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<Paper uid="P01-1018">
  <Title>Constraints on strong generative power</Title>
  <Section position="3" start_page="6" end_page="6" type="metho">
    <SectionTitle>
2 Characterizing structural descriptions
</SectionTitle>
    <Paragraph position="0"> First we define context-free rewriting systems.</Paragraph>
    <Paragraph position="1"> What these formalisms have in common is that their derivation sets are all local sets (that is, generable by a CFG). These derivations are taken as structural descriptions. The following definitions are adapted from Weir (1988).</Paragraph>
    <Paragraph position="3"> A generalized CFG G generates a set T(G)of terms, which are interpreted as derivations under some formalism. In this paper we require that G be free of spurious ambiguity, that is, that each term be uniquely generated.</Paragraph>
    <Paragraph position="4"> Definition 2 We say that a formalism F is a context-free rewriting system (CFRS)if its derivation sets can be characterized by generalized CFGs, and its derived structures are produced by a function~ dblbracketright F from terms to strings such that for each function symbol f, there is a yield function</Paragraph>
    <Paragraph position="6"> (A linear CFRS is subject to further restrictions, which we do not make use of.) As an example, Figure 3 shows a simple TAG with a corresponding GCFG and interpretation.</Paragraph>
    <Paragraph position="7"> A nice property of CFRS is that any formalism which can be defined as a CFRS immediately lends itself toseveral extensions, which arise when we give additional interpretations to the function symbols. For example, we can interpret the functions as ranging over probabilities, creating a stochastic grammar; or we can interpret them as yield functions of another grammar, creating a synchronous grammar.</Paragraph>
    <Paragraph position="8"> Now we define strong generative capacity as the relationship between strings and structural descriptions. null  This is similar in spirit, but not the same as, the notion of derivational generative capacity (Becker et al., 1992). Definition 3 The strong generative capacity of a grammar G aCFRSF is the relation fh~tdblbracketright F ;tijt2T(G)g: For example, the strong generative capacity of the grammar of Figure 3 is</Paragraph>
    <Paragraph position="10"> whereas any equivalent CFG must have a strong generative capacity of the form</Paragraph>
    <Paragraph position="12"> Thatis, ina CFGthe nbsandcs must appear later in the derivation than the m asandds, whereas in our example they appear in parallel.</Paragraph>
  </Section>
  <Section position="4" start_page="6" end_page="11" type="metho">
    <SectionTitle>
3 Simulating structural descriptions
</SectionTitle>
    <Paragraph position="0"> We now take a closer look at some examples of &amp;quot;squeezed&amp;quot; context-free formalisms to illustrate how a CFG can be used to simulate formalisms with greater strong generative power than CFG.</Paragraph>
    <Section position="1" start_page="6" end_page="6" type="sub_section">
      <SectionTitle>
3.1 Motivation
</SectionTitle>
      <Paragraph position="0"> Tree substitution grammar (TSG), tree insertion grammar (TIG),and regular-form TAG(RF-TAG) are all weakly context free formalisms which can additionally be parsed in cubic time (with a caveat for RF-TAGbelow). For each of these formalisms aCKY-styleparser can bewritten whose itemsare of the form [X;i;j] and are combined in various ways, but always according to the schema [X;i;j][Y;j;k] [Z;i;k] just as in the CKY parser for CFG. In e ect the parser dynamically converts theTSG,TIG,or RF-TAG into an equivalent CFG--each parser rule of the above form corresponds to the rule schema Z!XY.</Paragraph>
      <Paragraph position="1"> More importantly, given a grammar G and a string w, a parser can reconstruct all possible derivations of w under G by storing inside each chart item how that item was inferred. If we think of the parser as dynamically converting G into a  is not itself a parser. Thus these three formalisms have a special relationship to CFG that is independent of any particular parsing algorithm: for any TSG, TIG, or RF-TAG G, there is a CFGthat simulates G. We make this notion more precise below.</Paragraph>
    </Section>
    <Section position="2" start_page="6" end_page="6" type="sub_section">
      <SectionTitle>
3.2 Excursus: regular form TAG
</SectionTitle>
      <Paragraph position="0"> Strictly speaking, the recognition algorithm Rogers gives cannot be extended to parsing; that is, it generates all possible derived trees for a given string, but not all possible derivations. It is correct, however, as a parser for a further restricted subclass of TAGs: Definition 4 We say that a TAG is in strict regular form if there exists some partial ordering over the nonterminal alphabet such that for every auxiliary tree , if the root and foot of are labeled X, then for every node along 's spine where adjunction is allowed, X label( ), and X=label( ) only if is a foot node. (In this variant adjunction at foot nodes is permitted.) Thus the only kinds of adjunction which can occur to unbounded depth are o -spine adjunction and adjunction at foot nodes.</Paragraph>
