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<Paper uid="E89-1029">
  <Title>EXTENDED GRAPH UNIFICATION</Title>
  <Section position="1" start_page="0" end_page="0" type="metho">
    <SectionTitle>
EXTENDED GRAPH UNIFICATION
</SectionTitle>
    <Paragraph position="0"/>
  </Section>
  <Section position="2" start_page="0" end_page="0" type="metho">
    <SectionTitle>
Abstract
</SectionTitle>
    <Paragraph position="0"> We propose an apparently minor extension to Kay's (1985} notation for describing directed acyclic graphs (DAGs}. The proposed notation permits concise descriptions of phenomena which would otherwise be difficult to describe, without incurring significant extra computational overheads in the process of unification. We illustrate the notation with examples from a categorial description of a fragment of English, and discuss the computational properties of unification of DAGs specified in this way.</Paragraph>
    <Paragraph position="1"> argue that our extension makes it possible to describe any phenomena which could not have been described at all using the existing notations, just that the descriptions using the extension are more concise.</Paragraph>
  </Section>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 GRAPH SPECIFICATION
</SectionTitle>
    <Paragraph position="0"> We start by defining a language GSL (graph specification language} for describing graphs, and by specifying the conditions under which two graphs unify.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1 INTRODUCTION
</SectionTitle>
    <Paragraph position="0"> Much recent work on specifying grammars for fragments of natural languages, and on producing computational systems which make use of these grammars, has used partial descriptions of complex feature structures {Gazdar 1988}. Grammars are specified in terms of partial descriptions of syntactic structures; programs that depend on these grammars perform some variant of unification in order to investigate the relationship between specific strings of words and the syntactic structures permitted by the grammarmis some sentence grammatical, what actually is its syntactic structure, how can some partially specified structure be realised as a string of words, and so on. Nearly all existing unification grammars of this kind use either term unification (the kind of unification used in resolution theorem provers, and hence provided as a primitive in PROLOG) or some version of the graph unification proposed by Kay {1985) and Shieber (1984). We propose an extension to the languages used by Kay and Shieber for describing graphs, and to the specification of the conditions under which graphs unify. This extension enables us to write concise descriptions of syntactic phenomena which would be awkward to specify using the originM notations. We do not</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.1 GSL: syntax
</SectionTitle>
      <Paragraph position="0"> The syntax of GSL has been kept as close as possible to that of FUG (Kay 1985) in order to facilitate comparisons. It is not, unfortunately, possible to keep it close to both FUG and PATR (Shieber 1984), but it should be possible for readers familiar with PATR to see roughly what the relation between the two is.</Paragraph>
      <Paragraph position="1"> A node descriptor consists of either an atomic symbol, e.g. agr, cat, bar, or of two atomic symbols separated by a slash, e.g. cat/C, head/OBJECT. In the first case the symbol is the value of the described node, in the second the symbol before the slash is the node's value and the symbol after it is its name. We will generally use lower case words for values and upper case ones for names, but the distinction between upper and lower case has no significance in GSL.</Paragraph>
      <Paragraph position="2"> A path descriptor consists of a sequence of node descriptors separated by equals signs, e.g.</Paragraph>
      <Paragraph position="3"> head---major=cat=prep. The path described by such a descriptor consists of the sequence of described nodes. The first node in a path is called its initial node and the final node is called its terminal node. The descriptor of the terminal node in a path may be followed by an exclamation mark, - 212 as in head=major=cat=prep/, in which case the node is said to be mandatory.</Paragraph>
      <Paragraph position="4"> A graph descriptor consists of a set of path descriptors separated by commas. The graph consists of the set of described paths. If two node descriptors in a graph descriptor specify the same name, they refer to the same node.</Paragraph>
