposium on Logic in Computer Science, 1994.

....we show how to extract a small monotone circuit computing clique on many inputs, from a small CP proof. The lower bound for CP then follows using known monotone lower bounds for the clique function [Razb1] AB] This lower bound method can be viewed as an extension of the method in [IPU]. 1 We also show how our lower bound method can be applied to obtain exponential lower bounds for several generalizations of CP . In particular, our method works for any propositional proof system consisting of a sound family of inference rules, each of which takes a constant number of ....

....to a single formula. In the above definition, we have fixed the constant to be 2. 3 Methods and Results In this paper, we will use the well known lower bounds for monotone complexity [Razb1, AB] to prove lower bounds for the length of CP proofs. Our result is inspired by the result of [IPU], who prove an exponential lower bound for the length of tree like CP proofs, for some tautology. Below we give the main ideas of their proof. Given a monotone boolean function, f , a minterm x of f , and a maxterm y of f , there must be at least one coordinate i, such that x i = 1, and y i = 0. ....

[Article contains additional citation context not shown here]

Impagliazzo, R., Pitassi, T., Urquhart, A., "Upper and lower bounds for tree-like Cutting Planes proofs," Proceedings from Logic in Computer Science, 1994.

....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 [BPR1] Cutting Planes [IPU, BPR1, Pud1, CH] generalizations of Cutting Planes [BPR1, K1, K2] relativized bounded arithmetic [Razb] Hilbert s Nullstellensatz [PS] the polynomial calculus [PS] and the Lovasz Schriver proof system [Pud2] On the other hand, in a separate sequence of papers beginning with a key idea due ....

Impagliazzo, R., Pitassi, T., Urquhart, A., "Upper and lower bounds for tree-like Cutting Planes proofs," Proceedings from the IEEE Symposium on Logic in Computer Science, 1994.

No context found.

R. Impagliazzo, T. Pitassi and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proceedings of 9th Annual IEEE Symposium on Logic in Computer Science, 1994. pp. 220--228.

No context found.

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In 9th IEEE Symposium on Logic in Computer Science, pages 220--228, 1994.

No context found.

R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In 9th IEEE Symposium on Logic in Computer Science, pages 220--228, 1994.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC