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<Paper uid="W04-2907">
  <Title>tioning: Low latency real-time broadcast news tran-</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Preliminaries
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    <Paragraph position="0"> Definition 1 A system (K,[?],[?],0,1) is a semiring (Kuich and Salomaa, 1986) if: (K,[?],0) is a commutative monoid with identity element 0; (K,[?],1) is a monoid with identity element 1; [?] distributes over [?]; and 0 is an annihilator for [?]: for all a [?]K,a[?]0 = 0[?]a = 0.</Paragraph>
    <Paragraph position="1"> Thus, a semiring is a ring that may lack negation. Two semirings often used in speech processing are: the log semiring L = (R[?] {[?]},[?]log,+,[?],0) (Mohri, 2002) which is isomorphic to the familiar real or probability semiring (R+,+,x,0,1) via a log morphism with, for all a,b [?] R[?]{[?]}: a[?]log b = [?]log(exp([?]a) + exp([?]b)) and the convention that: exp([?][?]) = 0 and [?]log(0) = [?], and the tropical semiring T = (R+ [?] {[?]},min,+,[?],0) which can be derived from the log semiring using the Viterbi approximation.</Paragraph>
    <Paragraph position="2"> Definition 2 A weighted finite-state transducer T over a semiring K is an 8-tuple T = (S,[?],Q,I,F,E,l,r) where: S is the finite input alphabet of the transducer; [?] is the finite output alphabet; Q is a finite set of states; I [?] Q the set of initial states; F [?] Q the set of final states; E [?] Qx(S[?]{epsilon1})x([?][?]{epsilon1})xKxQ a finite set of transitions; l : I - K the initial weight function; and r : F - K the final weight function mapping F to K.</Paragraph>
    <Paragraph position="3"> A Weighted automaton A = (S,Q,I,F,E,l,r) is defined in a similar way by simply omitting the output labels. We denote by L(A) [?] S[?] the set of strings accepted by an automaton A and similarly by L(X) the strings described by a regular expression X. We denote by |A |= |Q|+|E |the size of A.</Paragraph>
    <Paragraph position="4"> Given a transition e [?] E, we denote by i[e] its input label, p[e] its origin or previous state and n[e] its destination state or next state, w[e] its weight, o[e] its output label (transducer case). Given a state q [?] Q, we denote by E[q] the set of transitions leaving q.</Paragraph>
    <Paragraph position="5"> A path pi = e1 ***ek is an element of E[?] with consecutive transitions: n[ei[?]1] = p[ei], i = 2,...,k. We extend n and p to paths by setting: n[pi] = n[ek] and p[pi] = p[e1]. A cycle pi is a path whose origin and destination states coincide: n[pi] = p[pi]. We denote by P(q,qprime) the set of paths from q to qprime and by P(q,x,qprime) and P(q,x,y,qprime) the set of paths from q to qprime with input label x [?] S[?] and output label y (transducer case). These definitions can be extended to subsets R,Rprime [?] Q, by: P(R,x,Rprime) = [?]q[?]R,qprime[?]RprimeP(q,x,qprime). The labeling functions i (and similarly o) and the weight function w can also be extended to paths by defining the label of a path as the concatenation of the labels of its constituent transitions, and the weight of a path as the [?]-product of the weights of its constituent transitions:</Paragraph>
    <Paragraph position="7"> also extend w to any finite set of paths P by setting: w[P] = circleplustextpi[?]P w[pi]. The output weight associated by A to each input string x [?] S[?] is:</Paragraph>
    <Paragraph position="9"> [[A]](x) is defined to be 0 when P(I,x,F) = [?]. Similarly, the output weight associated by a transducer T to a pair of input-output string (x,y) is:</Paragraph>
    <Paragraph position="11"> path in a weighted automaton or transducer M is a path from an initial state to a final state. M is unambiguous if for any string x [?] S[?] there is at most one successful path labeled with x. Thus, an unambiguous transducer defines a function.</Paragraph>
    <Paragraph position="12"> For any transducer T, denote by P2(T) the automaton obtained by projecting T on its output, that is by omitting its input labels.</Paragraph>
    <Paragraph position="13"> Note that the second operation of the tropical semiring and the log semiring as well as their identity elements are identical. Thus the weight of a path in an automaton A over the tropical semiring does not change if A is viewed as a weighted automaton over the log semiring or viceversa. null Given two strings u and v in S[?], v is a factor of u if u = xvy for some x and y in S[?]; if y = epsilon1 then v is also a suffix of u. More generally, v is a factor (resp. suffix) of L [?] S[?] if v is a suffix (resp. factor) of some u [?] L. We denote by |x |the length of a string x [?] S[?].</Paragraph>
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