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<Paper uid="P06-1139">
  <Title>Stochastic Language Generation Using WIDL-expressions and its Application in Machine Translation and Summarization</Title>
  <Section position="4" start_page="1105" end_page="1106" type="intro">
    <SectionTitle>
2 The WIDL Representation Language
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="1105" end_page="1106" type="sub_section">
      <SectionTitle>
2.1 WIDL-expressions
</SectionTitle>
      <Paragraph position="0"> In this section, we introduce WIDL-expressions, a formal language used to compactly represent probability distributions over finite sets of strings.</Paragraph>
      <Paragraph position="1"> Given a finite alphabet of symbols a1 , atomic WIDL-expressions are of the form a2 , with a2a4a3 a1 .</Paragraph>
      <Paragraph position="2"> For a WIDL-expression a5a7a6a8a2 , its semantics is a probability distribution a9a11a10a13a12a15a14a17a16a19a18a20a5a22a21a24a23a26a25a28a27a30a29a32a31a34a33</Paragraph>
      <Paragraph position="4"> other WIDL-expressions, by employing the following four operators, as well as operator distribution functions a51a53a52 from an alphabet a54 .</Paragraph>
      <Paragraph position="5"> Weighted Disjunction. If a5a56a55 a37a53a57a53a57a53a57a58a37 a5a60a59 are WIDL-expressions, then a5a61a6 a62a42a63a19a64a65a18a20a5a42a55 a37a53a57a53a57a53a57a45a37 a5a66a59a30a21 , with a51a17a67 a23a44a43 a38a68a37a53a57a53a57a53a57a58a37 a0 a46 a33 a35a36a30a37a39a38a41a40 , specified such that a1a70a69a45a71a73a72a75a74a77a76a60a78a63a64a75a79a51a17a67a80a18a82a81a47a21a83a6 a38 , is a WIDLexpression. Its semantics is a probability distribution a9a13a10a13a12a15a14a17a16a48a18a20a5a22a21a84a23a85a25a28a27a30a29 a31 a33 a35a36a30a37a39a38a41a40 , where</Paragraph>
      <Paragraph position="7"> ity values are induced by a51a39a67 and a9a49a10a11a12a15a14a41a16a19a18a20a5a66a52a82a21 ,  a25a28a27a30a29a133a31 is the set of all strings that obey the precedence imposed over the arguments, and the probability values are induced by a9a47a10a13a12a15a14a17a16a48a18a20a5a134a55a41a21 and a9a47a10a13a12a15a14a17a16a82a18a20a5a66a129a45a21 . For example, if</Paragraph>
      <Paragraph position="9"> Weighted Interleave. If a5a56a55 a37a53a57a53a57a53a57a58a37 a5a60a59 are WIDLexpressions, then a5a153a6a95a154a41a63a19a64a65a18a20a5a42a55 a37 a5a66a129 a37a53a57a53a57a53a57a58a37 a5a60a59a30a21 , with</Paragraph>
      <Paragraph position="11"> specified such that a1a101a179 a71a73a72a75a74a77a76a66a78a63a19a64 a79a51a17a67a65a18a48a2a49a21a146a6 a38 , is a WIDL-expression. Its semantics is a probability distribution a9a13a10a13a12a15a14a17a16a82a18a20a5a22a21a180a23a85a25a42a27a30a29 a31 a33 a35a36a30a37a39a38a41a40 , where a25a28a27a30a29 a31 consists of all the possible interleavings of strings from a25a28a27a30a29 a31a80a89 , a38a181a90a182a92a183a90 a0 , and the probability values are induced by a51a45a67 and a9a49a10a11a12a15a14a41a16a19a18a20a5a66a52a82a21 . The distribution function a51 a67 is defined either explicitly, over a173a180a174a184a176a49a177a68a178a185a29a112a59 (the set of all permutations of a0 elements), or implicitly, as a51a45a67a80a18a48a27a68a186a77a187a73a177a68a178a189a188a80a177a68a178a185a29a32a190a77a21 . Because the set of argument permutations is a sub-set of all possible interleavings, a51a45a67 also needs to specify the probability mass for the strings that are not argument permutations, a51a45a67a80a18a191a190a53a187a193a192a39a194a195a177a17a190a41a21 . For example, if a5 a6 a154a77a63a64 a18a48a96 a131 a97a167a37 a141a45a21 , a99 a64 a100a95a102a53a103a196a110a24a104 a105a45a107a108a41a105a39a109a42a158a77a159a119a160a162a161a162a163a53a164a17a161a77a163a88a165a167a166a198a197a200a199a20a201a202a203a200a204a205a104 a105a45a107a206a103a75a207a53a109a140a166a140a160a53a168a98a169a56a161a119a166a196a197a20a199a200a201a202a203a200a204a205a104 a105a45a107a105a41a207a17a113 , its semantics is a probability distribution a9a124a10a13a12a15a14a17a16a48a18a20a5a22a21 , with domain a25a28a27a30a29 a31 a6a93a43a45a96 a97 a141 a37 a141a53a96 a97a167a37 a96a65a141 a97 a46 , defined</Paragraph>
      <Paragraph position="13"> a18a20a5a28a220a88a21 is a WIDL-expression. The semantic mapping a9a47a10a13a12a15a14a17a16a48a18a20a5a22a21 is the same as a9a13a10a13a12a15a14a17a16a82a18a20a5 a220a21 , except that a25a28a27a30a29 a31 contains strings in which no additional symbol can be interleaved. For exam-</Paragraph>
      <Paragraph position="15"> a105a45a107a108a41a105a39a109a124a158a162a159a119a160a162a161a77a163a53a164a41a161a77a163a88a165a167a166a223a104a182a105a45a107a110a77a105a17a113 , its semantics is a probability distribution a9a13a10a13a12a15a14a17a16a82a18a20a5a22a21 , with domain a25a28a27a30a29 a31 a6</Paragraph>
      <Paragraph position="17"> In Figure 1, we show a more complex WIDLexpression. The probability distribution a51 a55 associated with the operator a154a41a63a126a137 assigns probability 0.2 to the argument order a225 a38a111a226 ; from a probability mass of 0.7, it assigns uniformly, for each of the remaining a226a189a227a147a228a135a38 a6a230a229 argument permutations, a permutation probability value of a67a41a231a232a233 a6 a36a30a57a88a38a17a234 . The</Paragraph>
