| { | |
| "paper_id": "C02-1001", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T12:18:01.722157Z" | |
| }, | |
| "title": "Disambiguation of Finite-State Transducers", | |
| "authors": [ | |
| { | |
| "first": "N", | |
| "middle": [], | |
| "last": "Smaili", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "[email protected]" | |
| }, | |
| { | |
| "first": "P", | |
| "middle": [], | |
| "last": "Cardinal", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "[email protected]" | |
| }, | |
| { | |
| "first": "G", | |
| "middle": [], | |
| "last": "Boulianne", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "[email protected]" | |
| }, | |
| { | |
| "first": "P", | |
| "middle": [], | |
| "last": "Dumouchel", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "[email protected]" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "The objective of this work is to disambiguate transducers which have the following form: T = R \u2022 D and to be able to apply the determinization algorithm described in (Mohri, 1997). Our approach to disambiguating T = R \u2022 D consists first of computing the composition T and thereafter to disambiguate the transducer T. We will give an important consequence of this result that allows us to compose any number of transducers R with the transducer D, in contrast to the previous approach which consisted in first disambiguating transducers D and R to produce respectively D and R , then computing T = R \u2022 D where T is unambiguous. We will present results in the case of a transducer D representing a dictionary and R representing phonological rules.", | |
| "pdf_parse": { | |
| "paper_id": "C02-1001", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "The objective of this work is to disambiguate transducers which have the following form: T = R \u2022 D and to be able to apply the determinization algorithm described in (Mohri, 1997). Our approach to disambiguating T = R \u2022 D consists first of computing the composition T and thereafter to disambiguate the transducer T. We will give an important consequence of this result that allows us to compose any number of transducers R with the transducer D, in contrast to the previous approach which consisted in first disambiguating transducers D and R to produce respectively D and R , then computing T = R \u2022 D where T is unambiguous. We will present results in the case of a transducer D representing a dictionary and R representing phonological rules.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "The task of speech recognition can be decomposed into several steps, where each step is represented by a finitestate transducer (Mohri et al., 1998) . The search space of the recognizer is defined by the composition of transducers T = A \u2022 C \u2022 R \u2022 D \u2022 M . Transducer A converts a sequence of observations O to a sequence of context-dependent phones.", | |
| "cite_spans": [ | |
| { | |
| "start": 128, | |
| "end": 148, | |
| "text": "(Mohri et al., 1998)", | |
| "ref_id": "BIBREF5" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "Transducer C converts a sequence of context-dependent phones to a sequence of context-independent phones. Transducer R is a mapping from phones to phones which implements phonological rules. Transducer D is the pronunciations dictionary. It converts a sequence of contextindependent phones to a sequence of words. Transducer M represents a language model: it converts sequences of words into sequences of words, while restricting the possible sequences or assigning a score to the sequences. The speech recognition problem consists of finding the path of least cost in transducer O \u2022 T , where O is a sequence of acoustic observations.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "The pronunciations dictionary representing the mapping from pronunciations to words can show an inherent ambiguity: a sequence of phones can correspond to more than one word, so we cannot apply the transducer determinization algorithm (an operation which reduces the redundancy, search time and possibly space). This problem is usually handled by adding special symbols to the dictionary to remove the ambiguity in order to be able to apply the determinization algorithm (Koskenniemi, 1990) . Nevertheless, when we compose the dictionary with the phonological rules, we must take into account special symbols. This complicates the construction of transducers representing these rules and leads to size explosion. It would be simpler to compose the rules with the dictionary, then remove the ambiguity in the result and then apply the determinization algorithm.", | |
| "cite_spans": [ | |
| { | |
| "start": 471, | |
| "end": 490, | |
| "text": "(Koskenniemi, 1990)", | |
