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<Paper uid="J91-3003">
  <Title>The Generative Power of Categorial Grammars and Head-Driven Phrase Structure Grammars with Lexical Rules</Title>
  <Section position="3" start_page="303" end_page="305" type="metho">
    <SectionTitle>
3. Adding Lexical Rules
</SectionTitle>
    <Paragraph position="0"> The form of lexical rules that we propose to add to the basic categorial system are what Keenan and Faltz (1985) have termed valency affecting operations. These operations allow the permutation, addition, or subtraction of complements and the modification of the head or functor category. Our operations do not have any overt morphological effects, and are thus often referred to as zero morphemes. The same general lexical rule format has been proposed by virtually everyone considering the lexicon from a categorial perspective (see Dowty 1978, 1979; Bach 1984; Keenan and Faltz 1985; Keenan and Timberlake 1988; and Hoeksema and Janda 1988). Moortgat (1987) and Aone and Wittenburg (1990) have presented systems that allow extended categorial grammars to operate at both the morphological and syntactic levels, but the operations that can be carried out by their extended sets of rules produce results very similar to the lexical operations we allow here. Our results are especially relevant in light of recent work in HPSG, which admits lexical rules that do the same work as the ones employed here (see Flickinger 1987 and Pollard and Sag 1987). In the tradition of Montague, Dowty (1979) allowed arbitrary well-defined operations to be applied to lexical entries, but none of the rules he considered fall outside of the scope of the system presented here. The lexical generalizations studied by Bach (1984) led him to employ lexical redundancy rules that are expressed by composing basic functions that pick out the head or tail of a subcategorization list as well as the subcategorization list with either the head or tail removed. More formally, Bach allowed arbitrary concatenations and compositions of the following functions to be applied to subcategorization lists:</Paragraph>
    <Paragraph position="2"> After we present our lexical rule system, it should be obvious that Bach's system allows exactly the same operations to be expressed as we do. The only difference is that we recast Bach's functional rules in terms of simple pattern-matching. It is conjectured in Hoeksema and Janda (1988) that the resulting system is a proper subset of the context-sensitive languages. We show here that this conjecture could not be further from the truth, as all of the recursively enumerable languages can be generated using these operations. Keenan and Timberlake (1988) also present a collection of lexical redundancy rules that would seem to admit the system presented here as a natural generalization. In particular, they present an analysis of passive almost identical to the one presented below.</Paragraph>
    <Paragraph position="3"> The restriction placed on the form of our rules is similar to the restriction on GPSG metarules; we only allow one variable over sequences of categories (see Gazdar et al.</Paragraph>
    <Paragraph position="4"> 1985). This allows the manipulation of arguments a specified distance from either end of the subcategorization list, but not arbitrary arguments. With obliqueness specified in terms of subcategorization order (see Dowty 1982 and Pollard and Sag 1987), lexical rules are able to specify operations on arguments based on their obliqueness. In  Carpenter CG and HPSG with Lexical Rules general, a lexical rule is of the form: Definition 5 b\[co,. ...... ,Cn,$,do, * ,dm\] ::~ b'\[c'o,. ,%;, $,d~o,... ,d~m ,\] where $ is taken to be a variable ranging over sequences of categories (as in Ades and Steedman 1982). More precisely, we assume that the lexical rules are given by a finite relation: Definition 6 LexRule C (BasCat x Cat* x Cat*) x (BasCat x Cat* x Cat*).</Paragraph>
    <Paragraph position="5"> Thus a rule of the form in (5) would be formally represented as: Definition 7 lib, (Co,. ,Cn), (do, .,dm)), (b', (Co, , ' ' .... ' ... Ca,), (do, ...,d~,))) E LexRule.  The intended interpretation of an element of LexRule is that if a basic expression is assigned to a category that matches its left-hand side, then it is also assigned to a category that matches its right-hand side. We generate the final (possibly infinite) lexicon LexRule(BasLex) by closing the basic lexicon under the lexical rules. More formally, we define LexRule(BasLex) to be the least relation such that:</Paragraph>
    <Paragraph position="7"> Thus, a grammar is determined by the specification of finite sets BasLex and lexRule, over some given finite sets BasExp and BasCat of basic expressions and categories.</Paragraph>
