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<?xml version="1.0" standalone="yes"?> <Paper uid="C92-4171"> <Title>I)YNAMICS, DEPENI)ENCY GRAMMAR AND INCREMENTAL INTERPRETATION*</Title> <Section position="3" start_page="0" end_page="0" type="metho"> <SectionTitle> 1 Dynamics </SectionTitle> <Paragraph position="0"> Dynantics can roughly be described ms the study (ff systems which consist of a set of states (cognitive, physical etc.) and a family of binary lmnsil~on relalionships between states, corresponding to actions which can be performed to change from one state to another (van Benthem, 1990).</Paragraph> <Paragraph position="1"> This paper introduces a notion of dynamic yram..</Paragraph> <Paragraph position="2"> Slat, where each word ill a sentence is treated a~s an action which ha.s the potential to produce a change in state, and each state encodes (in some form) the syntactic or semantic dependencies of the words which |lave been absorbed so far. There is no requirement for tile number of states to he finite. (ln fact, since dependency grammar allows centre embedding of arbitrary depth, the corresponding dynamic grammar provides an unlimited number of states).</Paragraph> <Paragraph position="3"> Dynamic grammars are specified using very simple logics, and a sentence is accepted ,~s grammatical if and only if there is some proof that it perforn~s a transition between some suitable initial and final *This resen.rch w~.s nUplmrted by ml SERC research fellowship.</Paragraph> <Paragraph position="4"> states, It is worth noting at this early stage that dynamic grammars are not lexicalised rehashes of Augmented Transition Networks (Woods, 1973). A'I'Ns use a finite number of states combined with a recursion mechanism, and act ea'~entially in the same way ms a top down parser. They are not particularly suited to increment,'d interpretation.</Paragraph> <Paragraph position="5"> To get an idea of how logics (instead of the more usual algebra.s) can be used to specify dynamic systenLs in general, it is worth considering a reformulation of the {bllowing finite state machine (FSM):</Paragraph> <Paragraph position="7"> This accepts .'~ grammatical any string which maps from the initial state, 0, to the final state, 3 (i.c.</Paragraph> <Paragraph position="8"> strings of the form: nb*eb). The FSM cart be reformulated using a logic where the notation,</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> StateO Str Statel </SectionTitle> <Paragraph position="0"> is used to state that the string, Str, perfornm a transition from StateO to Statel. The axioms (or atomic proofs) in tile logic are provided by the transitions peribrnted by the individual letters. Thus the following are mssumed, t 0 &quot;a&quot; t 1 &quot;b&quot; I 2 &quot;b&quot; 3 1 &quot;c&quot; 2 The transitions given by the single letter strings are put together using a deduction rule, Sequencing, 2 which states that, provided there is a proof that String, takes us from some state, So, to a state S t and a proof that Stringb takes us from St to $2, then there is a proof that the concatenation of the strings takes us from S0 to $2. The rule puts to aether strings of letters if the final state reached by tile first string matches an initial state for tile secontl string. For example, the rule may be instantiated as: A string is grammatical according to the logic if and only if it is possible to construct a proof of the statement 0 Str 3 using the axioms and the Sequencing Rule. For example, the string &quot;abbcb&quot; performs the transitions, l &quot;a&quot; is a ~trin~ coxudsting of the single letter, a. 2Notation: capital lettet~ will be used to denote variables throughout this paper. 'a' will be used to denote eoncatena~ tion. For example, if Stringa = &quot;kl&quot; mad String b = &quot;atilt&quot;, then StrlngaeStrlngb = ukltllllll.</Paragraph> <Paragraph position="1"> ACRES DE C()LING. 92, NAN-rES. 23-28 AOt'rr 1992 l 0 9 5 l)Roe, ov COLING-92, NA~I'ES AUG. 23-28, 1992 and has the following proof, amongst others:</Paragraph> <Paragraph position="3"> Each leaf of the tree is an axiom, and the subproofs are put together using instantiations of the Sequencing Rule.</Paragraph> </Section> </Section> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 2 Lexiealised Dependency Grammar </SectionTitle> <Paragraph position="0"> &quot;lYaditional dependency grammar is not concerned with constituent structure, but with links between individual words. For example, an analysis of the sentence John thought Mary showed Ben ~o Sue might be represented as follows: John thought Mary showed Ben to Sue The word thought is the head of the whole sentence, and it has two dependents, John and showed, showed is the head of the embedded sentence, with three dependents, Mary, Ben and to. A dependency graph is said to respect adjacency if no word is separated from its head except by its own dependents, or by another dependent of the same head and its dependents (i.e.