      <Paragraph position="19"> remaining probability mass of 0.1 is left for the 12 shuffles associated with the unlocked expression a2a15a7 a131 a2a122a178a24a16a3a25 , for a shuffle probability of a67a41a231a15a55</Paragraph>
      <Paragraph position="21"> a18a191a190a98a186a41a178a17a2a23a7 a4 a21a18a41 pairs that belong to the probability distribution defined by our example: rebels fighting turkish government in iraq 0.130 in iraq attacked rebels turkish goverment 0.049 in turkish goverment iraq rebels fighting 0.005 The following result characterizes an important representation property for WIDL-expressions.</Paragraph>
      <Paragraph position="22"> Theorem 1 A WIDL-expression a5 over a1 and a54 using a0 atomic expressions has space complexity O(a0 ), if the operator distribution functions of a5 have space complexity at most O(a0 ).</Paragraph>
      <Paragraph position="23"> For proofs and more details regarding WIDLexpressions, we refer the interested reader to (Soricut, 2006). Theorem 1 ensures that highcomplexity hypothesis spaces can be represented efficiently by WIDL-expressions (Section 5).</Paragraph>
    </Section>
    <Section position="2" start_page="1106" end_page="1106" type="sub_section">
      <SectionTitle>
2.2 WIDL-graphs and Probabilistic
Finite-State Acceptors
</SectionTitle>
      <Paragraph position="0"> WIDL-graphs. Equivalent at the representation level with WIDL-expressions, WIDL-graphs allow for formulations of algorithms that process them. For each WIDL-expression a5 , there exists an equivalent WIDL-graph a42 a31 . As an example, we illustrate in Figure 2(a) the WIDL-graph corresponding to the WIDL-expression in Figure 1.</Paragraph>
      <Paragraph position="1"> WIDL-graphs have an initial vertex a43a45a44 and a final vertex a43a37a46 . Vertices a43a80a67 , a43a48a47 , and a43a65a129a98a67 with in-going edges labeled a49 a55</Paragraph>
      <Paragraph position="3"> , respectively, result from the expansion of the a154a41a63a126a137 operator. Vertices a43a193a232</Paragraph>
      <Paragraph position="5"> spectively, and vertices a43a49a55a185a129 and a43a189a55a53a55 with out-going edges labeled a21 a55</Paragraph>
      <Paragraph position="7"> , respectively, result from the expansion of the a62a28a63a185a139 operator. With each WIDL-graph a42 a31 , we associate a probability distribution. The domain of this distribution is the finite collection of strings that can be generated from the paths of a WIDL-specific traversal of a42 a31 , starting from a43a56a44 and ending in a43a37a46 . Each path (and its associated string) has a probability value induced by the probability distribution functions associated with the edge labels of a42 a31 . A WIDL-expression a5 and its corresponding WIDL-graph a42 a31 are said to be equivalent because they represent the same distribution a9a11a10a11a12a15a14a41a16a19a18a20a5a22a21 . WIDL-graphs and Probabilistic FSA. Probabilistic finite-state acceptors (pFSA) are a well-known formalism for representing probability distributions (Mohri et al., 2002). For a WIDL-expression a5 , we define a mapping, called UNFOLD, between the WIDL-graph a42 a31 and a pFSA a57 a31 . A state a58 in a57 a31 is created for each set of WIDL-graph vertices that can be reached simultaneously when traversing the graph. State a58 records, in what we call a a154 -stack (interleave stack), the order in which a49 a52  graph), and then reaching vertex a43 a52 (inside the a49 a129</Paragraph>
      <Paragraph position="9"> -bordered sub-graph).</Paragraph>
      <Paragraph position="10"> A transition labeled a2 between two a57 a31 states a58a80a55 and a58a58a129 in a57 a31 exists if there exists a vertex a43a3a61 in the description of a58 a55 and a vertex a43 a69 in the description of a58a73a129 such that there exists a path in a42 a31 between a43a5a61 and a43 a69 , and a2 is the only a1 -labeled transitions in this path. For example, transition  in the description of a58 a55 and vertex a43 a69 in the description of a58 a129 , such  [v v v ,&lt;32][v v v ,&lt;32][v v v ,&lt;3] [v v v ,&lt;3] [v v v ,&lt;32] [v v v ,&lt;0] [v v v ,&lt;32] [v v v ,&lt;2] [v v v ,&lt;2]</Paragraph>
      <Paragraph position="12"> probabilistic finite-state acceptor (pFSA) that corresponds to the WIDL-graph is shown in (b).</Paragraph>
      <Paragraph position="13"> are responsible for adding and removing, respectively, the a38 a63 ,a41a126a63 symbols in the a154 -stack. The probabilities associated with a57 a31 transitions are computed using the vertex set and the a154 -stack of each a57a32a31 state, together with the distribution functions of the a62 and a154 operators. For a detailed presentation of the UNFOLD relation we refer the reader to (Soricut, 2006).</Paragraph>
    </Section>
  </Section>
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