    <Paragraph position="8"> In the undecidability proof that follows, it is only really necessary to consider lexical rules that modify the subcategorization list; a slightly modified proof goes through if lexical rules are prohibited from affecting the basic head category.</Paragraph>
    <Paragraph position="9"> We now take the time to motivate the full power of this lexical rule system. A standard example of the application of a lexical rule is passivization, which in our system can be stated in the form: Example 9 s{np\[\]l, $, np\[\]2\] ~ s{$, ppby\[\]2, np{\]l\] (Passive) The intuitive reading of this rule is that the first argument of any verb can become the last argument (subject) and the last argument becomes a prepositional by-phrase; the subscripts, while not an actual part of the rule, indicate this swapping of arguments and their syntactic markings. For instance, with a ditransitive verb category such as s\[np\[\]l, np{\]2, rip\[\]3\], the result of applying the passive rule would be s\[np\[\]2, Ppby\[\]3, np\[\]l\]. While we cannot account for the fact that the subject occurs before the verb rather than after it in the simplified system presented here, this category reflects the fact that the number, order, and category of arguments is permuted as a result of passivization.</Paragraph>
    <Paragraph position="10"> With the stripped-down system presented here, a second lexical rule is required for  Computational Linguistics Volume 17, Number 3 passives without the prepositional argument. An operation such as detransitivization can be stated by the rule: Example 10 s\[npN, $\] ~ s\[$\] (Detransitivization) The effect of detransitivization is simply to remove the most oblique argument from the subcategorization list. A nominalization rule might be stated in the form:</Paragraph>
    <Paragraph position="12"> Applying nominalization to a verbal lexical entry produces a nominal lexical entry with the same arguments. We can capture causative verbs with the following lexical rule: Example 12 siS\] ~ s\[$, np\[\]\] (Causative) The causative rule adds another noun phrase argument for the subject in the least oblique position and increases the obliqueness of the existing arguments. A rule for dative shift could be expressed as:</Paragraph>
    <Paragraph position="14"> The point of these examples is that very simple lexical regularities such as passives, causatives, and detransitives motivate a system of lexical rules that switch, add, and delete elements from the subcategorization lists of lexical entries.</Paragraph>
    <Paragraph position="15"> Lexical rules of the form we have here could also be used for what are traditionally considered to be syntactic operations, such as a rule for headless relatives (see Carpenter 1991): Example 14 s\[$, np~\]\] =~ n\[$, n\[\]\] (Headless Relativization) This rule has the effect of transforming a verbal category into a nominal modifier category with the same complements. Of course, to achieve the effect we desire, we must be able to mark verbs for their inflectional form and number. It should be apparent from these examples that the formalization of lexical rules presented here is particularly simple and motivated by a wide variety of seemingly lexical regularities. Further applications of lexical rules of the form we use here may be found in Dowty (1978, 1979), Bach (1984), Hoeksema and Janda (1988), Keenan and Timberlake (1988), Flickinger (1987), and Pollard and Sag (1987).</Paragraph>
  </Section>
  <Section position="4" start_page="305" end_page="306" type="metho">
    <SectionTitle>
4. Finite Closure
</SectionTitle>
    <Paragraph position="0"> We will now consider a restriction on the application of lexical rules that has the result of restricting the resulting system to a finite set of lexical entries. In the context of GPSG, Thompson (1982) restricted metarules to be nonrecursive so that they could not apply to rules that they had a hand in generating. This means that starting with a finite set of rules and metarules, only a finite number of rules would result. A similar  Carpenter CG and HPSG with Lexical Rules restriction allows Aone and Wittenburg (1990) to pre-compile the results of closing a categorial lexicon under a set of morphological operations. Let FinCIos(LexRule.BasLex) be the finite closure of the set of basic lexical entries under the metarules with the restriction that no rule can apply to its own output. More formally, we define FinCIos(LexRule,Bast.ex) inductively to be the least relation such that:  Trivial by induction on the cardinality of LexRule. \[\] The force of this result is that if we are willing to restrict our lexical rules to nonrecursive applications, then we have a finite lexicon and hence generate only a context-free language. But Carpenter (1991) argues for recursive lexical rules and provides examples from the English verbal system for which recursive rule application seems necessary.</Paragraph>
  </Section>
  <Section position="5" start_page="306" end_page="307" type="metho">
    <SectionTitle>
