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<?xml version="1.0" standalone="yes"?> <Paper uid="P86-1038"> <Title>A LOGICAL SEMANTICS FOR FEATURE STRUCTURES</Title> <Section position="4" start_page="257" end_page="257" type="metho"> <SectionTitle> 3 Logical Formulas for </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="257" end_page="257" type="sub_section"> <SectionTitle> Feature Structures </SectionTitle> <Paragraph position="0"> The feature structure of Figure 1 can also be represented by a type of logical formula: die = case : (hOrn V acc)</Paragraph> <Paragraph position="2"> This type of formula differs from standard propositional logic in that a theoretically unlimited set of atomic values is used in place of boolean values. The labels of attributes bear a superficial resemblance to modal operators. Note that no information is added or subtracted by rewriting the feature matrix of Figure 1 as a logical formula.</Paragraph> <Paragraph position="3"> These two forms may be regarded as notational variants for expressing the same facts. While feature matrices seem to be a more appealing and natural notation for displaying linguistic descriptions, logical formulas provide a precise interpretation which can be useful for computational and mathematical purposes.</Paragraph> <Paragraph position="4"> Given this intuitive introduction we proceed to a more complete definition of this logic.</Paragraph> </Section> </Section> <Section position="5" start_page="257" end_page="261" type="metho"> <SectionTitle> 4 A Logical Semantics </SectionTitle> <Paragraph position="0"> As Pereira and Shieber \[11\] have pointed out, a grammatical formalism can be regarded in a way similar to other representation languages. Often it is useful to use a representation language which is disctinct from the objects it represents. Thus, it can be useful to make a distinction between the domain of feature structures and the domain of their descriptions. As we shall see, this distinction allows a variety of notational devices to be used in descriptions, and interpreted in a consistent way with a uniform kind of structure.</Paragraph> <Section position="1" start_page="257" end_page="258" type="sub_section"> <SectionTitle> 4.1 Domain of Feature Structures </SectionTitle> <Paragraph position="0"> The PATR-II system uses directed acyclic graphs (dags) as an underlying representation for feature structures. In order to build complex feature structures, two primitive domains are required: null</Paragraph> <Paragraph position="2"> The elements of both domains are symbols, usually denoted by character strings. Attribute I~ belt (e.g., acase~) are used to mark edges in a dag, and atoms (e.g., &quot;gen z) are used as primitive values at vertices which have no outgoing edges.</Paragraph> <Paragraph position="3"> A dag may also be regarded as a transition graph for a partially specified deterministic finite automaton (DFA). This automaton recognises strings of labels, and has final states which are atoms, as well as final states which encode no information. An automaton is formally described by a tuple</Paragraph> <Paragraph position="5"> where L is the set of labels above, 6 is a partial function from Q x L to Q, and where certain elements of F may be atoms from the set A. We require that ~ be connected, acyclic, and have no transitions from any final states.</Paragraph> <Paragraph position="6"> DFAs have several desirable properties as a domain for feature structures: 1. the value of any defined path can be denoted by a state of the automaton; 2. finding the value of a path is interpreted by running the automaton on the path string; 3. the automaton captures the crucial properties of shared structure: (a) two paths which axe unified have the same state as a value, (b) unification is equivalent to a statemerge operation; 4. the techniques of automata theory become available for use with feature structures. A consequence of item 3 above is that the dis- ,&quot; tinction between type identity and token identity it clearly revealed by an automaton; two objects are necessarily the same token, if and only if they are represented by the same state.</Paragraph> <Paragraph position="7"> One construct of automata theory, the Nerode relation, is useful to describe equivalent paths. If #q is an automaton, we let P(A) be the set of all paths of ~4, namely the set {z E L* : 5(q0, z) is defined }. The Nerode relation N(A) is the equivalence relation defined on paths of P(~) by</Paragraph> </Section> <Section position="2" start_page="258" end_page="259" type="sub_section"> <SectionTitle> letting 4.2 Domain of Descriptions: Logical Formulas </SectionTitle> <Paragraph position="0"> We now define the domain FML of logical formulas which describe feature structures. Figure 3 defines the syntax of well formed formulas. In the following sections symbols from the Greek alphabet axe used to stand for arbitrary formulas in FML. The formulas NIL and TOP axe intended to convey gno information z and ~inconsistent information s respectively. Thus, NIL corresponds to a unification variable, and TOP corresponds to unification failure. A formula l : ~b would indicate that a value has attribute l, which is itself a value satisfying the condition ~b.