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    Lower Bounds for Cutting Planes Proofs with Small Coefficients.Maria Bonet, Toniann Pitassi & Ran Raz - 1997 - Journal of Symbolic Logic 62 (3):708-728.
    We consider small-weight Cutting Planes (CP * ) proofs; that is, Cutting Planes (CP) proofs with coefficients up to $\operatorname{Poly}(n)$ . We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP * proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following (...)
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  2. Resolution Over Linear Equations and Multilinear Proofs.Ran Raz & Iddo Tzameret - 2008 - Annals of Pure and Applied Logic 155 (3):194-224.
    We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using interpolation we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on interpolants (...)
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