5. Argument Complexity Bounds
</SectionTitle>
    <Paragraph position="0"> Before moving on to the undecidability result, we define a notion of category complexity and show that the result of closing a finite lexicon under a finite number of lexical rules leads to a lexicon for which there is an upper bound on the complexity of the complements in lexical categories. Placing an upper bound on the complexity of entire categories in the lexicon restricts the system to context-free languages and ensures decidability.</Paragraph>
    <Paragraph position="1"> Our complexity metric is based purely on the number of complements for which a category subcategorizes and not the complexity of the complements themselves. The complexity of a category b\[Cl~...~ Cn\] is defined to be n, which we write as: Definition 16 Comp(b\[q~..., cn\]) = n.</Paragraph>
    <Paragraph position="2"> While it is possible to have a lexical rule such as b\[$\] ~ b\[c, $\], which allows the derivation of categories of unbounded complexity, the complements of categories derived by lexical rules are bounded in their complexity.</Paragraph>
    <Paragraph position="3"> Theorem 2 For any finite base lexicon BasLex and finite set LexRule of lexical rules, there is a bound k such that if e := b\[q~..., c~\] c LexRule(BasLex) then Comp(ci) &lt; k for 1 &lt; i &lt; n.  Because BasLex is finite, there is an upper bound on argument complexity for any complement category assigned by BasLex. Similarly, since there are only a finite number of rules in LexRule there is an upper bound on the complexity of complements specified in the outputs of any rule. Taken together, these facts imply that there is a bounded complement complexity in the result of closing BasLex under LexRule, since any complement category assigned by LexRule(BasLex) must have been a complement category either in BasLex or in the complement list of one of the output rules in LexRule. * This result shows that the lexical rules cannot modify the structure of complements other than by completely replacing them with one of a finite number of alternatives. We now note that if we restrict the complexity of lexical categories themselves, we wind up with a context-free grammar. Let LexRule(13aslex)(n) be the set of lexical entries with categories of complexity less than or equal to n, so that (e, c) E LexRule(BasLex)(n) if and only if (e,c) E kexRule(BasLex)(n) and Comp(c) &lt; n.</Paragraph>
    <Paragraph position="4"> Theorem 3 The language generated by LexRule(BasLex)(n) is context-free.</Paragraph>
    <Paragraph position="5"> Proof Using the previous theorem, we know that there is a bound k on the size of the complements in any lexical entry in LexRule(BasLex), and thus there must only be a finite number of lexical entries with complexity of less than or equal to n. Consequently, I_exRule(Baslex)(n) is finite and thus a standard finitary categorial grammar lexicon that generates a context-free language. *</Paragraph>
  </Section>
  <Section position="6" start_page="307" end_page="310" type="metho">
    <SectionTitle>
6. Generative Power
</SectionTitle>
    <Paragraph position="0"> As we said in the introduction, we characterize the generative power of our system by the reduction of generalized rewriting systems to our head-complement grammars with lexical rules. Before doing this,, we review the basic definition of a generalized rewriting system. A generalized rewriting system is a quadruple G = (V, s, T,R) where V is a finite set of nonterminal category symbols, s E V is the start symbol, T is a set of terminal symbols, and R C_ (V* x V*) U (V x T) is a finite set of rewriting rules and lexical insertion rules, which are usually expressed in the forms:  if p, p' E (V U T)* and if (~ * ~-) E R is a rule. The language L(G) generated by a general rewriting system G is defined to be</Paragraph>
    <Paragraph position="2"> Carpenter CG and HPSG with Lexical Rules where s is the start symbol and _L, is the usual transitive closure of the It is well known that:  We now present the fundamental result of this paper, which states that the languages that can be characterized by categorial grammars with lexical rules are exactly the recursively enumerable languages.</Paragraph>
    <Paragraph position="3"> Theorem 5 (R.E.-Completeness) A language S is recursively enumerable if and only if there is a finite lexicon BasLex and finite set LexRule of lexical rules such that S is the set of strings assigned to the category s\[\] by LexRule(BasLex).</Paragraph>
    <Paragraph position="4"> Proof It is trivial to show that the languages generated by our system are recursively enumerable; standard breadth-first search mechanisms that interleave lexical and syntactic derivations in order of complexity can be seen to enumerate all analyses. Conversely, suppose that we have a recursively enumerable language S and that the generalized rewriting system G = (V~s~ T~R I is such that S = L(G) is the set of strings generated by G. We show how to construct a categorial grammar using lexical rules that assigns the set S of expressions to some distinguished basic category. We begin by assuming that:</Paragraph>
