Abstract
We propose matchgate tensors as a natural and proper language to develop Valiant’s new theory of Holographic Algorithms. We give a treatment of the central theorem in this theory—the Holant Theorem—in terms of matchgate tensors. Some generalizations are presented.
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Dodson, C.T.J., Poston, T.: Tensor Geometry, Graduate Texts in Mathematics, 2nd edn., vol. 130. Springer, New York (1991)
Jerrum, M., Snir, M.: Some Exact Complexity Results for Straight-Line Computations over Semirings. J. ACM 29(3), 874–897 (1982)
Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961)
Kasteleyn, P.W.: Graph Theory and Crystal Physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, London (1967)
Strassen, V.: Gaussian Elimination is Not Optimal. Numerische Mathematik 13, 354–356 (1969)
Tardos, É.: The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8(1), 141–142 (1988)
Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics –an exact result. Philosophical Magazine 6, 1061–1063 (1961)
Valiant, L.G.: Negation can be Exponentially Powerful. Theor. Comput. Sci. 12, 303–314 (1980)
Valiant, L.G.: Holographic Algorithms (Extended Abstract). In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 306–315 (2004) A more detailed version appeared in Electronic Colloquium on Computational Complexity Report TR05-099
Valiant, L.G.: Holographic circuits. In: Proc. 32nd International Colloquium on Automata, Languages and Programming, pp. 1–15 (2005)
Valiant, L.G.: Completeness for parity problems. In: Proc. 11th International Computing and Combinatorics Conference (2005)
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© 2006 Springer-Verlag Berlin Heidelberg
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Cai, JY., Choudhary, V. (2006). Valiant’s Holant Theorem and Matchgate Tensors. In: Cai, JY., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2006. Lecture Notes in Computer Science, vol 3959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11750321_24
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DOI: https://doi.org/10.1007/11750321_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34021-8
Online ISBN: 978-3-540-34022-5
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