File Information
File: 05-lr/acl_arc_1_sum/cleansed_text/xml_by_section/metho/06/p06-1137_metho.xml
Size: 23,524 bytes
Last Modified: 2025-10-06 14:10:22
<?xml version="1.0" standalone="yes"?> <Paper uid="P06-1137"> <Title>Sydney, July 2006. c(c)2006 Association for Computational Linguistics Highly constrained unification grammars</Title> <Section position="4" start_page="1089" end_page="1090" type="metho"> <SectionTitle> 2 Preliminary notions </SectionTitle> <Paragraph position="0"> A CFG is a four-tuple Gcf = <VN,Vt,Rcf,S> where Vt is a set of terminals, VN is a set of non1The term mildly context-sensitive was coined by Joshi (1985), in reference to a less formally defined class of languages. Strictly speaking, what we call MCSL here is also known as the class of tree-adjoining languages.</Paragraph> <Paragraph position="1"> terminals, including the start symbol S, and Rcf is a set of productions, assumed to be in a normal form where each rule has either (zero or more) non-terminals or a single terminal in its body, and where the start symbol never occurs in the right hand side of rules. The set of all such context-free grammars is denoted CFGS.</Paragraph> <Paragraph position="2"> In a linear indexed grammar (LIG),2 strings are derived from nonterminals with an associated stack denoted A[l1 ...ln], where A is a nonterminal, each li is a stack symbol, and l1 is the top of the stack. Since stacks can grow to be of unbounded size during a derivation, some way of partially specifying unbounded stacks in LIG productions is needed. We use A[l1 ...ln [?]] to denote the nonterminal A associated with any stack e whose top n symbols are l1,l2 ...,ln. The set of all nonterminals in VN, associated with stacks whose symbols come from Vs, is denoted VN[V [?]s ].</Paragraph> <Paragraph position="3"> Definition 1. A Linear Indexed Grammar is a five tuple Gli = <VN,Vt,Vs,Rli,S> where Vt, VN and S are as above, Vs is a finite set of indices (stack symbols) and Rli is a finite set of productions in one of the following two forms:</Paragraph> <Paragraph position="5"> where Ni,Nj [?] VN, p1 ...pn,q1 ...qm [?] Vs, n,m [?] 0 and a,b [?] (Vt [?]VN[V [?]s ])[?].</Paragraph> <Paragraph position="6"> A crucial characteristic of LIG is that only one copy of the stack can be copied to a single element in the body of a rule. If more than one copy were allowed, the expressive power would grow beyond MCSL.</Paragraph> <Paragraph position="7"> Definition 2. Given a LIG <VN,Vt,Vs,Rli,S> , the derivation relation '=li' is defined as follows: for all Ps1,Ps2 [?] (VN[V [?]s ][?]Vt)[?] and e [?] V [?]s , The language generated by Gli is L(Gli) = {w [?] V [?]t |S[ ] [?]=li w}, where ' [?]=li' is the reflexive, transitive closure of '=li'.</Paragraph> <Paragraph position="8"> Unification grammars are defined over feature structures (FSs) which are directed, connected, rooted, labeled graphs, usually depicted as attribute-value matrices (AVM). A feature structure A can be characterized by its set of paths, PA, an assignment of atomic values to the ends of some paths, ThA(*), and a reentrancy relation 'squiggleleftright' relating paths which lead to the same node. A sequence of feature structures, where some nodes may be shared by more than one element, is a multi-rooted structure (MRS).</Paragraph> <Paragraph position="9"> Definition 3. Unification grammars are defined over a signature consisting of a finite set ATOMS of atoms; a finite set FEATS of features and a finite set WORDS of words. A unification grammar is a tuple Gu = <Ru,As,L> where Ru is a finite set of rules, each of which is an MRS of length n [?] 1, L is a lexicon, which associates with every word w [?] WORDS a finite set of feature structures, L(w), and As is a feature structure, the start symbol.