      <Paragraph position="1"> This stricter definition still has greater strong generative capacity than CFG. For example, the TAG in Figure 3 is in strict regular form, because the only nodes along spines where adjunction is allowed are foot nodes.</Paragraph>
    </Section>
    <Section position="3" start_page="6" end_page="11" type="sub_section">
      <SectionTitle>
3.3 Simulability
</SectionTitle>
      <Paragraph position="0"> So far we have not placed any restrictions on how these structural descriptions are computed.</Paragraph>
      <Paragraph position="1"> Even though we might imagine attaching arbitrary functions to the rules of a parser, an algorithm like CKY is only really capable of computing values of bounded size, or else structure-sharing in the chart will be lost, increasing the complexity of the algorithm possibly to exponential complexity.</Paragraph>
      <Paragraph position="2"> For a parser to compute arbitrary-sized objects, such as the derivations themselves, it must use back-pointers, references to the values of sub-computations but not the values themselves. The only functions on a back-pointer the parser can compute online are the identity function (by copying the back-pointer) and constant functions (by replacing the back-pointer); any other function would have to dereference the back-pointer and destroy the structure of the algorithm. Therefore such functions must be computed o ine.</Paragraph>
      <Paragraph position="3"> Definition 5 A simulating interpretation ~ dblbracketrightis a bijection between two recognizable sets of terms such that</Paragraph>
      <Paragraph position="5"> can take one of the following forms:</Paragraph>
      <Paragraph position="7"> 3. Furthermore, we require that for any recognizable set T,~Tdblbracketrightis also a recognizable set. We say that~ dblbracketrightis trivial if every  is definable as</Paragraph>
      <Paragraph position="9"> where is a permutation off1;:::;ng.</Paragraph>
      <Paragraph position="10">  The rationale for requirement (3) is that it should not be possible, simply by imposing local constraints on thesimulating grammar, toproduce a simulated grammar which does not even come from a CFRS.</Paragraph>
      <Paragraph position="11">  As an example, a CFG which simulates the TAG of Figure 3 is shown in Figure 4. Note that if we give additional interpretations to the simulated yield functions ,  might take their arguments inadi erent order thanG,and there might be several yield functions of G  which correspond to a single yield function ofG used in several di erent contexts. In fact, for technical reasons we will use this notion instead of strong equivalence for testing the strong generative power of a formal system. null Thus the original problem, which was, given a formalism F, to find a formalism that has as much strong generative power as possible but remains weakly equivalent to F, is now recast as  Simulating interpretations and trivial simulating interpretations are similar to the generalized and &amp;quot;ungeneralized&amp;quot; syntax-directed translations, respectively, of Aho and Ullman (1969; 1971).</Paragraph>
      <Paragraph position="12">  Without this requirement, there are certain pathological cases that cause the construction of Section 4.2 to produce infinite MM-TAGs.</Paragraph>
      <Paragraph position="13">  of Figure 3. Here we leave the yield functions anonymous; y mapsfrom x denotes the function which maps x to y.</Paragraph>
      <Paragraph position="14"> the following problem: find a formalism that trivially simulates as many grammars as possible but remains simulable byF.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="11" end_page="11" type="metho">
    <SectionTitle>
3.4 Results
</SectionTitle>
    <Paragraph position="0"> The following is easy to show: Proposition 1 Simulability is reflexive and transitive. null Because of transitivity, it is impossible that a formalism which is simulable by F could simulate a grammar that is not simulable byF.Soweare looking for a formalism that can trivially simulate exactly those grammars thatF can.</Paragraph>
    <Paragraph position="1"> In Section 4.1 we define a formalism called multicomponent multifoot TAG (MMTAG), and then in Section 4.2 we prove the following result: Proposition 2 A grammar G from a CFRS is simulable by a CFG if and only if it is trivially simulable by an MMTAG in regular form.</Paragraph>
    <Paragraph position="2"> The &amp;quot;if&amp;quot; direction (() implies (because simulability is reflexive) that RF-MMTAG is simulable by a CFG, and therefore cubic-time parsable. (The proof below does give an e ective procedure for constructing a simulating CFG for any RF-MMTAG.)The &amp;quot;only if&amp;quot; direction ())shows that, in the sense we have defined, RF-MMTAG is the most powerful such formalism.</Paragraph>
    <Paragraph position="3"> We can generalize this result using the notion of a meta-level grammar (Dras, 1999).</Paragraph>