| "ref_id": "BIBREF4" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "Formally, a weighted transducer over a semiring K = (K, \u2295, \u2297,0,1) is defined as a 6-tuple", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "T = (Q, I, \u03a3 1 , \u03a3 2 , E, F )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "where Q is a finite set of states, I \u2286 Q is a finite set of initial states, \u03a3 1 is the input alphabet, \u03a3 2 is the output alphabet, E is a finite set of transitions and F \u2286 Q is a finite set of final states. A transition is an element of", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Q \u00d7 \u03a3 1 \u00d7 \u03a3 2 \u00d7 Q \u00d7 K.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Transitions are of the form", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "t = (p(t), i(t), o(t), n(t), w(t)), t \u2208 E", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "where p(t) denotes the transition's origin state, i(t) its input label, o(t) its output label, n(t) the transition's destination state and w(t) \u2208 K is the weight of t. The tropical semiring defined as (R + \u222a \u221e, min, +, \u221e, 0) is commonly used in speech recognition, but our results are applicable to the case of general semirings as well.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "A path \u03c0 = t 1 \u2022 \u2022 \u2022 t n of T is an ele- ment of E * verifying n(t i\u22121 ) = p(t i ) for 2 \u2264 i \u2264 n.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "We can easily extend the functions p and n to those paths:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "p(\u03c0) = p(t 1 ),", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "(1) n(\u03c0) = n(t n ).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "(2)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "We denote by P (r, s) the set of paths whose origin is state r and whose destination is state s. We can also extend the function P to the sets R \u2282 Q and S \u2282 Q:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "P (R, S) = r\u2208R, s\u2208S P (r, s)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "We can extend the functions i and o to the paths by taking the concatenations of the input and output symbols:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "i(\u03c0) = i(t 1 ) \u2022 \u2022 \u2022 i(t n ), (3) o(\u03c0) = o(t 1 ) \u2022 \u2022 \u2022 o(t n ).", | |
| "eq_num": "(4)" | |
| } | |
| ], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Definition 1 (unambiguous transducer, (Berstel, 1979) ) A transducer T is said to be unambiguous if for each w \u2208 \u03a3 * 1 , there exists at most one path \u03c0 in T such that i(\u03c0) = w.", | |
| "cite_spans": [ | |
| { | |
| "start": 38, | |
| "end": 53, | |
| "text": "(Berstel, 1979)", | |
| "ref_id": "BIBREF0" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Definition 2 (ambiguous paths) Two paths \u03c0 and \u03b1 are ambiguous if \u03c0 = \u03b1 and i(\u03c0) = i(\u03b1).", | |
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| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Remark 1 : To remove the ambiguity between two paths \u03c0 and \u03b1, it suffices to modify i(\u03c0) by changing the first input label of the path \u03c0. This is done by introducing an auxiliary symbol such that: i(\u03c0) = i(\u03b1). Figure 1a shows an ambiguous transducer.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 210, | |
| "end": 219, | |
| "text": "Figure 1a", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "It is ambiguous since for the input string \"s e [z]\", there are two paths representing the output strings {ces, ses}. In this figure, \"eps\" stands for epsilon or null symbol.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "To disambiguate a transducer, we first group the ambiguous paths; we then remove the ambiguity in each group by adding auxiliary labels as shown in Figure 1b . Unfortunately, it is infeasible to enumerate all the paths in a cyclic transducer. However, in (Smaili, 2001) it is shown that cyclic transducers of the type studied in this work can be disambiguated by transforming to a corresponding acyclic subtransducer such that T \u2282 T . This fundamental property is described in detail in section 2.1. Accordingly, we apply the appropriate transformation to the input transducer.", | |
| "cite_spans": [ | |
| { | |
| "start": 255, | |
| "end": 269, | |
| "text": "(Smaili, 2001)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 148, | |
| "end": 157, | |
