## On the Determinization of Weighted Finite Automata (1998)

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Venue: | SIAM J. Comput |

Citations: | 17 - 0 self |

### BibTeX

@ARTICLE{Buchsbaum98onthe,

author = {Adam L. Buchsbaum and Raffaele Giancarlo and Jeffery R. Westbrook},

title = {On the Determinization of Weighted Finite Automata},

journal = {SIAM J. Comput},

year = {1998},

volume = {30},

pages = {2000}

}

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### Abstract

. We study determinization of weighted finite-state automata (WFAs), which has important applications in automatic speech recognition (ASR). We provide the first polynomial-time algorithm to test for the twins property, which determines if a WFA admits a deterministic equivalent. We also provide a rigorous analysis of a determinization algorithm of Mohri, with tight bounds for acyclic WFAs. Given that WFAs can expand exponentially when determinized, we explore why those used in ASR tend to shrink. The folklore explanation is that ASR WFAs have an acyclic, multi-partite structure. We show, however, that there exist such WFAs that always incur exponential expansion when determinized. We then introduce a class of WFAs, also with this structure, whose expansion depends on the weights: some weightings cause them to shrink, while others, including random weightings, cause them to expand exponentially. We provide experimental evidence that ASR WFAs exhibit this weight dependence. ...

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Citation Context ...e RAM model of computation as follows. In the general case, we charge constant time for each arithmetic-logic operation involving weights (which are real numbers). We refer to this model as the ℜ-RAM =-=[21]-=-. The relevant parameters for our analyses are n, m, and |Σ|. In the integer case, we also use a RAM, except that each arithmetic-logic operation now takes O(ρ) time. We refer to this model as the CO-... |

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Citation Context ...e relevant parameters for our analyses are n, m, and |Σ|. In the integer case, we also use a RAM, except that each arithmetic-logic operation now takes O(ρ) time. We refer to this model as the CO-RAM =-=[1]-=-. The relevant parameters for the analyses are n, m, |Σ|, and ρ. 3 Determinization of WFAs 3.1 An Algorithm for Testing the Twins Property Definition 1. Two states, q and q ′ , of a WFA G are twins if... |

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Citation Context ...uniformly at random from [1, 2 k − 1]Z. Then E[|dta(G)|] = Θ(2 k ). The proofs of Theorems 8 and 9 use the observation that the random functions defined by RG are essentially universal hash functions =-=[6]-=- to bound sufficiently low the probability that the remainders of two distinct strings are equal. Theorem 9 is motivated by the fact that the weights of ASR WFAs are negated log probabilities. 5.5 Ext... |

318 | Finite-State Transducers in Language and Speech Processing
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Citation Context ...heir relation to rational functions and power series have been extensively studied [2, 3, 12, 16] and widely applied in fields ranging from image compression [9–11, 14] to natural language processing =-=[17, 18, 24, 26]-=-. A subclass of finitestate machines, the weighted finite-state automata (WFAs), has recently assumed new importance, because WFAs provide a powerful method for manipulating models of human language i... |

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Citation Context ...fore, is a result of favorable weightings in addition to special topology. 1 Introduction Finite-state machines and their relation to rational functions and power series have been extensively studied =-=[2, 3, 12, 16]-=- and widely applied in fields ranging from image compression [9–11, 14] to natural language processing [17, 18, 24, 26]. A subclass of finitestate machines, the weighted finite-state automata (WFAs), ... |

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Citation Context ...ngs to K: the value of an accepted string is the semiring sum over accepting paths of the semiring product of the weights along each accepting path. Such a partial function is a rational power series =-=[25]-=-. An important example in ASR is the set of WFAs with the min-sum semiring, (ℜ + ∪ {0, ∞}, min, +, ∞, 0), which compute for each accepted string the minimum cost accepting path.2 In this paper, we st... |

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Citation Context ... transition of G can be seen as the “confidence” one has in taking that transition. The weights need not, however, satisfy stochastic constraints, as do the probabilistic automata introduced by Rabin =-=[22]-=-. Fix two states q and q ′ and a string v ∈ Σ∗ . Then c(q, v, q ′) is the minimum of c(t), taken over all transition sequences from q to q ′ generating v. We refer to c(q, v, q ′) as the optimal cost ... |

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Citation Context ...fore, is a result of favorable weightings in addition to special topology. 1 Introduction Finite-state machines and their relation to rational functions and power series have been extensively studied =-=[2, 3, 12, 16]-=- and widely applied in fields ranging from image compression [9–11, 14] to natural language processing [17, 18, 24, 26]. A subclass of finitestate machines, the weighted finite-state automata (WFAs), ... |