    <Paragraph position="4">  The &amp;quot;only if&amp;quot; direction ()) follows from the fact that the MMTAG constructed in the proof of Proposition 2 generates the same derived trees as the CFG. The &amp;quot;if&amp;quot; direction (() is a little trickier because the constructed CFG inserts and relabels nodes.</Paragraph>
  </Section>
  <Section position="6" start_page="11" end_page="11" type="metho">
    <SectionTitle>
4 Multicomponent multifoot TAG
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="11" end_page="11" type="sub_section">
      <SectionTitle>
4.1 Definitions
</SectionTitle>
      <Paragraph position="0"> MMTAG resembles a cross between set-local multicomponent TAG (Joshi, 1987) and ranked node rewriting grammar (Abe, 1988), a variant of TAG in which auxiliary trees may have multiple foot nodes. It also has much in common with d-tree substitution grammar (Rambow et al., 1995). Definition 8 An elementary tree set ~ is a finite set of trees (called the components of~ ) with the following properties: 1. Zero or more frontier nodes are designated foot nodes, which lack labels (following Abe), but are marked with the diacritic ; 2. Zero or more (non-foot) nodes are designated adjunction nodes, which are partitioned into one or more disjoint sets called adjunction sites. We notate this by assigning an index i to each adjunction site and marking each node of site i with the diacritic</Paragraph>
      <Paragraph position="2"> 3. Each component is associated with a symbol called its type. This is analogous to the left-hand side of a CFG rule (again, following Abe).</Paragraph>
      <Paragraph position="3"> 4. The components of ~ are connected by d-edges from foot nodes to root nodes (notated by dotted lines) to form a single tree structure. A single foot node may have multiple d-children, and their order issignificant. (See  are all X.</Paragraph>
      <Paragraph position="4"> 1. is a finite alphabet; 2. P is a finite set of tree sets; and 3. S 2 is a distinguished start symbol. Definition 9 A component is adjoinable at a node if is an adjunction node and the type of  equals the label of .</Paragraph>
      <Paragraph position="5"> Theresult ofadjoining acomponent atanode is the tree set formed by separating from its children, replacing with the root of ,andreplacing the ith foot node of with the ith child of . (Thus adjunction of a one-foot component is analogous to TAG adjunction, and adjunction of a zero-foot component is analogous to substitution.) null A tree set~ is adjoinable at an adjunction site ~ if there is a way to adjoin each component of~ at a di erent node of~ (with no nodes left over) such that the dominance and precedence relations within~ are preserved. (See Figure 5 for an example.) null We now define a regular form for MMTAGthat is analogous to strict regular form for TAG. A spine is the path from the root to a foot of a single component. Whenever adjunction takes place, several spines are inserted inside or concatenated with other spines. To ensure that unbounded insertion does not take place, we impose an ordering on spines, by means of functions</Paragraph>
      <Paragraph position="7"> the type of a component to the rank of that component's ith spine.</Paragraph>
      <Paragraph position="8"> Definition 10 We say that an adjunction node 2 ~ is safe in a spine if it is the lowest node (except the foot) in that spine, and if each component under that spine consists only of a member of~ and zero or more foot nodes.</Paragraph>
      <Paragraph position="9"> We say that an MMTAG G is in regular form if there are functions i from into the domain of some partial ordering such that for each component of type X, for each adjunction node 2 ,ifthejth child of dominates the ith foot node of (that is, another component's jth spine would adjoin into the ith spine), then</Paragraph>
      <Paragraph position="11"> is safe in the ith spine.</Paragraph>
      <Paragraph position="12"> Thus the only kinds of adjunction which can occur to unbounded depth are o -spine adjunction and safe adjunction. The adjunction shown in Figure 5 is an example of safe adjunction.</Paragraph>
    </Section>
    <Section position="2" start_page="11" end_page="11" type="sub_section">
      <SectionTitle>
4.2 Proof of Proposition 2
</SectionTitle>
      <Paragraph position="0"> (() First we describe how to construct a simulating CFG for any RF-MMTAG; then this direction of the proof follows from the transitivity of simulability.</Paragraph>
      <Paragraph position="1"> When a CFG simulates a regular form TAG, each nonterminal must encapsulate a stack (of bounded depth) to keep track of adjunctions. In the multicomponent case, these stacks must be generalized to trees (again, of bounded size).</Paragraph>
      <Paragraph position="2"> So the nonterminals of G  are of the form [ ;t], where t is a derivation fragment of G with a dot ( ) at exactly one node~ ,and is a node of~ .Let - be the node in the derived tree where ends up. A fragment t can be put into a normal form as follows:  1. For every~ above the dot, if - does not lie along a spine of~ , delete everything above ~ .</Paragraph>
      <Paragraph position="3"> 2. For every~ not above or at the dot, if - does not lie along a d-edge of ~ , delete ~ and everything below and replace it with &gt;if dominates~ ; otherwise replace it with?. 3. If there are two nodes ~  along a path which name the same tree set and - lies along the same spine or same d-edge in both of them, collapse~  , deleting everything in between.</Paragraph>
