A. Goerdt. Unrestricted resolution versus Nresolution. In Mathematical Foundations of Computer Science 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of this paper, we show that for each constant k, there is an # k 0, such that there is a family of unsatisfiable CNFs which have polynomial size Res(k 1) refutations but which require size 2 # k in Res(k) The unsatisfiable clauses are a variation of the graph ordering tautologies of [19, 10]. Definition 5.4 Let G be an undirected graph. For each vertex u of G, let N(u) denote the set of neighbors of u in G. For each ordered pair of vertices (u, v) v, let there be a propositional variable X u,v . The graph ordering principle for G, GOP (G) is the following set of clauses: ....

A. Goerdt. Unrestricted resolution versus Nresolution. In Mathematical Foundations of Computer Science 1990.

.... Jordi Girona Salgado 1 3 Barcelona Spain e mail bonet lsi.upc.es Nicola Galesi School of Mathematics Institute for Advanced Study Princeton 08540 New Jersey USA e mail: galesi ias.edu Abstract We consider a modification of the pigeonhole principle, MPHP , introduced by Goerdt in [7]. Using a technique of Razborov [9] and simplified by Impagliazzo, Pudl ak and Sgall [8] we prove that any Polynomial Calculus refutation of a set of polynomials encoding the MPHP , requires degree Omega Gammagre n) We also prove that the this lower bound is tight, giving Polynomial ....

....degree d for which refutations of PHP of degree d exist. The main contribution of our result is extending Razborov s technique to a combinatorial principle somewhat different from the pigeonhole principle. We consider a modification of the pigeonhole principle (MPHP ) introduced by Goerdt [7]. MPHP is defined over n pigeons and log n holes, and differs from PHP since it allows many holes to hold more than one pigeon (see Definition 2.1 for further details) In fact, MPHP is minimally unsatisfiable. Note that the PC degree lower bound of Razborov doesn t give lower bounds for this ....

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A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93:159-- 167, 1992.

....used resolution refinements: negative resolution, regular resolution and ordered resolution. We show an exponential separation between tree like resolution and each one of the above restrictions (Corollary 20 for negative resolution and Corollary 23 for both regular and ordered resolution) Goerdt [14, 13, 15] gave several superpolynomial separations between unrestricted resolution and some refinements of resolution, in particular he gave a superpolynomial separation between ordered resolution and unrestricted resolution. In this paper we consider the case of ordered resolution and we improve his ....

....problem would follow from a strongly exponential separation of monotone real formula size from monotone circuit size. Such a strong separation is not even known for monotone boolean circuits. 4. Can the superpolynomial separations of regular and negative resolution from unrestricted resolution [14, 15] be improved to exponential as well And is there an exponential speed up of regular over ordered resolution Acknowledgments We would like to thank Ran Raz for reading a previous version of this work and discovering an error, Andreas Goerdt for sending us copies of his papers, Sam Buss for ....

A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93:159--167, 1992.

....degree lower bound for a formula known to have polynomial size resolution refutations. Our degree lower bound proof extends the Polynomial Calculus lower bound technique introduced by Razborov in [21] to a formula obtained as a modification of the pigeon hole principle defined by Goerdt in [15]. It is hence of independent interest since this technique was known to work only for the PHP formula. We conjecture that our result can be improved, to show that the simulation of [12] is the best possible in the case the resolution proof is small and we use the standard translation of clauses ....

....for resolution refutations. He conjectured that GT n required long proofs in resolution. Stalmark in [23] refuted Krishnamurthy s conjecture giving polynomial size unrestricted resolution refutations. We will use a tautology encoding a modification of the pigeon hole principle defined in [15]. Let n be a natural number of the form 2 k , for some k, and let m = log 2 n. For each j = 1; m, let Part(j) be the partition of [n] induced by j the following way: Part(j) ffi; i 1; i (2 j Gamma 1)g j i = 1; 1 2 j ; 1 2 Delta 2 j ; 1 ( n 2 j ....

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A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93:159--167, 1992.

....that each node receives at least an edge from some other node (i.e. there is no source node) This formula was first formulated by Krishnamurthy in [Kr85] and then Stalmark in [St96] gave polynomial size resolution refutations. We consider a modification of the pigeon hole principle defined in [Go92] Let n be of the form 2 k , for some k and let m = log 2 n (all the log are in base 2) For each j = 1; m let Part(j) the partition of [n] induced by j the following way: Part(j) ffi; i 1; i (2 j Gamma 1)g j i = 1; 1 2 j ; 1 2 Delta 2 j ; 1 ( n 2 ....

.... clauses, where x i;j means that the pigeon i is sitting in the hole j: W m j=1 x i;j i 2 [n] x i;j x i 0 ;j j 2 [m] i 6= i 0 2 [n] not j compatible x i;j x i;k i 2 [n] j 2 [m] Observe that the clauses defining our MPHP n are a superset of the clasuses defining the MPHP n of [Go92] Goerdt gave in [Go92] polynomial size unrestricted refutations for MPHP n . 3 Optimality of the width size Method and its Consequences In this section we show the optimality of the size width trade off of [BW99] As formulated in [BW99] the question is the following: can one find an ....

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A. Goerdt. Unrestricted Resolution versus N-Resolution. Theoretical Computer Science 93 (1992) pp. 159-167.