| "text": "Figure 1b", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Notations and definitions", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "We are interested in the transducer T = (Q, I, \u03a3, \u2126, E, F ) with \u03a3 = \u03a3 0 \u03a3 1 verifying the following property:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Any cycle in T contains at least a transition t such that i(t) \u2208 \u03a3 1 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "We denote by E 0 and E 1 the following sets:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "E 0 = {t \u2208 E : i(t) \u2208 \u03a3 0 } and E 1 = {t \u2208 E : i(t) \u2208 \u03a3 1 }. Notice that E = E 0 E 1 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "We can give a characterization of the ambiguous paths verifying the fundamental property. Before, let's make the following remark:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Remark 2 Any path \u03c0 in T has the following form:", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "\u03c0 = f 0 \u03c0 0 f 1 \u03c0 1 \u2022 \u2022 \u2022 \u03c0 n\u22121 f n \u03c0 n with \u03c0 i \u2208 E + 0 , f i \u2208 E + 1 for 1 \u2264 i \u2264 n, f 0 \u2208 E * 1 and \u03c0 0 \u2208 E * 0 if n \u2265 1. If n = 0 then \u03c0 = f 0 \u03c0 0 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Proposition 1 (characterization of ambiguous paths) Let \u03c0 and \u03b1 be two paths such that:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "\u03c0 = f 0 \u03c0 0 f 1 \u03c0 1 \u2022 \u2022 \u2022 \u03c0 n\u22121 f n \u03c0 n and \u03b1 = g 0 \u03b1 0 g 1 \u03b1 1 \u2022 \u2022 \u2022 \u03b1 k\u22121 g k \u03b1 k .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "\u03c0 and \u03b1 are ambiguous if and only if \uf8f1 \uf8f2 \uf8f3 k = n \u03b1 i and \u03c0 i are ambiguous (0 \u2264 i \u2264 n). f i and g i are ambiguous (0 \u2264 i \u2264 n).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "We will assume that the first transition's path belongs to E 0 , i.e. f 0 = .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Recall that if we want to avoid cycles, we just have to remove from T all transitions t \u2208 E 1 . According to Proposition 1, ambiguity needs to be removed only in paths that use transitions t \u2208 E 0 , namely the path \u03c0 i that performs the decomposition given in Remark 2. Disambiguation consists only of introducing auxiliary labels in the ambiguous paths. We denote by A src the set of origin states of transitions belonging to E 1 and by A dst the set of destination states of transitions belonging to E 2 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "A src = {p(t) : t \u2208 E 1 } A dst = {n(t) : t \u2208 E 1 }", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "According to Proposition 1 and what precedes, it would be equivalent and simpler to disambiguate an acyclic transducer obtained from T in which we have removed all E 1 transitions. Therefore, we introduce the operator \u03a8 : {T in } \u2212\u2192 {T out } which accomplishes this construction.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Let T = (Q, I, \u03a3 1 , \u03a3 2 , E, F ). Then \u03a8(T ) = (Q, I 1 , \u03a3 1 , \u03a3 2 , E T , F 1 ) where: 1. I 1 = I \u222a A dst \u222a {i}, with i \u2208 Q. 2. F 1 = F \u222a A src \u222a {f }, with f \u2208 Q. 3. E T = E \\ E 1 \u222a {(i, q, , , 0), q \u2208 I 1 } \u222a {(q, f, , , 0), q \u2208 F 1 }.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "The third condition insures the connectivity of \u03a8(T ) if T is itself connected. It suffices to disambiguate the acyclic transducer \u03a8(T ), then reinsert the transitions of E 1 in \u03a8(T ). The set of paths in \u03a8(T ) is then P(I 1 , F 1 ).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Fundamental Property", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Input:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "T = (Q, i, X, Y, E, F )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "is an ambiguous transducer verifying the fundamental property.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "Output:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "T 1 = (Q, i, X \u222a X 1 , Y, E T , F )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "is an unambiguous transducer, X 1 is the set of auxiliary symbols.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "1. T acyclic \u2190 \u03a8(T ).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "2. Path \u2190 set of paths of T acyclic .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "3. Disambiguate the set Path (creating the set X 1 ).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "4. T 0 \u2190 build the unambiguous transducer which has unambiguous paths.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "5. T 1 \u2190 \u03a8 \u22121 (T 0 ) (consists of reinserting in T 0 the transitions of T which where removed).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "Now, we will study an important class of transducers verifying the fundamental property. This class is obtained by doing the composition of a transducer D verifying the fundamental property with a transducer R. The composition of two transducers is an efficient algebraic operation for building more complex transducers. We give a brief definition of composition and the fundamental theorem that insures the invariance of the fundamental property by composition.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "return T 1", | |