</Paragraph> <Paragraph position="1"> Conjunction and disjunction will have their ordinary logical meaning as operators in formulas. An interesting result is that conjunction can be used to describe unification. Unifying two structures requires finding a structure which has all features of both structures; the conjunction of two formulas describes the structures which satisfy all conditions of both formulas.</Paragraph> <Paragraph position="2"> One difference between feature structures and their descriptions should be noted. In a feature structure it is required that a particular attribute have a unique value, while in descriptions it is pouible to specify, using conjunction, several values for the same attribute, as in the formula s bj : (19e.so. : 3) ^ s bj: : A feature structure satisfying such a description will contain a unique value for the attribute, which can be found by unifying all of the values that are specified in the description.</Paragraph> <Paragraph position="3"> Formulas may also contain sets of paths, denoting equivalence classes. Each element of the set represents an existing path starting from the initial state of an automaton, and all paths in the set are required to have a common endpoint. If E = I< z >, < y >~, we will sometimes write E as < z >=< y >. This is the notation of PATR-II for pairs of equivalent paths. In subsequent sections we use E (sometimes with subscripts) to stand for a set of paths that belong to the same equivalence class.</Paragraph> </Section> <Section position="3" start_page="259" end_page="259" type="sub_section"> <SectionTitle> 4.3 Interpretation of Formulas </SectionTitle> <Paragraph position="0"> We can now state inductively the exact conditions under which an automaton Jl satisfies a formula: 1. A ~ NIL always; 2. 11 ~ TOP never; 3. /l ~ a C/=~ /I is the one-state automaton a with no transitions; 4. A ~ E C/=~ E is a subset of an equivalence class of N(~); 5. A ~ l : cb C/=~ A/l is defined and A/I ~ ~; where ~/I is defined by a subgraph of the automaton A with start state 5(qo, l), that is ira = (Q,L, 6, qo, F), then .~/l = (Q', L, 6, 6(qo, l), f'); where Qi and F' are formed from Q and F by removing any states which are unreachable from 6(q0, 0.</Paragraph> <Paragraph position="1"> Any formula can be regarded as a specification for the set of automata which satisfy it. In the case of conjunctive formulas (containing no occurences of disjunction) the set of automata satisfying the formula has a unique minimal element, with respect to subsumption.* For disjunctive formulas there may be several minimal elements, but always a finite number.</Paragraph> </Section> <Section position="4" start_page="259" end_page="261" type="sub_section"> <SectionTitle> 4.4 Calculus of Formulas </SectionTitle> <Paragraph position="0"> It is possible to write many formulas which have an identical interpretation. For example, the formulas given in the equation below are satisfied by the same set of automata.</Paragraph> <Paragraph position="1"> case : (gen V ace V dat) A case : ace = case : ace In this simple example it is clear that the right side of the formula is equivalent to the left side, and that it is simpler. In more complex examples it is not always obvious when two formulas are equivalent. Thus, we are led to state the laws of equivalence shown in Figure 4. Note that equivalence (26) is added only to make descriptions of cyclic structures unsatisfiable.</Paragraph> <Paragraph position="2"> 1A subsumption order can be defined for the domain of automata, just as it is defined for dags by Shieber \[15\]. A formal definition of subsurnption for this domain appears in \[12\].</Paragraph> <Paragraph position="4"> for any y such that 3z : z ~ El and zy E E2</Paragraph> </Section> </Section> <Section position="6" start_page="261" end_page="264" type="metho"> <SectionTitle> 5 Complexity of Disjunctive </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="261" end_page="261" type="sub_section"> <SectionTitle> Descriptions </SectionTitle> <Paragraph position="0"> To date, the primary benefit of using logical formulas to describe feature structures has been the clarification of several problems that arise with disjunctive descriptions.</Paragraph> </Section> <Section position="2" start_page="261" end_page="261" type="sub_section"> <SectionTitle> 5.1 NP-completeness of consistency </SectionTitle> <Paragraph position="0"> problem for formulas One consequence of describing feature structures by logical formulas is that it is now relatively easy to analyse the computational complexity of various problems involving feature structures. It turns out that the satisfiability problem for CNF formulas of propositional logic can be reduced to the consistency (or satisfiability) problem for formulas in FML. Thus, the consistency problem for formulas in FML is NPcomplete. It follows that any unification algorithm for FML formulas will have non-polynomial worst-case complexity (provided P ~ NP!), since a correct unification algorithm must check for consistency.