... the Association for Computational Linguistics:shortpapers, pages 1–5,Portland, Oregon, June 19-24, 2011.c2011 Association for Computational LinguisticsLexicographic Semirings for Exact Automata ... “otherwise” formulationof smoothing (Allauzen et al., 2003). For example,the typical backoff formulation of the probability ofa word w given a history h is as followsP(w | h) =P(w | h) if c(hw) > ... henceforth the T, T -lexicographicsemiring. For this semiring the operations ⊕ and ⊗are defined as follows (Golan, 1999, pp. 223–224 ): w1, w2 ⊕ w3, w4 =if w1<...