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<?xml version="1.0" standalone="yes"?> <Paper uid="P89-1033"> <Title>PARSING AS NATURAL DEDUCTION</Title> <Section position="3" start_page="272" end_page="273" type="metho"> <SectionTitle> 2 Lambek Calculus </SectionTitle> <Paragraph position="0"> In the following, we restrain ourselves to cut-free and product-free Lambek Calculus, a calculus which still allows us to infer infinitely many derived rules such as Geach-rule, functional composition etc. \[Zielonka 1981\]. The cut-free and product-free Lambek Calculus is given in figures 1 and 2.</Paragraph> <Paragraph position="1"> Be aware of the fact that we did not adopt Lambek's representation of complex categories. Proofs in Lambek Calculus can be represented as trees whose nodes are annotated with sequents. An example is given in figure 3. A lexical lookup step which replaces lexemes by their corresponding categories has to precede the actual theorem proving process. For this reason, the categories in the antecedens of the input sequent will also be called lezical categories. We introduce the notions of head, goal category, and current fanctor: The head of a category is its &quot;innermost&quot; value category: The head of a basic category is the category itself. The</Paragraph> <Section position="1" start_page="272" end_page="273" type="sub_section"> <SectionTitle> 2.1 Unification Lambek Calculus </SectionTitle> <Paragraph position="0"> Lambek Calculus, as such, is a propositional calculus. There is no room to express additional constraints concerning the combination of categories.</Paragraph> <Paragraph position="1"> Clearly, some kind of feature handling mechanism is needed to enable the grammar writer to state e.g.</Paragraph> <Paragraph position="2"> conditions on the agreement of morpho-syntactic features or to describe control phenomena. For the reason of linguistic expressiveness and to facilitate the description of the parsing algorithm below, we extend Lambek Calculus to Unification Lambek Calculus (ULC).</Paragraph> <Paragraph position="3"> First, the definition of basic category must be adapted: a basic category consists of an atomic category name and feature description. (For the definition of feature descriptions or feature terms see \[Smolka 1988\].) For complex categories, the same recursive definition applies as before. The syntax for categories in ULC is given informally in figure 4 which shows the category of a control verb like &quot;persuade&quot;. We assume that variable names for feature descriptions are local to each category in a sequent. The (/:left)- and (\:left)-inference rules have to take care of the substitutions which are involved in handling the variables in the extended categories (figure 5). Heed that the substitution function o&quot; has scope over a whole sequent, and therefore, over a complete subproof, and not only over a single category. In this way, correct variable bindings for hypothetic categories, which are introduced by &quot;right&quot;-rules, are guaranteed. np/n, n, (n\n)/np, np -- np np ---* np np/n, n, n\n ~ np n ---* n np/n, n --, np n ----* n np ~ np np/n, n, (n\n)/np, np ---, np .np--,np np/n, n, n\n ---* np n, n\n ~ n np --~ np n---~ n n----~ n</Paragraph> </Section> </Section> <Section position="4" start_page="273" end_page="275" type="metho"> <SectionTitle> 3 Normal Proof Trees </SectionTitle> <Paragraph position="0"> The sentence in figure 3 has two other proofs, which are listed in figure 6, although one would like to contribute only one syntactic or semantic reading to it. In this section, we show that such a set of a possibly abundant number of proofs for the same reading of a sequent possesses one distinguished member which can be regarded as the representative or the normal form proof tree for this set. In order to be able to use the notion of a &quot;reading&quot; more precisely, we undertake the following definition of structures which determine readings for our purposes. Because of their similarity to syntax trees as used with context-free grammars, we also call them &quot;syntax trees&quot; for the sake of simplicity. Since, on the semantic level, the use of a &quot;left'-rule in Lambek Calculus corresponds to the functional application of a functor term to some argument and the &quot;right&quot;-rules are equivalent to functional abstraction \[van Benthem 1986\], it is essential that in a syntax tree, a trace for each of these steps in a derivation be represented. Then it is guaranteed that the semantic representation of a sentence can be constructed from a syntax tree which is annotated by the appropriate partial semantic expressions of whatever semantic representation language one chooses. Structurally distinct syntax trees amount to different semantic expressions. null A syntax tree t condenses the information of a proof for a sequent s in the following way: 1. Labels of single.node trees, are either lexical categories or arguments of lexical categories.</Paragraph> <Paragraph position="1"> 2. The root of a non.trivial tree has either (a) one daughter tree whose root is labelled with the value category of the root's label. This case catches the application of a &quot;right'-inference rule; or (b) two daughter trees. The label of the root node is the value category, the label of the root of one daughter is the functor, and the label of the root of the other daughter is the argument category of an application of a &quot;left&quot;-inference rule.</Paragraph> <Paragraph position="2"> Since the size of a proof for a sequent is correlated linearily to the number of operators which occur in the sequent, different proof trees for the same sequent do not differ in terms of size - they are merely structurally distinct. The task of deft- null ning those relative normal forms of proofs, which we are aiming at, amounts to describing proof trees of a certain structure which can be more easily correlated with syntax trees as would possibly be the case for other proofs of the same set of proofs.