      <Paragraph position="5"> A set of paths with identical initial segments may be specified by writing the initial segment just once and including the divergent tails within nested brackets, so that  The sub-graph governed by a path is the set of all terminal sequences of paths whose initial sequence matches the given path. The last node in the given path is called the root of the sub-graph governed by the path. Thus in the above example the set of paths X=Y, W=V=U, W=Q=R is the sub-graph governed by the path A=B=C, and C is the root of this sub-graph.</Paragraph>
      <Paragraph position="6"> A macro is simply a symbol which has been specified as a shorthand for some other sequence of symbols. Macros are expanded by simple textual substitution, so that if NP were a macro for the sequence of symbols cat=n, bar=two then head=(NP) expands to head=(cat=n/, bar=two~).</Paragraph>
      <Paragraph position="7"> The parentheses are important--head=NP expands to head--cat=a~, bar=two~, which is very different from head=(cat=n!, bar=two/).</Paragraph>
      <Paragraph position="8"> The major differences between GSL and the languages used by Kay and Shieber axe that GSL distinguishes between optional and mandatory nodes, and that names (which function as the constraints for turning trees into graphs) can be attached to non-terminal nodes. GSL also differs from FUG in that it does not provide a facility for disjunctive graphs--disjunction is catered for by requiring the grammar and lexicon to contain explicit alternatives, rather than by permitting graphs themselves to contain options. Most of the other differences are cosmetic--the GSL path agr=num=sinq/ is equivalent to the PATR path \[aqr: Inure: siag\]\] and the FUG descriptor agr=num=sing. The GSL path aqr=num=sing is roughly equivalent to the PATR path \[agr: \[hum: \[sittg: &lt;Alpha&gt;\]\]\] and the FUG descriptor agr=num=sing=ANY. The fact that the second set of paths axe only =roughly ~ equivalent is a consequence of the new definition of unification given in the next section.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.2 CSL: unification
</SectionTitle>
      <Paragraph position="0"> The major operation that we are going to perform on graphs specified in GSL is unification. We define this, as usual, in terms of the common extension of sets of graphs. We start by defining the common extension of a pair of graphs. Two graphs G1 and G2 unify to produce a common eztettsion E under the following conditions: (i) Suppose V is the value of initial nodes in each of G1 and G2. Then the sub-graphs of G1 and G2 which axe governed by the path consisting of just the node V must have a common extension, say Ev. If they do have such a common extension, then the common extension E of G1 and G2 themselves must include all the paths obtained by adding V to the front of members of Ev. If they do not then G1 and G2 do not unify, and hence have no common extension.</Paragraph>
      <Paragraph position="1"> Furthermore, if any initial node in either graph with V as its value has a name, that name must be associated with a sub-graph which has a common extension with each of G1 and G2. All the paths which appear in any of these extensions must also be included in E. Again if the sub-graph associated with any such name fails to have a common extension with either G1 or G2 then G1 and G2 themselves do not unify.</Paragraph>
      <Paragraph position="2"> (ii) Suppose V appears as the value of one or more initial nodes in G1 but of none in G2. Then if V is a mandatory terminal node of any path in G1 of which it is the initial node then G1 and G2 do not have a common extension (since V is mandatory in G1, but does not appear as an initial node of any path in G2). Otherwise the common extension of G1 and G2, if it exists, must include all the paths in G1 for which V is an initial node. The same condition applies if V is the value of one or more initial nodes in G2 but of none in G1.</Paragraph>
      <Paragraph position="3"> (iii) The common extension of G1 and G2 contains no paths not explicitly required by conditions (i} and (ii}.</Paragraph>
      <Paragraph position="4"> The common extension of a set of graphs {G1, G2, ..., Gn} where n &gt; 2 is simply the common extension of G 1 with the common extension of the set {G2, ..., Gn}.</Paragraph>
      <Paragraph position="5"> This definition of the common extension of - 213 a set of graphs is rather non-constructive, and is neutral with respect to compatational mechanisms. We need to show that we can in fact compute common extensions, and to consider the complexity of the algorithm for doing so, but before that we ought to try to show that we can use GSL to give concise descriptions of syntactic rules. If we can't do that, there is no point in worrying about the efficiency of algorithms for comparing graphs described in GSL at all.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 SYNTACTIC DESCRIP-
TIONS USING GSL
