TR07-003 Authors: Jin-Yi Cai, Pinyan Lu

Publication: 8th January 2007 17:58

Downloads: 3008

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Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is $\#$P-complete,and counting mod 2 is NP-hard.

We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a bases folding theorem showing that in a natural setting the support vectors can be simulated by basis of dimension 2.