</Paragraph> <Paragraph position="3"> The outline of the proof for the existence of normal form proof trees in Lambek Calculus is the following: Each proof tree of the set of proof trees for one reading of a sentence, i.e. a sequent, is mapped onto the syntax tree which represents this reading. By a proof reconstruction procedure (PR), this syntax tree can be mapped onto exactly one of the initial proof trees which will be identified as being the normal form proof tree for that set of proof trees.</Paragraph> <Paragraph position="4"> It is obvious that the mapping from proof trees onto syntax trees (Syntax Tree Construction - SC) partitions the set of proof trees for all readings of a sentence into a finite number of disjoint subsets, i.e. equivalence classes of proof trees. Proof trees of one of these subsets share the property of having the same syntax tree, i.e. reading. Hence, the single proof tree which is reconstructed from such a syntax tree can be safely taken as a representative for the subset which it belongs to. In figure 7, this argument is restated more formally.</Paragraph> <Paragraph position="5"> proof syntax normal trees trees proofs</Paragraph> <Paragraph position="7"> We want to prove the following theorem: Theorem 1 The set of proofs for a sequent can be partitioned into equivalence classes according to their corresponding syntax trees. There is exactly one proof per equivalence class which can be identified as its normal proof.</Paragraph> <Paragraph position="8"> This theorem splits up into two lemmata, the first of which is: Lemma 1 For every proof tree, there exists exactly one syntax tree.</Paragraph> <Paragraph position="9"> The proof for lemma 1 consists of constructing the required syntax tree for a given proof tree.</Paragraph> <Paragraph position="10"> The preparative step of the syntax tree construction procedure SC consists of augmenting lexical categories with (partial) syntax trees. Partial syntax trees are represented by A-expressions to indicate which subtrees have to be found in order to make the tree complete. The notation for a category c paired with its (partial) syntax tree t is c : t. A basic category is associated with the tree consisting of one node labelled with the name of the category.</Paragraph> <Paragraph position="11"> Complex categories are mapped onto partial binary syntax trees represented by A-expressions. We omit the detailed construction procedure for partial syntax trees on the lexical level, and give an example (see fig. 8) and an intuitive characterization instead. Such a partial tree has to be built up in such a way that it is a &quot;nesting&quot; of functional applications, i.e. one distinguished leaf is labelled with the functor category which this tree is associated with, all other leaves are labelled with variables bound by A-operators. The list of node labels along the path from the distinguished node to the root node must show the &quot;unfolding&quot; of the functor category towards its head category. Such a path is dubbed projection line.</Paragraph> <Paragraph position="13"> On the basis of these augmented categories, the overall syntax tree can be built up together with the proof for a sequent. As it has already been discussed above, a &quot;left&quot;-rule performs a functional application of a function t/ to an argument expression to, which we will abbreviate by tf\[t~ \].</Paragraph> <Paragraph position="14"> &quot;right&quot;-rules turn an expression tv into a function (i.e. partial syntax tree) t/ = Atatv by means of A-abstraction over to. However, in order to retain the information on the category of the argument and on the direction, we use the functor category itself as the root node label instead of the afore mentioned A-expression.</Paragraph> <Paragraph position="15"> The steps for the construction of a syntax tree along with a proof are encoded as annotations of the categories in Lambek Calculus (see figure 9).</Paragraph> <Paragraph position="16"> An example for a result of Syntax Tree Construction is shown in figure 10 where &quot;input&quot; syntax trees are listed below the corresponding sequent, and &quot;output&quot; syntax trees are displayed above their sequents, if shown at all.</Paragraph> <Paragraph position="17"> Since there is a one-to-one correspondence between proof steps and syntax tree construction steps, exactly one syntax tree is constructed per successful proof for a sequent. This leads us to the next step of the proof for the existence of normal forms, which is paraphrased by lemma 2.</Paragraph> <Paragraph position="18"> Lemma 2 From every syntax tree, a unique proof tree can be reconstructed.</Paragraph> <Paragraph position="19"> The proof for this lemma is again a constructive one: By a recursive traversal of a syntax tree, we obtain the normal form proof tree. (The formulation of the algorithm does not always properly distinguish between the nodes of a tree and the node</Paragraph> <Paragraph position="21"/> <Paragraph position="23"> Input: A syntax tree t with root node label g.</Paragraph> <Paragraph position="24"> Output: A proof tree p whose root sequent s with antecedens A and goal category g, and whose i daughter proofs pi (i = 0, 1, 2) are determined by the following method: Method: * If t consists of the single node g, p consists of an s which is an instantiation of the axiom scheme with g --~ g. s has no daughters.</Paragraph> <Paragraph position="25"> * If g is a complex category z/y reap. z\y and has one daughter tree tl, the antecedens A is the list of all leaves of t without the leftmost resp. the rightmost leaf., s has one daughter proof which is determined by applying Proof Reconstruction to the daughter tree of g.