</Paragraph> <Paragraph position="10"> Definition 4. A unification grammar <Ru,As,L> over the signature <ATOMS, FEATS, WORDS> is non-reentrant iff for any rule ru [?] Ru, ru is non-reentrant. It is one-reentrant iff for every rule ru [?] Ru, ru includes at most one reentrancy, between the head of the rule and some element of the body. Let UGnr, UG1r be the sets of all non-reentrant and one-reentrant unification grammars, respectively.</Paragraph> <Paragraph position="11"> Informally, a rule is non-reentrant if (on an AVM view) no reentrancy tags occur in it. When the rule is viewed as a (multi-rooted) graph, it is non-reentrant if the in-degree of all nodes is at most 1. A rule is one-reentrant if (on an AVM view) at most one reentrancy tag occurs in it, exactly twice: once in the head of the rule and once in an element of its body. When the rule is viewed as a (multi-rooted) graph, it is one-reentrant if the in-degree of all nodes is at most 1, with the exception of one node whose in-degree can be 2, provided that the only two distinct paths that lead to this node leave from the roots of the head of the rule and an element of the body.</Paragraph> <Paragraph position="12"> FSs and MRSs are partially ordered by subsumption, denoted 'subsetsqequal'. The least upper bound with respect to subsumption is unification, denoted 'unionsq'. Unification is partial; when A unionsq B is undefined we say that the unification fails and denote it as AunionsqB = latticetop. Unification is lifted to MRSs: given two MRSs s and r, it is possible to unify the i-th element of s with the j-th element of r.</Paragraph> <Paragraph position="13"> This operation, called unification in context and denoted (s,i) unionsq (r,j), yields two modified variants of s and r: (sprime,rprime).</Paragraph> <Paragraph position="14"> In unification grammars, forms are MRSs. A form sA = <A1,...,Ak> immediately derives another form sB = <B1,...,Bm> (denoted by sA 1=u sB) iff there exists a rule ru [?] Ru of length n that licenses the derivation. The head of ru is matched against some element Ai in sA using unification in context: (sA,i) unionsq (ru,0) = (sprimeA,rprime). If the unification does not fail, sB is obtained by replacing the i-th element of sprimeA with the body of rprime. The reflexive transitive closure of ' 1=u' is denoted by ' [?]=u'.</Paragraph> <Paragraph position="15"> Definition 5. The language of a unification grammar Gu is L(Gu) = {w1***wn [?] WORDS[?] | As [?]=u <A1,...,An> }, where Ai [?] L(wi) for 1 [?] i [?] n.</Paragraph> </Section> <Section position="5" start_page="1090" end_page="1091" type="metho"> <SectionTitle> 3 Context-free unification grammars </SectionTitle> <Paragraph position="0"> We define a constraint on unification grammars which ensures that grammars satisfying it generate the class CFL. The constraint disallows any reentrancies in the rules of the grammar. When rules are non-reentrant, applying a rule implies that an exact copy of the body of the rule is inserted into the generated (sentential) form, not affecting neighboring elements of the form the rule is applied to. The only difference between rule application in UGnr and the analog operation in CFGS is that the former requires unification whereas the latter only calls for identity check. This small difference does not affect the generative power of the formalisms, since unification can be pre-compiled in this simple case.</Paragraph> <Paragraph position="1"> The trivial direction is to map a CFG to a non-reentrant unification grammar, since every CFG is, trivially, such a grammar (where terminal and non-terminal symbols are viewed as atomic feature structures). For the inverse direction, we define a mapping from UGnr to CFGS. The non-terminals of the CFG in the image of the mapping are the set of all feature structures defined in the source UG.</Paragraph> <Paragraph position="2"> Definition 6. Let ug2cfg : UGnr mapsto- CFGS be a mapping of UGnr to CFGS, such that if Gu = <Ru,As,L> is over the signature <ATOMS, FEATS, WORDS> then ug2cfg(Gu) = <VN,Vt,Rcf,Scf> , where: * VN = {Ai |A0 - A1 ...An [?] Ru,i [?] 