....used resolution re nements: negative resolution, regular resolution and ordered resolution. We show an exponential separation between tree like resolution and each one of the above restrictions (Corollary 20 for negative resolution and Corollary 23 for both regular and ordered resolution) Goerdt [19, 18, 20] gave several superpolynomial separations between unrestricted resolution and some re nements, in particular he gave a superpolynomial separation between ordered resolution and unrestricted resolution. We improve this result by giving an exponential separation between ordered and negative ....

....problem would follow from a strongly exponential separation of monotone real formula size from monotone circuit size. Such a strong separation is not even known for monotone boolean circuits. 4. Can the superpolynomial separations of regular and negative resolution from unrestricted resolution [19, 20] be improved to exponential as well And is there an exponential speed up of regular over ordered resolution Acknowledgments We would like to thank Ran Raz for reading a previous version of this work and discovering an error, Andreas Goerdt for sending us copies of his papers, Sam Buss for ....

A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93:159{ 167, 1992.

....DNF DNF. 2. Application of DP l is commutative (modulo r S ) 6) that is, for any bijection : f1; mg f1; mg we have: r S ffi DP l 1 ; l m = r S ffi DP l (1) l (m) 6) but in general not with respect to restriction R (see Remark 5) furthermore, as shown in [Go] at least for the unrestricted case and for DP resolution proof systems, the amount of work needed for computing DP l 1 ; l m by successive computations of DP l i may depend heavily on the order of the l 1 ; l m 33 Proof: Parts 1 and 2 follows from Lemma 7.2, part 1 respectively part ....

Goerdt, A.: Personal communication. The hard formulas from "Unrestricted resolution versus N-resolution", Theoretical Computer Science 93 (1992), 159--167, have polynomial DP-resolution proofs as (implicitely) shown in the article, but have only exponential DPresolution proofs for the reverse order.

....in [11] cannot be bether than quasipolynomial, in the case we start with a polynomial size resolution refutation. Our lower bound proof extends the PC lower bound technique introduced by Razborov in [17] to a formula obtained as a modification of the pigeonhole principle defined by Goerdt in [13]. It is hence of independent interest since this technique was known to work only for the PHP formula. We moreover conjecture that our result can be improved, to show that the simulation of [11] is the best possible in the case the resolution proof is small. Recall that without this last ....

....(3) say that each node receives at least an edge from some other node (i.e. there is no source node) This formula was first formulated by Krishnamurthy in [16] and then Stalmark in [19] gave polynomial size resolution refutations. We consider a modification of the pigeon hole principle defined in [13]. Let n be of the form 2 k , for some k and let m = log 2 n (all the log are in base 2) For each j = 1; m let Part(j) the partition of [n] induced by j the following way: Part(j) ffi; i 1; i (2 j Gamma 1)g j i = 1; 1 2 j ; 1 2 Delta 2 j ; 1 ( n 2 ....

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A. Goerdt. Unrestricted Resolution versus NResolution. Theoretical Computer Science 93 (1992) pp. 159-167.

....finding tree like resolution proofs cannot be an efficient strategy for finding resolution proofs. Until now only superpolynomial separations were known [30, 10] Many strategies for finding resolution proofs are described in [28] but very little theoretical work has been done until now. Goerdt [15, 14, 16] gave several superpolynomial separations between resolution and some restricted versions of it. In particular, he gave a separation between Davis Putnam resolution and unrestricted resolution. We improve this result by giving an exponential separation between Davis Putnam and unrestricted ....

A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93:159--167, 1992.

....three of the most commonly used restrictions of resolution: N resolution, Regular and Davis Putnam resolution. We show an exponential separation between tree like Resolution and each one of the above restrictions (Corollary 20 for N resolution and Corollary 23 for Regular and Davis Putnam) Goerdt [17, 16, 18] gave several superpolynomial separations between resolution and some restricted versions of it. In particular, he gave a separation between Davis Putnam resolution and unrestricted resolution. We improve this result by giving an exponential separation between Davis Putnam and N resolution ....

....problem would follow from a strongly exponential separation of monotone real formula size from monotone circuit size. Such a strong separation is not even known for monotone boolean circuits. 4. Can the superpolynomial separations of regular and negative resolution from unrestricted resolution [17, 18] be improved to exponential as well And is there an exponential speed up of regular over Davis Putnam resolution Acknowledgments We would like to thank Ran Raz for reading a previous version of this work and discovering an error, Andreas Goerdt for sending us copies of his papers, Sam Buss for ....

A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93:159-- 167, 1992.

....deserves to be mentioned. When applying provers that are based on resolution, it is quickly suggested to use (ordered) hyper resolution instead of ordinary binary resolution, as this yields bigger inference steps and reduction of the use of memory. This strongly contrasts with the results in [8] from which one can easily conclude that there is a class of formulas with polynomial proofs using ordinary resolution that only have exponential proofs in hyper resolution. HeerHugo clearly shows that the current situation is deploring. Without truly understanding why it appeared to be possible ....

A. Goerdt. Unrestricted resolution versus N--resolution. Technical Report KI--NRW 89--7. Universit at Duisburg. Germany. 1989.

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A. Goerdt. Unrestricted resolution versus Nresolution. In Mathematical Foundations of Computer Science 1990.

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A. Goerdt. Unrestricted resolution versus N-resolution. Theoretical Computer Science, 93(1):159--167, February 1992.

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