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<Paper uid="E89-1026">
  <Title>LAMBEK THEOREM PROVING AND FEATURE UNIFICATION</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 CATEGORIES
</SectionTitle>
    <Paragraph position="0"> In LTP categories and a set of inference rules constitute the calculus. The addition of FU necessitates the extension of these with respect to LTP without FU. Categories are for a start defined in the framework introduced by Gazdar et al. (1988). Gazdar et al. define category structure on a metatheoretical level as a pair &lt; ~., 6&amp;quot;&gt;. E is a quadruple&lt;F, A, % p&gt; where F is a finite set of features; A is a set of atoms; r is a function that divides the set of features into two sets, those that take atomic values (Type 0 features), and those that take categories as values (Type 1). p is a function that assigns a range of atomic values to each Type 0 feature. C is a set of constraints expressed in a language Lc. The reader is referred to Gazdar et al. (1988) for a precise definition of this language: we will merely use it here. For LTP-FU, the category structure in (2) and the constraints in (3) apply.</Paragraph>
    <Paragraph position="1"> (2)</Paragraph>
    <Paragraph position="3"> (3) (a) \[3(CONNECTIVE ~-, -1 LABEL) (b) n(DOMAIN ~ RANGE) (c) O(DOMAIN ~ CONNECTIVE:( / V \) ) (d) rT(FIRST *-* CONNECTIVE:*) (e) n(FIRST ~ LAST) (f) n(RANGE:f--- f/~ FEAT_NAMES)  The fact that ~category' is a central notion in CG justifies the division between features that express syntactic combinatorial possibilities ({DOMAIN,..., LABEL}) and other features (FEAT_NAMES) in (2) 1 In what follows we will use 'feature structure' to denote a set of feature-value combinations with *This view can for instance be found in the following citation from Calder et al. (1986): &amp;quot;(..) these \[categories\] can carry additionol feature specifications&amp;quot; (Calder et al., 1986, p. 7; my emphasis).</Paragraph>
    <Paragraph position="4"> features from FEAT_NAMES. We will use 'category' in the sense common in categorial linguistics. For a category with feature structure, we will use the term 'category specification'. Constraint (3)(a) ensures that a category is either complex or basic. Functor categories, those with the connective \ or / are specified by (3)(b), (3)(c); other complex categories are specified by (3)(d) and (e); (3)(f) describes the distribution of features from FEAT.NAMES. Here we follow Bouma (1988a) in the addition of features to complex categories. Firstly features are added to the argument (DOMAIN) in a complex category.</Paragraph>
    <Paragraph position="5"> This is &amp;quot;to express all kinds of subcategorization properties which an argument has to meet as it functions as the complement of the functor&amp;quot; (Bouma, 1988a, p. 27). Secondly, the category as a whole, rather than the RANGE carries features. &amp;quot;This has the advantage that complex categories can be directly characterized as finite, verbal etc.&amp;quot; (Bouma, 1988a, p. 27; of. Bach, 1983).</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 INFERENCE-RULES
</SectionTitle>
    <Paragraph position="0"> A sequent in the calculus is denoted with P :&gt; T, where P, called the antecedent, and T, the suceedent, are finite sequences of category specifications: P : K1 ... K,, and T : L. In LTP P and T are required to be non-empty; notice that the suceedent contains one and only one category specification. The axioms and inference rules of the calculus define the theorems of the categorial calculus. Recursive application of the inference rules on a sequent may result in the derivation of a sequent as a theorem of the calculus.</Paragraph>
    <Paragraph position="1"> In what follows, X, Y and Z are categories; A,B,C,D and E are feature structures; K,L,M,N are category specifications; P, T, Q, U, V are sequences of category specifications, where P, T and Q are non-empty. We use the notation category;feature structure:seraa~tics.</Paragraph>
    <Paragraph position="2"> Axioms are sequents of the form X;A:a =&gt; X;A:a.</Paragraph>
    <Paragraph position="3"> Note that identical letters for categories and semantic formulas denote identical categories and identical semantic formulas; identical letters for feature structures mean unified feature structures; and identical letters for category specifications mean category specifications with identical categories and unified features structures.</Paragraph>