</Paragraph> <Paragraph position="1"> there are no crossed links). Adjacency is a reasonably standard restriction, and has been proposed as a universal principle e.g. by Hudson (1988).</Paragraph> <Paragraph position="2"> Given adjacency, it is possible to extract bracketed strings (mid hence a notion of constituent structure) by grouping together each head with its dependents (and the dependents of its dependents). For example, the sentence above gets the bracketing: \[John thought \[Mary showed Ben \[to Sue\]\]\] A noun phrase can be thought of as a noun plus all its dependents, a sentence as a verb plus all its dependents. null In this paper we will assume adjacency, and, for simplicity, that dependents are fixed in their order relative to the head and to each other. Dependency grammars adopting these assumptions were formalised by Gaifman (Hays, 1964). Lexicalisation is relatively trivial given this formalisation, and the work on embedded dependency grammar within categorial grammar (Barry and Picketing, 1990).</Paragraph> <Paragraph position="3"> Lexiealised Dependency Grammar (LDG) treats each head as aflmction. For example, the head lhought is treated as a function with two arguments, a noun phrase and a sentence. Constituents are formed by combining functions with all their arguments. The example above gets the following bracketing and tree structure: \[John thought \[Mary showed Ben \[to Sue\]\]\]</Paragraph> <Paragraph position="5"> The tree structure is particularly flat, with all arguments of a function appearing at tile same level (this contrasts with standard phrase structure analyses where the subject of showed would appear at a different level from its objects).</Paragraph> <Paragraph position="6"> Lexical categories are feature structures with three main features, a base type (the type of the constituent formed by combining the lexical item with its arguments), a list of arguments which must appear to the left, and a list of arguments which must appear to the right. The arguments at the top of the lists must appear closest to the functor. For example, showed has the lexical entry, I 1 showed : llnp) L=(-p, ppl J and can combine with an np on its left and with an np and then a pp on its right to form a sentence.</Paragraph> <Paragraph position="7"> When left and right argument lists are empty, categories are said to be saturated, and may be written as their base type i.e. \[X)\] isidenticaltoX.</Paragraph> <Paragraph position="8"> L~clj A requirement inherited from dependency grammar is for arguments to be saturated categories, a LDGs will be specified more formally in Section 4.</Paragraph> <Paragraph position="9"> It is worth outlining the differences between the categories in LDG and those in a directed categorial grammar. Firstly, in LDG there is no specification of whether arguments to the right or to the left of the functor should be combined with first. Thus, the category, I(Y) , maps to both (X\Y)/Z and LrlZ~ J (X/Z)\Y, 4 Seeondly, arguments in LDG must be saturated, so there can be no functions of functions. 5 aIn dependency granunar it is not ponaible to specify that a head requirea a dependent with only some, but not all of its dependenta.</Paragraph> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> ACRES DE COLING-92. NANTES, 23-28 AOITI&quot; 1992 1 096 PROC. OF COLING-92, NANTES, AUO. 23-28, 1992 3 Dynamic Dependency Grammar </SectionTitle> <Paragraph position="0"> Lexicali~d Dependency Grammar can be reformulated as the dynamic grammar, Dynamic Dependency Grammar (DDG). Each state in DDG encodes the syntactic context, and is labelled by the typc of the string absorbed so far. For example, a possible set of transitions for the string of words &quot;Sue saw Ben&quot; is as follows: ' Ii 1 S Sue \]J'etl L~lsl j ~l ~,up, , L~-Inplj The state after absorbing &quot;Sue saw&quot; is of type sentence mi~ing a noun phrase, and that after absorblug &quot;Sue saw Ben&quot; is of type sentence.