    <Paragraph position="6"> We take a basic category for every nonterminal symbol in the generalized rewriting system along with two special symbols; the # is used as a delimiter in representing sequences of nonterminals by means of circular queues, while the s is used as the distinguished category of the grammar (note that s, not s, is the distinguished start symbol of G). Our claim is that the lexicon in (21) and lexical rules in (22) generate exactly the same language as G.</Paragraph>
    <Paragraph position="7">  Computational Linguistics Volume 17, Number 3 We represent an arbitrary string vlv2...v, E V* by means of the categorial grammar category v\[#\[\], vl \[\],..., vn \[\]\]. In what follows, we omit empty subcategorization lists, so that the above category would be abbreviated to v\[#,vl .... ,Vn\]. Note that with this encoding, there are as many representations of a string as there are nonterminals vEV.</Paragraph>
    <Paragraph position="8"> By repeated application of the first lexical rule in (22), from a |exical entry</Paragraph>
    <Paragraph position="10"> The application of this rule is the key to allowing an arbitrary string in the middle of a subcategorization list to move to the end so that it may be modified by a lexical rule. The # symbol keeps track of the true beginning of the sequence being derived.</Paragraph>
    <Paragraph position="11"> Suppose that the generalized rewriting system allows the one-step derivation: xl . . . XiVl . . . Vnyl &amp;quot;&amp;quot; &amp;quot; yj ~ Xl &amp;quot; &amp;quot; &amp;quot; XiUl &amp;quot; &amp;quot; &amp;quot; Umyl &amp;quot; &amp;quot; &amp;quot; yj. If this rewriting is possible, then (v~ ... vn ~ Ul &amp;quot;'&amp;quot; Um) E R so that by a combination of the first lexical rule and second lexical rule we can carry out the following lexical derivation:</Paragraph>
    <Paragraph position="13"> v\[#,x~,...,xi,vl .... ,v,,yl,...,yj\] v\[yl,.., Yi, #, xl .... , x;, vl, * * *, v,\] ?d\[yl,..., yj, #, Xl .... , xi, Ul .... , Urn\] V\[#, X~, . . . , Xi, U~ .... , Urn, yl,..., yj\]. A simple induction then gives us the result that if xl . . . xi &amp;quot; ~ yl &amp;quot; &amp;quot; yj then we can derive a lexical entry of the form</Paragraph>
    <Paragraph position="15"> from a lexical entry of the form</Paragraph>
    <Paragraph position="17"> Carpenter CG and HPSG with Lexical Rules according to the generalized rewriting system, then we can derive the lexical entry</Paragraph>
    <Paragraph position="19"> from the lexical entry t:= v\[#, s\] if (v , t) E R. Now suppose that (vi ~ ti) C R for 1 &lt; i &lt; n so that h'&amp;quot; t, c L(G). Beginning with the basic lexical entry</Paragraph>
    <Paragraph position="21"> representing the lexical rewriting (Vl ----* tl) E R, we can derive the lexical entry:</Paragraph>
    <Paragraph position="23"> From this last entry and our last lexical rule, we may derive a lexical entry:</Paragraph>
    <Paragraph position="25"> Furthermore, since we have ti :-= vi for 2 &lt; i &lt; n because (Vi ~ ti) E R, we can assign tit2.., tn to the category s\[\] by repeated application of the categorial grammar rule scheme. In fact, for 1 &lt; i &lt; n we have tl.' .ti assigned to s\[vi+l,... ,vn\]. Thus every string that belongs to the language generated by G can also be generated using our categorial grammar with lexical rules.</Paragraph>
    <Paragraph position="26"> It simply remains to notice that we have tl := s\[v2,...,vn\] if and only if tl := vl \[#, Vl,..., vn\], which holds if and only if s * ~vl ... Vn and vl ---* h- The only way to derive viii from ti is by using a lexical entry which is only possible if (vi , ti) E R.</Paragraph>
    <Paragraph position="27"> Furthermore, the only derivations of category s\[\] in the categorial grammar must be derived in this manner, as the only way s can arise is by lexical rule application in the above situation. Also note that while the # category can be removed by applying some lexical rule, there are no expressions that are assigned to #\[\] by our grammar. Thus the only way that the string tl ... tn can be assigned to category s\[\] is by following a lexical derivation that directly mirrors a derivation in G. * The fundamental idea behind our reduction is that the complement specification on a categorial grammar category can be used to simulate the intermediate stages of a generalized rewriting system derivation. The only complication arises from the fact that categorial grammar lexical rules operate on the ends of subcategorization lists, while generalized rewriting systems are allowed to operate on arbitrary substrings.</Paragraph>
  </Section>
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