<?xml version="1.0" standalone="yes"?>
<Paper uid="P99-1070">
  <Title>Relating Probabilistic Grammars and Automata</Title>
  <Section position="3" start_page="543" end_page="543" type="intro">
    <SectionTitle>
2 Probabilistic and Weighted
Grammars
</SectionTitle>
    <Paragraph position="0"> For the remainder of the paper, we fix a terminal alphabet E and a nonterminal alphabet N, to which we may add auxiliary symbols as needed.</Paragraph>
    <Paragraph position="1"> A weighted context-free grammar (WCFG) consists of a distinguished start symbol S E N plus a finite set of weighted productions of the form X -~ a, (alternately, u : X --~ a), where X E N, a E (Nt2E)* and the weight u is a non-negative real number. A probabilistic context-free grammar (PCFG) is a WCFG such that for all X, )-~u:x-~a u = 1. Since weights are nonnegative, this also implies that u &lt;_ 1 for any individual production.</Paragraph>
    <Paragraph position="2"> A PCFG defines a stochastic process with sentential forms as states, and leftmost rewriting steps as transitions. In the more general case of WCFGs, we can no longer speak of stochastic processes; but weighted parse trees and sets of weighted parse trees are still well-defined notions.</Paragraph>
    <Paragraph position="3"> We define a parse tree to be a tree whose nodes are labeled with productions. Suppose node ~ is labeled X -~ a\[Y1,...,Yn\], where we write a\[Y1,...,Yn\] for a string whose nonterminal symbols are Y1,...,Y~. We say that ~'s nonterminal label is X and its weight is u. The subtree rooted at ~ is said to be rooted in X. ~ is well-labeled just in case it has n children, whose nonterminal labels are Y1,..., Yn, respectively.</Paragraph>
    <Paragraph position="4"> Note that a terminal node is well-labeled only if a is empty or consists exclusively of terminal symbols. We say a WCFG G admits a tree d just in case all nodes of d are well-labeled, and all labels are productions of G. Note that no requirement is placed on the nonterminal of the root node of d; in particular, it need not be S.</Paragraph>
    <Paragraph position="5"> We define the weight of a tree d, denoted Wa(d), or W(d) if G is clear from context, to be the product of weights of its nodes. The depth r(d) of d is the length of the longest path from root to leaf in d. The root production it(d) is the label of the root node. The root symbol p(d) is the left-hand side of ~r(d). The yield a(d) of the tree d is defined in the standard way as the string of terminal symbols &amp;quot;parsed&amp;quot; by the tree. It is convenient to treat the functions 7r, p, a, and r as random variables over trees. We write, for example, {p = X} as an abbreviation for {dip(d)= X}; and WG(p = X) represents the sum of weights of such trees. If the sum diverges, we set WG(p = X) = oo. We call IIXHG = WG(p = X) the norm of X, and IIGII = IISlla the norm of the grammar.</Paragraph>
    <Paragraph position="6"> A WCFG G is called convergent if \[\[G\[\[ &lt; oo.</Paragraph>
    <Paragraph position="7"> If G is a PCFG then \[\[G\[\[ = WG(p &amp;quot;- S) &lt; 1, that is, all PCFGs are convergent. A PCFG G is called consistent if \]\]GII = 1. A sufficient condition for the consistency of a PCFG is given in (Booth and Thompson, 1973). If (I) and * are two sets of parse trees such that 0 &lt; WG(~) &lt; co we define PG((I)\]~) to be WG(~Nqt)/WG(kO).</Paragraph>
    <Paragraph position="8"> For any terminal string y and grammar G such that 0 &lt; WG(p -- S) &lt; co we define PG(Y) to be Pa(a = YIP = S).</Paragraph>
  </Section>
class="xml-element"></Paper>