</SectionTitle>
    <Paragraph position="0"> We will illustrate the use of GSL with elements of a categorial grammar for a fragment of English. GSL is not specifically designed for categorim grammar, but the complexity of the category structures of any non-trivial categorial grammar means that such grammars provide a good testbed for notations for describing categories. Although categorial grammars have recently received considerable attention (Pareschi &amp; Steedman (1987), Klein &amp; van Benthem (1987), Oehrle, Bach &amp; Wheeler (1987)), computational treatments have been hindered by the need to develop and manipulate large category descriptions. The expressive power of GSL is therefore well illustrated by the ease with which we can develop the category descriptions required for a non-trivial categorial grammar.</Paragraph>
    <Paragraph position="1"> We start with the basic categorial rules:</Paragraph>
    <Paragraph position="3"> The first of these is an extended version of the normal categorial rule for combining something which requires an argument to its right with an argument of the appropriate type, namely: A ~ A/B B We have been forced to complicate this rule, as have others trying to produce categorial grammars for non-trivial fragments, in order to take into account intrinsic syntactic functions such as case and number agreement, and to deal with the fine details of sub-categorisation rules. In our extended version of the basic rule, the A of the basic version is replaced by (major/X, minor/Y, subcat/SUB, slash/SLASH) and the B of the basic version by (major/X1, minor/Y1, subcat/SUB1, slash/SLASH). The major features of a category are simply its main category (noun, verb, preposition, conj) and its bar level (zero, one, two). The minor features are the intrinsic syntactic features such as agr and auz. subcat specifies what arguments (lslash and rslash) are required and what the head (head) of the local tree described by the rule is like. slash, as usual in unification grammars, carries information about unbounded dependencies. The category A/B of the basic rule is replaced by:</Paragraph>
    <Paragraph position="5"> This describes a structure which will join with a (major/X, minor/Y, subcat/SUB, dash/SLASH) to its right to make a (major/Xl, minor/Yl, sub- cat/SUBl, slash/SLASH).</Paragraph>
    <Paragraph position="6"> We have made very little use of the extra facilities provided by GSL in specifying this rule, beyond the convenience of the abbreviations HEAD for subcat=head and RSLASH for subcat=rslaah.</Paragraph>
    <Paragraph position="7"> Apart from that, we have used names for specifying constraints, but that could easily have been done in any of the standard formalisms; and we have used the exclamation mark to constrain the value of slash on the first element of the right hand side to be null. The second of the basic rules is sufficiently similar that it requires no further discussion. null To show how the extra power of GSL can help us construct concise descriptions, we will consider two specific examples. The first is the definition - 214 of the lexical entry for an auxiliary. This requires the, fr,ll,,wing three macro definitions:</Paragraph>
    <Paragraph position="9"> The definition of A UX says that it is a special type of VERB, namely one that will combine with a VP to its right. The head of the A UX inherits any constraints on the subject of its own rslash.</Paragraph>
    <Paragraph position="10"> The definition of VERB says that it is something which does not require anything to its left, and that it will participate in local trees dominated by objects of type VP, with the constraint that the VERB has the same minor features as the VP.</Paragraph>
    <Paragraph position="11"> The definition of VP is fairly similar, but it does make use of the facility for placing names in non-terminal positions to enforce two constraints--one between the entire set of minor features of the VP and the minor features of its head, and another between the agr features of the VP and the agr features of its subject.</Paragraph>
    <Paragraph position="12"> Although this set of abbreviations appears only to call upon the facility for including names for non-terminal nodes once, we can see that if we were to expand the macros inside the definition of A UX there would be two other places where this was done (the definition below still has some macros unexpanded to help keep it readable):  It is worth noting that nowhere in either the expanded definition or in the three abbreviations is the major category of the subject specified. This information may be inherited from the main verb of the VP argument of the auxiliary, but otherwise its major category is unconstrained, in order to permit sentences like Eating people i8 going out of fa.qhion and For me to eat you u, oulJ be the h*icht of impropriety. It is assumed that the \[exical entries for verbs will sub-categorise for NP, VP or S subjects as required, just as they sub-categorise for complements.</Paragraph>