      <Paragraph position="4"> Basically this process removes all unboundedly long paths, sothat theset ofnormal formsis finite. In the rule schemata below, the terms in the left-hand sides range over normalized terms, and their corresponding right-hand sides are renormalized. Let up(t) denote the tree that results from moving the dot in t up one step.</Paragraph>
      <Paragraph position="5"> The value of a subderivation t  is a tuple of partial derivations of G, one for each &gt;symbol in the root label of t  , in order. Where we do not define a yield function for a production below, the identity function is understood. For every set ~ with a single, S-type component rooted by , add the rule</Paragraph>
      <Paragraph position="7"> This last rule skips over deleted parts of the derivation tree, but this is harmless in a regular form MMTAG, because all the skipped adjunctions are safe.</Paragraph>
      <Paragraph position="8"> ()) First we describe how to decompose any given derivation t  . (Note the convention that primed variables always pertain to the simulating grammar, unprimed variables to the simulated grammar.) If, during the computation of t, a node  creates the node , we say that  is productive and produces . Without loss of generality, let us assume that there is a one-to-one correspondence between productive nodes and nodes of t.  To start, let be the root of t,and  or any of its descendants is in the domain of i , and any non-productive node whose parent is in the domain of</Paragraph>
      <Paragraph position="10"> , excise each connected component of the domain of i . This operation is the reverse of adjunction (see Figure 6): each component gets  does not have this property, it can be modified so that it does. This may change the derived trees slightly, which makes the proof of Proposition 3 trickier.  mar of Figure 4, and first step of decomposition. Non-adjunction nodes are shown with the place-holder (because the yield functions in the original grammar were anonymous), the Greek letters indicating what is produced by each node. Adjunction nodes are shown with labels Q i in place of the (very long) true labels.</Paragraph>
      <Paragraph position="11">  ure 4 (cf. the original TAG in Figure 3). Each components' type is written to its left. foot nodes to replace its lost children, and the components are connected by d-edges according to their original configuration.</Paragraph>
      <Paragraph position="12"> Meanwhile an adjunction node is created in place of each component. This node is given a label (which also becomes the type of the excised component) whose job is to make sure the final grammar does not overgenerate; we describe how the label is chosen below. The adjunction nodes are partitioned such that the ith site contains all the adjunction nodes created when removing</Paragraph>
      <Paragraph position="14"> The tree set that is left behind is the elementary tree set corresponding to (rather, the function symbol that labels ); this process is repeated recursively on the children of ,ifany.</Paragraph>
      <Paragraph position="15"> Thus any derivation of G  can be decomposed into elementary tree sets. Let ^ G be the union of the decompositions of all possible derivations of  are the signatures of the foot nodes with respect to their d-children. Note that the number of possible adjunction labels is finite, though large.  set, to obtain a grammar whose simulated derivation set is non-recognizable. The idea is that multicomponent tree sets give rise to dependent paths in the derivation set, so if there is no bound on the number of components in a tree set, neither is there a bound on the length of dependent paths. This contradicts the requirement that a simulating interpretation map recognizable sets to recognizable sets.</Paragraph>
      <Paragraph position="16"> Suppose that the number of nodes per component is unbounded. If the number of components per tree set is bounded, so must the number of adjunction nodes per component; then it is possible, again by intersecting G  with a recognizable set, to obtain a grammar which is infinitely ambiguous with respect to simulated derivations, which contradicts the requirement that simulating interpretations be bijective.</Paragraph>
      <Paragraph position="17">  fields from several subderivations and processes them, combining some into a larger structure and copying some straight through to the root. Let i (X) be the number of fields that a component of type X copies from its ith foot up to its root. This information is encoded in X, in the signature of the root. Then ^ G satisfies the regular form constraint, because when adjunction inserts one spine into another spine, the the inserted spine must copy at least as many fields as the outer one. Furthermore, if theadjunction site isnot safe, then the inserted spine must additionally copy the value produced by some lower node.</Paragraph>
    </Section>
  </Section>
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