</Paragraph> <Paragraph position="26"> * If g is a basic category and has two daughter trees tt and t~_, then A is the list of all leaves of t. s has two daughter proof trees Pt and P2- C is the label of the leaf whose projection line ends at the root g. tl is the sister tree of this leaf. Pl is obtained by applying PR to tl. P2 is the result of applying PR to t2 which remains after cutting off the two subtrees C and tt from t.</Paragraph> <Paragraph position="27"> Thus, all proofs of an equivalence class are mapped onto one single proof by a composition of the two functions Syntax Tree Construction and Proof Reconstruction. \[:\]</Paragraph> </Section> <Section position="5" start_page="275" end_page="276" type="metho"> <SectionTitle> 4 The Parser </SectionTitle> <Paragraph position="0"> We showed the existence of relative normal form proof trees by the detour on syntax trees, assuming that all possible proof trees have been generated beforehand. This is obviously not the way one wants to take when parsing a sentence. The goal is to construct the normal form proof directly.</Paragraph> <Paragraph position="1"> For this purpose, a description of the properties which distinguish normal form proofs from non-normal form proofs is required.</Paragraph> <Paragraph position="2"> The essence of a proof tree is its nesting of current functors which can be regarded as a partial order on the set of current functors occuring in this specific proof tree. Since the current functors of two different rule applications might, coincidently, be the same form of category, obviously some kind of information is missing which would make all current functors of a proof tree (and hence of a syntax tree) pairwise distinct. This happens by stating which subsequence the head of the current functor spans over. As for information on a subsequence, it is sufficient to know where it starts and where it ends.</Paragraph> <Paragraph position="3"> Here is the point where we make use of the expressiveness of ULC. We do not only add the start and end position information to the head of a complex category but also to its other basic subcategories, since this information will be used e.g. for making up subgoals. We make use of obvious constraints among the positional indices of subcategories of the same category. The category in figure 11 spans from position 2 to 3, its head spans from 1 to 3 if its argument category spans from 1 to 2.</Paragraph> <Paragraph position="4"> whom mary loves 'tel'( 'rel/(s/np)', 's/n/( 's'('n/, 's\np'( '(s\np)lnp', 'np' )))) rel/(s/np), np, (s\np)/np ---, rel Az 'tel'( x ), 'np', AzlAz2 's'( z2, 's\np'( '(s\np)/np', zl )) 's/n/( 's'( 'rip', ' s\np'( '(s\np)/np', 'rip' ))) np, (s\np)/np ---* slnP '.p', ~1~2 's'( x2, 's\np'('(s\np)/np', xl )) np, (s\np)/np, np --* s We can now formulate what we have learned from the Proof Reconstruction (PR) procedure.</Paragraph> <Paragraph position="5"> Since it works top-down on a syntax tree, the characteristics of the partial order on current functors given by their nesting in a proof tree are the following null</Paragraph> <Section position="1" start_page="276" end_page="276" type="sub_section"> <SectionTitle> Nesting Constraints: </SectionTitle> <Paragraph position="0"/> </Section> </Section> <Section position="6" start_page="276" end_page="277" type="metho"> <SectionTitle> 1. Right.Rule Preference: Complex categories on </SectionTitle> <Paragraph position="0"> th.e righthand side of the arrow become current functors before complex categories on the lefthand side.</Paragraph> <Paragraph position="1"> 2. Current Functor Unfolding: Once a lefthand side category is chosen for current functor it has to be &quot;unfolded&quot; completely, i.e. in the next inference step, its value category has to become current functor unless it is a basic category. null 3. Goal Criter~um: A lefthand side functor category can only become current functor if its head category is unifiable with the goal category of the sequent where it occurs.</Paragraph> <Paragraph position="2"> Condition 3 is too weak if it is stated on the background of propositional Lambek Calculus only. It would allow for proof trees whose nesting of current functors does not coincide with the nesting of current functors in the corresponding syntax tree (see figure 12).</Paragraph> <Paragraph position="3"> The outline of the parsing/theorem proving algorithm P is: * A&quot; sequent is proved if it is an instance of the axiom scheme.</Paragraph> <Paragraph position="4"> * Otherwise, choose an inference rule by obeying the nesting constraints and try to prove the premises of the rule.</Paragraph> <Paragraph position="5"> Algorithm P is sound with respect to LC because it has been derived from LC by adding restrictions, and not by relaxing original constraints. It is also complete with regard to LC, because the restrictions are just as many as needed to rule out proof trees of the &quot;spurious ambiguity&quot; kind according to theorem 1.</Paragraph> <Section position="1" start_page="277" end_page="277" type="sub_section"> <SectionTitle> 4.1 Further Improvements </SectionTitle> <Paragraph position="0"> The performance of the parser/theorem prover can be improved further by adding at least the two following ingredients: The positional indices can help to decide where sequences in the &quot;left&quot;-rules have to be split up to form the appropriate subsequences of the premises.</Paragraph> <Paragraph position="1"> In \[van Benthem 1986\], it was observed that theorems in LC possess a so-called count invariant, which can be used to filter out unpromising suggestions for (sub-)proofs during the inference process.</Paragraph> </Section> </Section> class="xml-element"></Paper>