    <Paragraph position="13"> The second example of the use of GSL features comes from a group of rules which describe alternative sub-categorisation frames--rules which say, for instance, that a typical ditransitive verb has a case frame requiring two NP's rather than an NP and a PP. The rule below generates the %uxinverted&amp;quot; case frame for A UX's:  This rule again specifies names for non-terminal nodes, with VFORM twice being used as a name for a non-terminal node. The effect of this is to constrain the relevant item to be tensed and to share the same value for agr as its &amp;quot;inverted&amp;quot; subject. The rule also contains a number of mandatory features. The path minor=~form=finite=tensed!, for instance, restricts the rule to cases of tensed auxiliaries.</Paragraph>
    <Paragraph position="14"> We cannot use examples to &amp;quot;prove&amp;quot; that GSL makes it possible to write more concise specifications than we could write in FUG or PATR. This is particularly clear when the examples are culled from a grammar whose overall structure imposes constraints which can only be motivated by considering the grammar as a whole (which we do not have space for), rather than by looking at the examples in isolation. The best we can hope for is that the examples do seem to describe the constructions they are aimed at fairly concisely; and perhaps that it is not all that obvious how you would describe them in PATR or FUG.</Paragraph>
    <Paragraph position="15"> ~_~ - 215 -</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 COMPUTATIONAL COM-
PLEXITY
</SectionTitle>
    <Paragraph position="0"> We end by briefly considering the complexity of the task of seeing whether two graphs with named non-terminal nodes have a common extension. It is well-known that disjunctive unification is NP-complete (Kasper 1987). What is the status of unification of structures with constraints on subgraphs? null The definition of unification given in Section 2 looks very non-deterministic--full of phrases like ~Suppose V is the value of initial nodes in each of G1 and G2 ~ and ~Suppose V appears as the value of one or more initial nodes in G1 but of none in G2&amp;quot;. We can make it much more constrained by imposing a normal form on graphs. The first thing we need for this is an arbitrary ordering on features, which we can easily find since features are just alphanumeric strings, and these can be ordered lexicographically. If we were working with trees rather than DAGS, and we had such an ordering, we could impose a normal form by ordering the sub-trees of a node by the lexicographic ordering of their own root nodes, so that the normal form of the tree</Paragraph>
    <Paragraph position="2"> Unification of trees in this kind of normal form is of complexity o(M x N), where M is the maximum branching factor for the tree and N is the maximum depth. It is clear that we can impose a very similar normal form on DAGs without constraints on non-terminal nodes. For DAGs which do have constraints on non-terminal nodes, we have to split the representation of the graph into two pieces. We represent the basic structure of the graph in terms of sets of nodes and their successors; but where a node has a name, we include the name rather than the node itself. For each such named node, we store the sub-graph rooted at the node separately as the value of the name (this sub-graph itself, of course, may contain named nodes, in which case we just do the same again). We now effectively have a set of DAGs each of which has no constraints on internal nodes.</Paragraph>
    <Paragraph position="3"> We can therefore put each of these into normal form as before. The theoretical time for unification is again o(M x N), though N is now the length of the longest path through the graph you would get if you replaced names by the sub-graphs they name. The practical time is such as to make it perfectly sensible to use it as the basis of a computational system. Quoting times for analysing specific texts is a fairly meaningless way of comparing parsers, let alone unification algorithms, since there are so many unspecified parameters-size of the grammar, degree of ambiguity in the lexicon, speed of the basic machine, ... All I can say is that left-corner chart parsing with categorial rules specified via GSL descriptions of categories is markedly quicker than naive top-down left-right parsing of grammars of comparable coverage written as DCGs.</Paragraph>
  </Section>
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