Algorithms for Acyclic Weighted Finite-State Automata with Failure Arcs
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Weighted finite-state automata (WSFAs) are commonly used in NLP. Failure transitions are a useful extension for compactly representing backoffs or interpolation in n-gram models and CRFs, which are special cases of WFSAs. The pathsum in ordinary acyclic WFSAs is efficiently computed by the backward algorithm in time O(|E|), where E is the set of transitions. However, this does not allow failure transitions, and preprocessing the WFSA to eliminate failure transitions could greatly increase |E|. We extend the backward algorithm to handle failure transitions directly. Our approach is efficient when the average state has outgoing arcs for only a small fraction s ≪ 1 of the alphabet Σ. We propose an algorithm for general acyclic WFSAs which runs in O(|E| + s |Σ| |Q| T_maxlog|Σ|), where Q is the set of states and T_max is the size of the largest connected component of failure transitions. When the failure transition topology satisfies a condition exemplified by CRFs, the T_max factor can be dropped, and when the weight semiring is a ring, the log|Σ| factor can be dropped. In the latter case (ring-weighted acyclic WFSAs), we also give an alternative algorithm with complexity O(|E| + |Σ| |Q| min(1,sπ_max) ), where π_max is the size of the longest failure path.
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