0}[?] {A |A [?] L(a),a [?] ATOMS}[?]{As}. VN is the set of all the feature structures occurring in any of the rules or the lexicon of Gu.</Paragraph> <Paragraph position="3"> * Scf = As * Vt = WORDS * Rcf consists of the following rules: 1. Let A0 - A1 ...An [?] Ru and B [?] L(b). If for some i, 1 [?] i [?] n, AiunionsqB negationslash= latticetop, then Ai - b [?] Rcf 2. If A0 - A1 ...An [?] Ru and AsunionsqA0 negationslash= latticetop then Scf - A1 ...An [?] Rcf.</Paragraph> <Paragraph position="4"> 3. Let ru1 = A0 - A1 ...An and ru2 =</Paragraph> <Paragraph position="6"> for some i, 1 [?] i [?] n, Ai unionsq B0 negationslash= latticetop, then the rule Ai - B1 ...Bm [?] Rcf The size of ug2cfg(Gu) is polynomial in the size of Gu. By inductions on the lengths of the derivation sequences, we prove the following theorem: null Theorem 1. If Gu = <Ru,As,L> is a non-reentrant unification grammar and Gcf = ug2cfg(Gu), then L(Gcf) = L(Gu).</Paragraph> <Paragraph position="7"> Corollary 2. Non-reentrant unification grammars are weakly equivalent to CFGS.</Paragraph> </Section> <Section position="6" start_page="1091" end_page="1094" type="metho"> <SectionTitle> 4 Mildly context-sensitive UG </SectionTitle> <Paragraph position="0"> In this section we show that one-reentrant unification grammars generate exactly the class MCSL.</Paragraph> <Paragraph position="1"> In such grammars each rule can have at most one reentrancy, reflecting the LIG situation where stacks can be copied to exactly one daughter in each rule.</Paragraph> <Section position="1" start_page="1091" end_page="1092" type="sub_section"> <SectionTitle> 4.1 Mapping LIG to UG1r </SectionTitle> <Paragraph position="0"> In order to simulate a given LIG with a unification grammar, a dedicated signature is defined based on the parameters of the LIG.</Paragraph> <Paragraph position="1"> Definition 7. Given a LIG <VN,Vt,Vs,Rli,S> , let t be <ATOMS, FEATS, WORDS> , where ATOMS = VN [?]Vs [?]{elist}, FEATS = {HEAD, TAIL}, and</Paragraph> <Paragraph position="3"> We use t throughout this section as the signature over which UGs are defined. We use FSs over the signature t to represent and simulate LIG symbols. In particular, FSs will encode lists in the natural way, hence the features HEAD and TAIL. For the sake of brevity, we use standard list notation when FSs encode lists. LIG symbols are mapped to FSs thus: Definition 8. Let toFs be a mapping of LIG symbols to feature structures, such that: 1. If t [?] Vt then toFs(t) = <t> 2. If N [?] VN and pi [?] Vs,1 [?] i [?] n, then</Paragraph> <Paragraph position="5"> The mapping toFs is extended to sequences of symbols by setting toFs(ab) = toFs(a)toFs(b).</Paragraph> <Paragraph position="6"> Note that toFs is one to one.</Paragraph> <Paragraph position="7"> When FSs that are images of LIG symbols are concerned, unification is reduced to identity: Lemma 3. Let X1,X2 [?] VN[V [?]s ] [?] Vt. If</Paragraph> <Paragraph position="9"> When a feature structure which is represented as an unbounded list (a list that is not terminated by elist) is unifiable with an image of a LIG symbol, the former is a prefix of the latter.</Paragraph> <Paragraph position="10"> Lemma 4. Let C = <p1,...,pn, i > be a non-reentrant feature structure, where p1,...,pn [?] Vs, and letX [?] VN[V [?]s ][?]Vt. Then CunionsqtoFs(X) negationslash= latticetop iff toFs(X) = <p1,...,pn,a> , for some a [?] V [?]s .</Paragraph> <Paragraph position="11"> To simulate LIGs with UGs we represent each symbol in the LIG as a feature structure, encoding the stack of LIG non-terminals as lists. Rules that propagate stacks (from mother to daughter) are simulated by means of reentrancy in the UG.</Paragraph> <Paragraph position="12"> Theorem 5. If Gli = <VN,Vt,Vs,Rli,Sli> is a LIG and Gu = lig2ug(Gli) then L(Gu) = L(Gli).