    <Paragraph position="4"> From the form of the axiom it may follow that feature structures in antecedent and succedent should unify. This principle is the Axiom Feature Convention (AFC).</Paragraph>
    <Paragraph position="5"> In (4) the inference rules of LTP-FU are pre- null - 191 sented 2. \[\ _ el denotes a rule that eliminates a \-connective. i denotes introduction. The 'active type' in a sequent is the category from which the connective is removed.</Paragraph>
    <Paragraph position="6"> (4) \[/-el U,(X/Y;t);B:b,T,V =&gt; Z if T =&gt; Y;A:a and U,X;B:b(a),V =&gt; Z \[\-el U,T,(Y;t\X);B:b,V =&gt; Z if T =&gt; Y;A:a and U,X;B:b(a),V =&gt; Z \[*-el U, K:a*L:b,V =&gt; M if U,K:a,L:b,V =&gt; M \[/-i\] T =&gt; (X/Y;A);B:'v.b if T,Y;A:v =&gt; X;B:b \[\-i\] T =&gt; (Y;t\X);B;'v.b if Y;A:v, T =&gt; X;B:b \[*-i\] P:a,Q:b =&gt; K*L:c*d if P:a =&gt; K:C/ and Q:b =&gt; L:d  Certain feature structures are required to unify in inference rules. We formulate the so-called Active Functor Feature Convention (AFFC) to control the distribution of features. This convention is comparable to Head Feature Convention (Gasdar et al., 1985) and Functor Feature Convention (Bouma, 1988a). The AFFC states that the feature structure of an active functor type must be unified with the feature structure on the RANGE of the functor in the subsequent.</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5 AN EXAMPLE
</SectionTitle>
    <Paragraph position="0"> This paragraph limits itself to some observations concerning reflexives because this sheds light on a remaining question: are there principles other than AFFC and AFC necessary to account for 'FOOT' phenomena? There are two properties of reflexive pronouns that have to be accounted for in the theory.</Paragraph>
    <Paragraph position="1"> ~To envisage the rules without FU, just leave out all feature structures Firstly, the reflexive pronoun has to agree in number, person, and gender with some antecedent in the sentence (Chierchia, 1988), mostly the subject. Secondly, the reflexive pronoun is not necessarily the head of a constituent (Gazdar et al., 1985).</Paragraph>
    <Paragraph position="2"> The HFC in GPSG (Gazdar et al., 1985) cannot instantiate the antecedent information of a reflexive pronoun on a mothernode in cases where the reflexive is not the head of a constituent. Therefore in GPSG the so-called FOOT Feature Principle (FFP) is formulated. Together with the Control Agreement Principle (CAP) and the HFC, the FFP ensures that agreement between the demanded antecedent and the reflexive pronoun is obtained. Inclusion of a principle similar to FFP, and the use of category-valued features could be a solution for CUG. However, a solution that makes use of means supplied by categorial theory would keep us from 'stipulating axioms and principled', and as we will see, has as a consequence that we can avoid the use of category-valued features.</Paragraph>
    <Paragraph position="3"> For an account of reflexives in LTP-FU we will make use of reduction laws, other than the inference rules in (4). These reduction laws (like 1) normally have to be stipulated within categorial theory, but in LTP they can be derived as theorems within the calculus presented in (4) (Moortgat, 1987b). Feature distribution for these laws in LTP-FU can also be derived within the calculus with the application of AFFC and AFC and thus feature unification within these reduction laws also falls out as 'theorem' of the calculus: it is not necessary to include other principles than AFFC and AFC. In (5) a derivation for the reduction law composition is given (cf. Moortgat, 1987, p. 6).</Paragraph>
    <Paragraph position="5"> The cut rule (6) is not an inference rule, but a structural rule that is used to include proofs from a 'data base' into other proofs, for instance to include the results of the application of composition to part of a sequent. The cut rule is added to the inference rules of the calculus s. In (7(d)) the cut rule is used once to include a partial proof derived with the composition rule. The lexical category we assume the reflexive to have (see 7(b)) takes a verb with two arguments as its argument, and results in a verb with one argument. The verb requires, in the example, its subject to carry two feature-value pairs: \[num#sing,pers#3\]. (In (7(d)), all feature structures containing these features are abbreviated with the notation 3S.) These features are instantiated for the subject of the resulting oneargument verb. (7) gives a derivation where the reflexive is embedded in a prepositional phrase.</Paragraph>