| "sec_num": "6." | |
| }, | |
| { | |
| "text": "The transducer T created by the composition of two transducers R and D, denoted T = R \u2022 D, performs the mapping of word x to word z if and only if R maps x to y and D maps y to z. The weight of the resulting word is the \u2297-product of the weights of y and z (Pereira and Riley, 1997) .", | |
| "cite_spans": [ | |
| { | |
| "start": 256, | |
| "end": 281, | |
| "text": "(Pereira and Riley, 1997)", | |
| "ref_id": "BIBREF7" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Composition", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "Definition 3 (Transitions) Let t = (q, a, b, q 1 , w 1 ) and e = (r, b, c, r 1 , w 2 ) be two transitions. We define the composition t with e by:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Composition", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "t \u2022 e = ((q, r), a, c, (q 1 , r 1 ), w 1 \u2297 w 2 ).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Composition", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "Note that, in order to make the composition possible, we must have o(t) = i(e).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Composition", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": ") Let R = (Q R , I R , X, Y, E R , F R ) and S = (Q S , I S , Y, Z, E S , F S ) be two transducers.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The composition of R with S is a transducer R \u2022 S = (Q, Q, X, Z, E, F ) defined by:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "1. i = (i R , i S ), 2. Q = Q R \u00d7 Q S , 3. F = F R \u00d7 F S , 4. E = {e R \u2022e S : e R \u2208 E R , e S \u2208 E S }. Let D = (Q D , I D , Y, Z, E D , F D )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "be a transducer verifying the fundamental property. We can write", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Y = Y 0 Y 1 where Y 0 = {i(t) : t \u2208 E 0 } and Y 1 = {i(t) : t \u2208 E 1 }. Theorem 1 (Fundamental) Let R = (Q R , I R , X, Y, E R , F R ) verifying the following condition: (C) \u2200t \u2208 E R , o(t) \u2208 Y 1 \u21d2 i(t) \u2208 Y 1 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Then the transducer T = R \u2022 D verifies the fundamental property.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Proof : Let X 1 = {i(t) : t \u2208 E R and o(t) \u2208 Y 1 } \u2282 Y 1 and X 0 = X \\ X 1 . We will prove that any path in T contains at least a transition t such that i(t) \u2208 X 1 . Let \u03c0 be a cycle in T . Then, there exists two cycles \u03c0 R and \u03c0 D in R and in D respectively such that \u03c0 = \u03c0 R \u2022 \u03c0 D . The paths \u03c0 R and \u03c0 D have the following form:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u03c0 D = g 1 \u2022 \u2022 \u2022 g n , with g i \u2208 E D for 1 \u2264 i \u2264 n; \u03c0 R = f 1 \u2022 \u2022 \u2022 f n , with f i \u2208 E R for 1 \u2264 i \u2264 n; \u03c0 = \u03c0 R \u2022 \u03c0 D = (f 1 \u2022 g 1 ) \u2022 \u2022 \u2022 (f n \u2022 g n ).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "There is an index k such that i(g k ) \u2208 Y 1 since D verifies the fundamental property. We also necessarily have i(g k ) = o(f k ) . According to condition (C) of Theorem 1, we deduce that", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "i(f k ) \u2208 Y 1 . Knowing that f k \u2208 E R , we deduce that i(f k ) \u2208 X 1 , which implies i(f k \u2022 g k ) = i(f k ) \u2208 X 1 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 4 (Composition", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The restriction to the case X = Y allows us to build a large class of transducers verifying the fundamental property. In fact, if two transducers R = (Q R , I R , Y, Y, E R , F R ) and S = (Q S , I S , Y, Y, E S , F S ) verify the condition (C) of Theorem 1, then S \u2022 R verifies the condition (C), associativity of \u2022 implies:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "S \u2022 (R \u2022 D) = (S \u2022 R) \u2022 D.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "Suppose that we have m transducers R i ( 1 \u2264 i \u2264 m ) verifying the condition (C) of Theorem 1 and that we want to reduce the size of the transducer:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "T m = R m \u2022 R m\u22121 \u2022 \u2022 \u2022 R 1 \u2022 D.