</Paragraph> </Section> <Section position="2" start_page="1092" end_page="1094" type="sub_section"> <SectionTitle> 4.2 Mapping UG1r to LIG </SectionTitle> <Paragraph position="0"> We are now interested in the reverse direction, namely mapping UGs to LIG. Of course, since UGs are more expressive than LIGs, only a sub-set of the former can be correctly simulated by the latter. The differences between the two formalisms can be summarized along three dimensions: The basic elements UG manipulates feature structures, and rules (and forms) are MRSs; whereas LIG manipulates terminals and non-terminals with stacks of elements, and rules (and forms) are sequences of such symbols.</Paragraph> <Paragraph position="1"> Rule application In UG a rule is applied by unification in context of the rule and a sentential form, both of which are MRSs, whereas in LIG, the head of a rule and the selected element of a sentential form must have the same non-terminal symbol and consistent stacks.</Paragraph> <Paragraph position="2"> Propagation of information in rules In UG information is shared through reentrancies, whereas In LIG, information is propagated by copying the stack from the head of the rule to one element of its body.</Paragraph> <Paragraph position="3"> We show that one-reentrant UGs can all be correctly mapped to LIG. For the rest of this section we fix a signature <ATOMS, FEATS, WORDS> over which UGs are defined. Let NRFSS be the set of all non-reentrant FSs over this signature.</Paragraph> <Paragraph position="4"> One-reentrant UGs induce highly constrained (sentential) forms: in such forms, there are no reentrancies whatsoever, neither between distinct elements nor within a single element. Hence all the FSs in forms induced by a one-reentrant UG are non-reentrant.</Paragraph> <Paragraph position="5"> Definition 10. Let A be a feature structure with no reentrancies. The height of A, denoted |A|, is the length of the longest path in A. This is well-defined since non-reentrant feature structures are acyclic. Let Gu = <Ru,As,L> [?] UG1r be a one-reentrant unification grammar. The maximum height of the grammar, maxHt(Gu), is the height of the highest feature structure in the grammar. This is well defined since all the feature structures of one-reentrant grammars are non-reentrant.</Paragraph> <Paragraph position="6"> The following lemma indicates an important property of one-reentrant UGs. Informally, in any FS that is an element of a sentential form induced by such grammars, if two paths are long (specifically, longer than the maximum height of the grammar), they must have a long common prefix.</Paragraph> <Paragraph position="7"> Lemma 6. Let Gu = <Ru,As,L> [?] UG1r be a one-reentrant unification grammar. Let A be an element of a sentential form induced by Gu. If pi * <Fj> *pi1,pi*<Fk> *pi2 [?] PA, where Fj, Fk [?] FEATS, j negationslash= k and |pi1 |[?] |pi2|, then |pi1 |[?] maxHt(Gu). Lemma 6 facilitates a view of all the FSs induced by such a grammar as (unboundedly long) lists of elements drawn from a finite, predefined set. The set consists of all features in FEATS and all the non-reentrant feature structures whose height is limited by the maximal height of the unification grammar. Note that even with one-reentrant UGs, feature structures can be unboundedly deep. What lemma 6 establishes is that if a feature structure induced by a one-reentrant unification grammar is deep, then it can be represented as a single &quot;core&quot; path which is long, and all the sub-structures which &quot;hang&quot; from this core are depth-bounded. We use this property to encode such feature structures as cords.</Paragraph> <Paragraph position="8"> Definition 11. Let Ps : NRFSS x PATHS mapsto(FEATS [?] NRFSS)[?] be a mapping such</Paragraph> <Paragraph position="10"> for 1 [?] i [?] n +1, Ai are non-reentrant FSs such that:</Paragraph> <Paragraph position="12"> fined).