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• W2890817997 abstract "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 {em 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 $n^{Omega(k)}$ lower bound for the size of refutations of the binary $k$-Clique Principle in $Res(lfloor frac{1}{2}log log nrfloor)$. This improves the result of Lauria, Pudlak et al. [24] who proved the lower bound for Resolution, that is $Res(1)$. Our second lower bound proves that in $RES(s)$ for $sleq log^{frac{1}{2-epsilon}}(n)$, the shortest proofs of the $BinPHP^m_n$, requires size $2^{n^{1-delta}}$, for any $delta>0$. Furthermore we prove that $BinPHP^m_n$ can be refuted in size $2^{Theta(n)}$ in treelike $Res(1)$, contrasting with the unary case, where $PHP^m_n$ requires treelike $RES(1)$ refutations of size $2^{Omega(n log n)}$ [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 $Pi_2$-formula and involving {em total variable comparisons}. We then show that this is true, from binary to unary, when one considers the emph{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." @default.
• W2890817997 created "2018-09-27" @default.
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• W2890817997 date "2018-09-08" @default.
• W2890817997 modified "2023-09-27" @default.
• W2890817997 title "Resolution and the binary encoding of combinatorial principles" @default.
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