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In this paper, we propose an algebraic formalization of the two importantclasses of dynamic programming algorithms called forward and forward-backwardalgorithms. They are generalized extensively in this study so that a wide rangeof other existing algorithms is subsumed. Forward algorithms generalized inthis study subsume the ordinary forward algorithm on trellises for sequencelabeling, the inside algorithm on derivation forests for CYK parsing, aunidirectional message passing on acyclic factor graphs, the forward mode ofautomatic differentiation on computation graphs with addition andmultiplication, and so on. In addition, we reveal algebraic structuresunderlying complicated computation with forward algorithms. By the aid of therevealed algebraic structures, we also propose a systematic framework to designcomplicated variants of forward algorithms. Forward-backward algorithmsgeneralized in this study subsume the ordinary forward-backward algorithm ontrellises for sequence labeling, the inside-outside algorithm on derivationforests for CYK parsing, the sum-product algorithm on acyclic factor graphs,the reverse mode of automatic differentiation (a.k.a. back propagation) oncomputation graphs with addition and multiplication, and so on. We also proposean algebraic characterization of what can be computed by forward-backwardalgorithms and elucidate the relationship between forward and forward-backwardalgorithms.

An Algebraic Formalization of Forward and Forward-backward Algorithms