Abstract.
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 Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.
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Manuscript received 2 July 2007
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Cai, JY., Lu, P. Basis Collapse in Holographic Algorithms. comput. complex. 17, 254–281 (2008). https://doi.org/10.1007/s00037-008-0249-x
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DOI: https://doi.org/10.1007/s00037-008-0249-x