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We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHP cn n and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system(More)
An exponential lower bound for the size of tree-like Cutting Planes refutations of a certain family of CN F formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known(More)
We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former deenition 11] the space required for the resolution of any unsatissable formula in CNF is linear in the number of clauses. The new deenition allows a much ner analysis of the space in the refutation, ranging from constant to(More)
We prove an exponential lower bound for tree-like Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both Cutting Planes and resolution; in both cases only superpoly-nomial separations were known before [30, 20, 10]. In order to(More)
We analyze size and space complexity of Res(k), a family of propo-sitional proof systems introduced by Kraj cek in 21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreover Resolution,(More)
y Abstract. We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHP cn n and random unsatissable CNF formulas require exponential-size proofs in this system. This is the strongest(More)
The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use(More)