</Paragraph> <Paragraph position="13"> We also define last(Ps(A,pi)) = An+1. The height of a cord is defined as |Ps(A,pi) |= max1[?]i[?]n+1(|Ai|). For each cord Ps(A,pi) we refer to A as the base feature structure and to pi as the base path. The length of a cord is the length of the base path.</Paragraph> <Paragraph position="14"> The function Ps is one to one: given Ps(A,pi), both A and pi are uniquely determined.</Paragraph> <Paragraph position="15"> Lemma 7. Let Gu be a one-reentrant unification grammar and let A be an element of a sentential form induced by Gu. Then there is a path pi [?] PA such that |Ps(A,pi) |< maxHt(Gu).</Paragraph> <Paragraph position="16"> be represented as a height-limited cord. This mapping resolves the first difference between LIG and UG, by providing a representation of the basic elements. We use cords as the stack contents of LIG non-terminals: cords can be unboundedly long, but so can LIG stacks; the crucial point is that cords are height limited, implying that they can be represented using a finite number of elements. We now show how to simulate, in LIG, the unification in context of a rule and a sentential form. The first step is to have exactly one non-terminal symbol (in addition to the start symbol); when all non-terminal symbols are identical, only the content of the stack has to be taken into account. Recall that in order for a LIG rule to be applicable to a sentential form, the stack of the rule's head must be a prefix of the stack of the selected element in the form. The only question is whether the two stacks are equal (fixed rule head) or not (unbounded rule head). Since the contents of stacks are cords, we need a property relating two cords, on one hand, with unifiability of their base feature structures, on the other. Lemma 8 establishes such a property. Informally, if the base path of one cord is a prefix of the base path of the other cord and all feature structures along the common path of both cords are unifiable, then the base feature structures of both cords are unifiable. The reverse direction also holds.</Paragraph> <Paragraph position="17"> Lemma 8. Let A,B [?] NRFSS be non-reentrant feature structures and pi1,pi2 [?] PATHS be paths such that pi1 [?] PB, pi1*pi2 [?] PA, Ps(A,pi1*pi2) =</Paragraph> <Paragraph position="19"> <F|pi1|+1> negationslash[?] Ps|pi1|+1. Then A unionsq B negationslash= latticetop iff for all i, 1 [?] i [?] |pi1|+1, si unionsqti negationslash= latticetop.</Paragraph> <Paragraph position="20"> The length of a cord of an element of a sentential form induced by the grammar cannot be bounded, but the length of any cord representation of a rule head is limited by the grammar height. By lemma 8, unifiability of two feature structures can be reduced to a comparison of two cords representing them and only the prefix of the longer cord (as long as the shorter cord) affects the result. Since the cord representation of any grammar rule's head is limited by the height of the grammar we always choose it as the shorter cord in the comparison.</Paragraph> <Paragraph position="21"> We now define, for a feature structure C (which is a head of a rule) and some path pi, the set that includes all feature structures that are both unifiable with C and can be represented as a cord whose height is limited by the grammar height and whose base path is pi. We call this set the compatibility set of C and pi and use it to define the set of all possible prefixes of cords whose base FSs are unifiable with C (see definition 13). Crucially, the compatibility set of C is finite for any feature structure C since the heights and the lengths of the cords are limited.</Paragraph> <Paragraph position="22"> Definition 12. Given a non-reentrant feature structure C, a path pi = <F1,..., Fn> [?] PC and a natural number h, the compatibility set, G(C,pi,h), is defined as the set of all feature structures A such that C unionsq A negationslash= latticetop, pi [?] PA, and |Ps(A,pi) |[?] h.