</Paragraph> <Paragraph position="1"> States are labelled by complex categories which arc similar to tile lexical categories of LDG, but without the restriction that arguments must be saturated (for example, the state after absorbing &quot;Sue&quot; has an unsaturated argument on its right list). A string of words, Str, is grammatical provided tile following statement can be proven:</Paragraph> <Paragraph position="3"> The initial state is labelled with an identity type i.e.</Paragraph> <Paragraph position="4"> a sentence missing a sentence. This cat, be thought of as a context in whic}~ a sentence is expected, or as a suitable type for a string of words of length zero.</Paragraph> <Paragraph position="5"> The final state is just of type sentence.</Paragraph> <Paragraph position="6"> DDG is specified nsing a logic consisting of a set of axioms and a deduction rule. The logic is similar, but more general, than that used in Axiomatic Grammar (Milward, 1990)f The deduction rule is again called Sequencing. The rule is identical in form to the Sequencing Rule used in the reformulation of the FSM, though here it puts together strings of words rather tiian strings of letters. The rule is as follows, ~ and is restricted to non-empty strings, s menks h&s been developed, and this also e2tn be reformulated as a dynamic ~,m*mma&quot; (Milward, 1992).</Paragraph> <Paragraph position="7"> 6Axiomatic Grmnm~r is a particular dynamic grammar designed for English, which take~ relationslfips between states tm a primary phenomenon, to be justified solely by linguistic data (rather thmt by an existing formalism such as dependency granmlar).</Paragraph> <Paragraph position="8"> ZHere 'o' concatenates strings of words e.g.</Paragraph> <Paragraph position="9"> &quot;John&quot;o&quot;~leepa&quot; = &quot;John sleeps&quot;. SThis re~trictlon is not actually necessary as far as the equivalence between LDGs and DDGa is concerned. However its inclusion makes it trivial to show certain fontud properties of DDGs, such a.~ termination of proofs.</Paragraph> <Paragraph position="10"> The set of axioms is infinite since we need to consider transitions between an arbitrary number of categories. 9 The set can be described using just two axiom schemata, Prediction and Application. Prediction is given below, but is best understood by considering various instantiations) deg</Paragraph> <Paragraph position="12"> Prediction is usually used when the category of tile word encomitered does not match the category expected by the current state. Consider the following instantiation:</Paragraph> <Paragraph position="14"> The current state expects a seutence and encounters a noun phrase with \]exical entry Sue:rip. The resulting state expects a sentence missing a noun phrase on its left e.g. a verb phrase.</Paragraph> <Paragraph position="15"> Application gets its name from its similarity to function application (though it actually plays the role of both application and composition). The</Paragraph> <Paragraph position="17"> An example instantiation is when a noun phrase i~ both expected and encountered e.g. \[sl</Paragraph> <Paragraph position="19"> Given a word and a particular current state, the only non-determinism in forming a resnlting state is due to lexical ambiguity or from a choice between using Prediction or Application (Prediction is possible whenever Application is). Non-determinism is gendegAn infinite nmnber of distinguishable stat~ is required to deal with centre embedding.</Paragraph> <Paragraph position="21"> erally kept low due to states being labelled by types as opposed to explicit tree structures. This is easiest to illustrate using a verb final language. Consider a pseudo English where the strings, &quot;Ben Sue saw&quot; and &quot;Ben Sue sleeps believes&quot; are acceptable sentences, and have the following LDG analyses:</Paragraph> <Paragraph position="23"> Despite the differences in the LDG tree structures, the initial fragment of each sentence, &quot;Ben Sue&quot;, can ..be treated identically by the corresponding DDG.</Paragraph> <Paragraph position="24"> The proof of the transition performed by the string &quot;Ben Sue&quot; involves two applications of Prediction put together using the Sequencing Rule. The transitions are as follows:</Paragraph> <Paragraph position="26"> The transitions for the two sentences diverge when we consider the words saw and sleeps. In the former case, Application is used, in the latter, Prediction then Application.</Paragraph> <Paragraph position="27"> Efficient parsing algorithms can be based upon DDGs due to this relative lack of non-determinism in choosing between states. H The simplest algorithm is merely to non-deterministically apply Prediction and Application to the initial category. Derivations of algorithms from more complex logics, and the use of normalised proofs and equivalence classes of proofs are described in Milward (1991).