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "To this end, we proceed as follows: we add the auxiliary symbols to disambiguate the transducer; then we apply determinization and finally we remove the auxiliary labels. These three operations are denoted by \u03c8.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "T i = \u03c8(D) if i = 0. \u03c8(R i \u2022 \u03c8(T i\u22121 )) if i \u2265 1.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "The size of transducer T m can also be reduced by computing:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "T m = \u03c8(R m \u2022 R m\u22121 \u2022 \u2022 \u2022 R 1 \u2022 D).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "The old approach:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "T m = R m \u2022 R m\u22121 \u2022 \u2022 \u2022 R 1 \u2022 D .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "has several disadvantages. The size of R i for 1 \u2264 i \u2264 m increases considerably since the auxiliary labels introduced in each transducer have to be taken into account in all others. This fact limits the number of transducers that can be composed with D.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Consequence", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "We will now apply our algorithm to transducers involved in speech recognition. Transducer D represents the pronunciation dictionary and possesses the fundamental property. The set of transitions of D is defined as", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Application and Results", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "E = E 0 {(f, #, x, 0, w)}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Application and Results", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "where f is the unique final state of D, 0 is the unique initial state of D, x is any symbol and # is a symbol representing the end of a word. All transitions t \u2208 E 0 are such that i(t) = #. Any path \u03c0 in E * 0 is acyclic. The transducer R representing a phonological rule is constructed to fulfill condition (C) of the fundamental theorem. The transducer D represents a French dictionary with 20000 words and their pronunciations. The transducer R represents the phonological rule that handles liaison in the French language. This liaison, which is represented by a phoneme appearing at the end of some words, must be removed when the next word begins with a consonant since the liaison phoneme is never pronounced in that case. However, if the next word begins with a vowel, the liaison phoneme may or may not be pronounced and thus becomes optional. Table 1 shows the results of our algorithm using the dictionary and the phonological rule previously described. As we can see in Table 1 , the operator \u03c8 produces a smaller transducer in all the cases considered here.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 852, | |
| "end": 859, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| }, | |
| { | |
| "start": 981, | |
| "end": 988, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Application and Results", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "We have been able to disambiguate an important class of cyclic and ambiguous transducers, which allows us to apply the determinization algorithm (Mohri, 1997) ; and then to reduce the size of those transducers. With our new approach, we do not have to take into account the number of transducers R i and their auxiliary labels as was the case with the approach used before. Thus, new transducers R i such as phonological rules can be easily inserted in the chain. The major disadvantage of our approach is that disambiguating a transducer increases its size systematically. Our future work will consist of developing a more effective algorithm for disambiguating an acyclic transducer.", | |
| "cite_spans": [ | |
| { | |
| "start": 145, | |
| "end": 158, | |
| "text": "(Mohri, 1997)", | |
| "ref_id": "BIBREF6" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusion and future work", | |
| "sec_num": "5" | |
| } | |
| ], | |
| "back_matter": [], | |
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| "ref_entries": { | |
| "FIGREF1": { | |
| "text": "(a) Ambiguous transducer (b) Disambiguated transducer", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF2": { | |
| "text": "Transducer used to handle the optional liaison rule.", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF3": { | |
| "text": "shows the transducer that handles this rule. In the figure, p denotes all phonemes, v the vowels and [x] the liaison phonemes.", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "TABREF0": { | |
| "text": "Size reduction on a French dictionary", | |
| "content": "<table/>", | |
| "html": null, | |
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