    <Paragraph position="6"> In the example only relevant feature structures have been given actual feature-value pairs. (7(b)) presents the category of the reflexive. (c) presents one reduction using the composition rule and (d) presents the reduction of the whole sequent. The derivation of the semantic structure is presented seperately (e-f) from the syntactic derivation to improve readability.</Paragraph>
    <Paragraph position="7"> The refiexive's semantics imposes equality upon the arguments of the verb (Szabolcsi, 1987; but see also Chierchia (1988) and Popowich (1987) for other proposals). Note that in all cases, the reflexive should combine with the verb before the subject comes into play: the refiexive's semantics can only deal with A-bound variables as arguments. null</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
6 IMPLEMENTATION
</SectionTitle>
    <Paragraph position="0"> In this section a Prolog implementation of LTP-FU is described. The implementation makes use of the interpreter described in Moortgat (1988).</Paragraph>
    <Paragraph position="1"> Categoriai calculi, described in the proper format, can be offered to this interpreter. The interpreter then uses the axioms, inference rules and reduction rules as data and applies them to an input sequent recursively, in order to see whether the input sequent is a theorem in the calculus. In order to 'implement' a calculus, firstly it has to be described in a proper format. --~ and ~-- are defined as Prolog operators and denote respectively derivability in the calculus and inference during theorem proving. So, for instance with respect to the axiom, we may say that we have shown that X;A reduces to X;B if feat_des_unify aFor consequences of the addition of this rule, see Moortgat (1988) between A and B holds and true holds. The list notation is equal to the usual Prolog list notation, and is used to find the proper number of arguments while unifying an actual sequent with a rule. For instance \[T\[R\] cannot be instantiated as an empty list, whereas U can be instantiated as one. The LTP-FU calculus is presented in (8) (semantics is left out for readability).</Paragraph>
    <Paragraph position="2">  Note that feature unification is added explicitely: identity statements are interpreted &amp;quot;as instructions to replace the substructures with their unifications&amp;quot; (Shieber, 1986, p. 23). Prolog, however, does not allow this so-called destructive unification and therefore unification is reformulated. The necessity for destructive unification becomes clear from (9), where it is necessary to let features percolate to the &amp;quot;mother node&amp;quot; of a constituent. Note that in (9) reentrance for the modifier her and the specifier kleine is necessary (cf. Bouma, 1988a) to let the feature-value pair sex#fern percolate to the np. Reentrance is denoted with a number followed by a hook. It is represented uJithin lexical items; it is therefore not necessary to stipulate principles to account for percolation through reentrance.</Paragraph>
    <Paragraph position="4"> her kleine meisj e the little girl (np/n;l&gt;C) ;I&gt;D (n/n;9-&gt;A) ;2&gt;B n; \[sex#fem\] Within the ITI-TNO parser project (see footnote on first page), an attempt is made to develop a parser based on the mechanisms described here, using standard software development methods and techniques. During the so-called information analysis and the design stage (Van Berkel et al., 1988), several prototypes ofa Lambek Theorem Prover have been developed (Van Paassen, 1988). Implementation in C is currently undertaken, including semantic representation. Addition of Feature unification to this parser is scheduled for 1989. Lexical software for this purpose (in C) is available (Van der Linden, 1988b).</Paragraph>
  </Section>
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