Resolution and the binary encoding of combinatorial principles
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We investigate the size complexity of proofs in Res(s) -- an extension of Resolution working on s-DNFs instead of clauses -- for families of contradictions given in the unusual binary encoding. A motivation of our work is size lower bounds of refutations in Resolution for families of contradictions in the usual unary encoding. Our main interest is the k-Clique Principle, whose Resolution complexity is still unknown. Our main result is a n^Ω(k) lower bound for the size of refutations of the binary k-Clique Principle in Res(1/2 n). This improves the result of Lauria, Pudlák et al. [24] who proved the lower bound for Resolution, that is Res(1). Our second lower bound proves that in RES(s) for s≤^1/2-ϵ(n), the shortest proofs of the BinPHP^m_n, requires size 2^n^1-δ, for any δ>0. Furthermore we prove that BinPHP^m_n can be refuted in size 2^Θ(n) in treelike Res(1), contrasting with the unary case, where PHP^m_n requires treelike RES(1) refutations of size 2^Ω(n n) [9,16]. Furthermore we study under what conditions the complexity of refutations in Resolution will not increase significantly (more than a polynomial factor) when shifting between the unary encoding and the binary encoding. We show that this is true, from unary to binary, for propositional encodings of principles expressible as a Π_2-formula and involving total variable comparisons. We then show that this is true, from binary to unary, when one considers the functional unary encoding. Finally we prove that the binary encoding of the general Ordering principle OP -- with no total ordering constraints -- is polynomially provable in Resolution.
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