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Abstract: This paper stands at the intersection of two distinct lines of research. Oneline is -holographic algorithms,- a powerful approach introduced by Valiant forsolving various counting problems in computer science; the other is -normalfactor graphs,- an elegant framework proposed by Forney for representing codesdefined on graphs. We introduce the notion of holographic transformations fornormal factor graphs, and establish a very general theorem, called thegeneralized Holant theorem, which relates a normal factor graph to itsholographic transformation. We show that the generalized Holant theorem on theone hand underlies the principle of holographic algorithms, and on the otherhand reduces to a general duality theorem for normal factor graphs, a specialcase of which was first proved by Forney. In the course of our development, weformalize a new semantics for normal factor graphs, which highlights variouslinear algebraic properties that potentially enable the use of normal factorgraphs as a linear algebraic tool.



Author: Ali Al-Bashabsheh, Yongyi Mao

Source: https://arxiv.org/







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