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Citation Context ...ghted finite-state automata (WFAs), has recently assumed new importance, because WFAs provide a powerful method for manipulating models of human language in automatic speech recognition (ASR) systems =-=[19, 20]-=-. This new research direction also raises a number of challenging algorithmic questions [5]. A weighted finite-state automaton (WFA) is a nondeterministic finite automaton (NFA), A, that has both an a... |

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Citation Context ...ghted finite-state automata (WFAs), has recently assumed new importance, because WFAs provide a powerful method for manipulating models of human language in automatic speech recognition (ASR) systems =-=[19, 20]-=-. This new research direction also raises a number of challenging algorithmic questions [5]. A weighted finite-state automaton (WFA) is a nondeterministic finite automaton (NFA), A, that has both an a... |

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Citation Context ...he importance of determinization to ASR is well established [17, 19, 20]. As far as we know, Mohri [17] presented the first determinization procedure for WFAs, extending the seminal ideas of Choffrut =-=[7, 8]-=- and Weber and Klemm [27] regarding string-to-string transducers. Mohri gives a determinization procedure with three phases. First, A is converted to an equivalent unambiguous, trim WFA At, using an a... |

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Citation Context ...ation to ASR is well established [17, 19, 20]. As far as we know, Mohri [17] presented the first determinization procedure for WFAs, extending the seminal ideas of Choffrut [7, 8] and Weber and Klemm =-=[27]-=- regarding string-to-string transducers. Mohri gives a determinization procedure with three phases. First, A is converted to an equivalent unambiguous, trim WFA At, using an algorithm analogous to one... |

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Citation Context ...he importance of determinization to ASR is well established [17, 19, 20]. As far as we know, Mohri [17] presented the first determinization procedure for WFAs, extending the seminal ideas of Choffrut =-=[7, 8]-=- and Weber and Klemm [27] regarding string-to-string transducers. Mohri gives a determinization procedure with three phases. First, A is converted to an equivalent unambiguous, trim WFA At, using an a... |

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Citation Context ...heir relation to rational functions and power series have been extensively studied [2, 3, 12, 16] and widely applied in fields ranging from image compression [9–11, 14] to natural language processing =-=[17, 18, 24, 26]-=-. A subclass of finitestate machines, the weighted finite-state automata (WFAs), has recently assumed new importance, because WFAs provide a powerful method for manipulating models of human language i... |

13 |
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Citation Context ...w1s2) − c(w2s2), a contradiction. 6 Experimental Observations on ASR WFAs To determine whether ASR WFAs manifest weight dependence, we experimented on 100 WFAs generated by the AT&T speech recognizer =-=[23]-=-, using a grammar for the Air Travel Information System (ATIS), a standard test bed [4]. Each transition was labeled with a word and weighted by the recognizer with the negated log probability of real... |

13 |
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Citation Context ...to a WFA with all arcs weighted identically. Since acyclic WFAs satisfy the twins property, they can always be determinized. Altering the weights can only increase the expansion. Kintala and Wotschke =-=[15]-=- provide a set of NFAs that produces a hierarchy of expansion factors when determinized, providing additional examples of hot WFAs. 5 Weight-Dependent Automata In this section we study a simple family... |

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Citation Context ...he determinized equivalent. This family shrinks without weights, so any expansion is due to weighting. This study is related in spirit to previous works on measuring nondeterminism in finite automata =-=[13,15]-=-. Here, however, nondeterminism is encoded only in the weights. We first discuss the case of a binary alphabet and then generalize to arbitrary alphabets. 5.1 The Rail Graph We denote by RG(k) the k-l... |

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Citation Context ...heir relation to rational functions and power series have been extensively studied [2, 3, 12, 16] and widely applied in fields ranging from image compression [9–11, 14] to natural language processing =-=[17, 18, 24, 26]-=-. A subclass of finitestate machines, the weighted finite-state automata (WFAs), has recently assumed new importance, because WFAs provide a powerful method for manipulating models of human language i... |

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Citation Context ...powerful method for manipulating models of human language in automatic speech recognition (ASR) systems [19, 20]. This new research direction also raises a number of challenging algorithmic questions =-=[5]-=-. A weighted finite-state automaton (WFA) is a nondeterministic finite automaton (NFA), A, that has both an alphabet symbol and a weight, from some set K, on each transition. Let R = (K, ⊕, ⊗, 0, 1) b... |

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