</Paragraph> <Paragraph position="23"> The compatibility set is defined for a feature structure and a given path (when h is taken to be the grammar height). We now define two similar sets, FH and UH, for a given FS, independently of a path. When rules of a one-reentrant unification grammar are mapped to LIG rules (definition 14), FH and UH are used to define heads of fixed and unbounded LIG rules, respectively. A single unification rule is mapped to a set of LIG rules, each with a different head. The stack of the head is some member of the sets FH and UH. Each such member is a prefix of the stack of potential elements of sentential forms that the LIG rule can be applied to.</Paragraph> <Paragraph position="24"> Definition 13. Let C be a non-reentrant feature structure and h be a natural number. Then:</Paragraph> <Paragraph position="26"> This accounts for the second difference between LIG and one-reentrant UG, namely rule application. We now briefly illustrate our account of the last difference, propagation of information in rules. In UG1r information is shared between the rule's head and a single element in its body. Let ru = <C0,...,Cn> be a reentrant unification rule in which the path ue, leaving the e-th element of the body, is reentrant with the path u0 leaving the head. This rule is mapped to a set of LIG rules, corresponding to the possible rule heads induced by the compatibility set of C0. Let r be a member of this set, and let X0 and Xe be the head and the e-th element of r, respectively. Reentrancy in ru is modeled in the LIG rule by copying the stack from X0 to Xe. The major complication is the contents of this stack, which varies according to the cord representations of C0 and Ce and to the reentrant paths.</Paragraph> <Paragraph position="27"> Summing up, in a LIG simulating a one-reentrant UG, FSs are represented as stacks of symbols. The set of stack symbols Vs, therefore, is defined as a set of height bounded non-reentrant FSs. Also, all the features of the UG are stack symbols. Vs is finite due to the restriction on FSs (no reentrancies and height-boundedness). The set of terminals, Vt, is the words of the UG. There are exactly two non-terminal symbols, S (the start symbol) and N.</Paragraph> <Paragraph position="28"> The set of rules is divided to four. The start rule only applies once in a derivation, simulating the situation in UGs of a rule whose head is unifiable with the start symbol. Terminal rules are a straight-forward implementation of the lexicon in terms of LIG. Non-reentrant rules are simulated in a similar way to how rules of a non-reentrant UG are simulated by CFG (section 3). The major difference is the head of the rule, X0, which is defined as explained above. One-reentrant rules are simulated similarly to non-reentrant ones, the only difference being the selected element of the rule body, Xe, which is defined as follows.</Paragraph> <Paragraph position="29"> Definition 14. Let ug2lig be a mapping of UG1r to LIGS, such that if Gu = <Ru,As,L> [?] UG1r</Paragraph> <Paragraph position="31"> maxHt(Gu)}, and Rli is defined as follows:3 1. S[ ] - N[Ps(As,e)] 2. For every w [?] WORDS such that L(w) = {C0} and for every pi0 [?] PC0, the rule N[Ps(C0,pi0)] - w is in Rli.</Paragraph> <Paragraph position="32"> 3. If <C0,...,Cn> [?] Ru is a non-reentrant rule, then for every X0 [?] LIGHEAD(C0) the rule X0 - N[Ps(C1,e)]...N[Ps(Cn,e)] is in Rli.</Paragraph> <Paragraph position="33"> 4. Let ru = <C0,...,Cn> [?] Ru and (0,u0) rusquiggleleftright (e,ue), where 1 [?] e [?] n. Then for every is in Rli, where Xe is defined as follows. Let pi0 be the base path of X0 and A be the base feature structure of X0. Applying the rule ru to A, define (<A> ,0) unionsq (ru,0) =</Paragraph> </Section> </Section> class="xml-element"></Paper>