</Paragraph> <Paragraph position="28"> 4 LDGs --+ DDGs An LDG can he specified more formally as follows: 1. A finite set of base types To, .. &quot;In (such as s, up, and pp) llDetermlnism can also be increased by restricting the axioms according to the properties of a particular lexicon. For example, there is no point predicting categories missing two noun phrases to the left when pansing English. 2. An infinite set of lexical categories of the form, XL1 where X is a base type, and L and It are rR\] lists of base types. When L and R are empty, a category is identical to its base type, X 3. A finite lexicon, L, which assigns lexical categories to words 4. A distinguished base type, To. A string is grammatieal iff it has the category, To 5. A combination rule stating that, F \] if W has category, 1( Ti,.., TI) Jr( 7,+a ..... Ti+i) and String1 has category T1, String2 has category 7~ etc.</Paragraph> <Paragraph position="29"> then the string formed by concatenating String1, .. ,String~, &quot;W', Stringi+l, .. ,String/+j has category X corresponding DDG is as follows: ....</Paragraph> <Paragraph position="30"> L~Rj where X is a base type, and L and Yt are lists of categories 2. Two axiom schemata, Application and Prediction null 3. The lexicon, L (as above) 4. One deduction rule, Sequencing I \] 5. A distinguished pair of categories, ll) , 7~</Paragraph> <Paragraph position="32"> where To is as above. A string, Str, is grammatical iff it is possible to prove: \[</Paragraph> <Paragraph position="34"> A proof that any DDG is strongly equivalent to its corresponding LDG is given by Milward (1992). The proof is split into a soundness proof (that a DDG accepts the same strings of words and assigns them corresponding analysesl2), and a completeness proof (that a DDG accepts whatever strings are accepted by the corresponding LDG).</Paragraph> </Section> <Section position="6" start_page="0" end_page="0" type="metho"> <SectionTitle> 5 Incremental Interpretation </SectionTitle> <Paragraph position="0"> It is possible to augment each state with a semantic type and a semantic value. Adopting a 'standard' A-calculus semantics (c.f. Dowty et al, 1981) we obtain the following transitions for the string &quot;Sue saw&quot;: 12For this purpose, it is convenient to treat an analysis in a DDG as the traasition~ performed by each word. Each analysis is a label for all equivalence Class of proofs.</Paragraph> <Paragraph position="1"> The semantic types can generally be extracted fronl the syntactic types. The base types s and ap map to the semantic types l and e, stmtding for trutb-value and enl, ity respectively. Categories with argmnents map to corresponding flmctional types.</Paragraph> <Paragraph position="2"> Provided a close mapping between syntactic and semantic types is assumed, the addition of semantic values to the axiom schemata is relatively trivial, as is the addition of semantic vahtes to the lexicon. For example, the semantic value given to the verb saw is AYAX.saw'(X,Y), which has type e~(e-~t).</Paragraph> <Paragraph position="3"> It is worth contrasting the approach taken here with two otller al)proaches to incremental interpretation. Tim first is that of Pnlman (19851. Pulman's approach separates syntactic and senmntie analysis, driving semantic combinations off the actions of a parser for a phrase structnre grammar. The approach was important ill showing that hierarchical syntactic analysis and word by word incremental interpretation are not incompatihle. The second approach is that of Ades and Steedman (19821 who incorporate conlposition rules directly into a categorim granlmar. This allows a certain amount of incremental interpretation dim to tile possibility of forming constitnents for some initial substrings, flowew:r, the incorporation of composition into the grammar itself does haw: sonic unwanted side effects when nfixed with a use of functions of fimctions. For exampie, if the two types, N/N and N/N are composed to give tile type N/N, then this can be modified by an adjectival modifier of type (N/N)/(N/N). Thus, the phrase the very old green car&quot; can get the bracketing, \[the \[very \[old green\]\] car\]. Although tile Application schema used in DDGs does compose functions together, DI)Gs have identical strong generative capacity to the LDGs they are based upon (the coverage of the grammars is identical, and tile analyses are ill a one-to-one correspondence). 13</Paragraph> </Section> class="xml-element"></Paper>