</Paragraph> <Paragraph position="1"> Note that disjunction is the source of this complexity. If disjunction is eliminated from the domain of formulas, then the consistency problem is in P. Thus systems, such as the original PATR-II, which do not use disjunction in their descriptions of feature structures, do not have to contend with this source of NP-completeness.</Paragraph> </Section> <Section position="3" start_page="261" end_page="261" type="sub_section"> <SectionTitle> 5.2 Disjunctive Normal Form </SectionTitle> <Paragraph position="0"> A formula is in disjt, neti,~s normal form (DNF) if and only if it has the form ~1 V ... v ~bn, where each ~i is either 1. sEA 2. ~bx A ... A ~bm, where each ~bl is either (a) lx : ... : lk : a, where a E A, and no path occurs more than once (b) \[< pl >,...,< p~ >\], where each p~ E L*, and each set denotes an equivalence class of paths, and all such sets disjoint. The formal equivalences given in Figure 4 allow us to transform any satisfiable formula into its disjunctive normal form, or to TOP if it is not satisfiable. The algorithm for finding a normal form requires exponential time, where the exponent depends on the number of disjunctions in the formula (in the worst case).</Paragraph> </Section> <Section position="4" start_page="261" end_page="262" type="sub_section"> <SectionTitle> 5.3 Avoiding expansion to DNF </SectionTitle> <Paragraph position="0"> Most of the systems which are currently used to implement unification-based grammars depend on an expansion to disjunctive normal form in order to compute with disjunctive descriptions. 2 Such systems are exemplified by Definite Clause Grammar \[10\], which eliminates disjunctive terms by multiplying rules which contain them into alternative clauses. Kay's parsing procedure for Functional Unification Grammar \[8\] also requires expanding functional descriptions to DNF before they are used by the parser. This expansion may not create much of a problem for grammars containlng a small number of disjunctions, but if the grammar contains 100 disjunctions, the expansion is clearly not feasible, due to the exponential sise of the DNF.</Paragraph> <Paragraph position="1"> Ait-Kaci \[1\] has pointed out that the expansion to DNF is not always necessary, in work with type structures which are very similar to the feature structures that we have described here. Although the NP-completeness result cited above indicates that any unification algorithm for disjunctive formulas will have exponential complexity in the worst case, it is possible to develop algorithms which have an average complexity that is less prohibitive. Since the exponent of the complexity function depends on the number of disjunctions in a formula, one obvious way to improve the unification algorithm is to reduce the number of disjunctions in the formula be/ors ezpan.sion to DNF. Fortunately the unification of two descriptions frequently results in a reduction of the number of alternatives that remain consistent. Although the fully expanded formula may be required as a final result, it is expedient to delay the expansion whenever possible, until after any desired unifications are performed.</Paragraph> <Paragraph position="2"> The algebraic laws given in Figure 4 provide a sound basis for simplifying formulas containing disjunctive values without expanding to DNF. Our calculus differs from the calculus of Ait-Kaci by providing a uniform set of equivalences for formulas, including those that contain disjunction. These equivalences make it possible to ~ 2One exception is Karttunen's implementation, which was described in Section 2, but it handles only value disjunctions, and does not handle non-local path values embedded within disjunctions.</Paragraph> <Paragraph position="3"> eliminate inconsistent terms before expanding to DNF. Each term thus eliminated may reduce, by as much as half, the sise of the expanded formula.</Paragraph> </Section> <Section position="5" start_page="262" end_page="263" type="sub_section"> <SectionTitle> 5.4 Representing Non-local Paths </SectionTitle> <Paragraph position="0"> The logic contains no direct representation for non-local paths of the type described in Section 2. The reason is that these cannot be interpreted without reference to the global context of the formula in which they occur. Recall that in Functional Unification Grammar a non-local path denotes the value found by extracting each of the attributes labeled by the path in successively embedded feature structures, beginning with the entire structure currently under consideration. Stated formally, the desired interpretstion of I :< p > is A~l:<p> in the context of~</Paragraph> <Paragraph position="2"> This interpretation does not allow a direct comparison of the non-local path value with other values in the formula. It remains an unknown quantity unless the environment is known.</Paragraph> <Paragraph position="3"> Instead of representing non-local paths directly in the logic, we propose that they can be used within a formula as a shorthand, but that all paths in the formula must be expanded before any other processing of the formula. This path expansion is carried out according to the equiva~ lences 9 and 6.</Paragraph> <Paragraph position="4"> After path expansion all strings of labels in a formula denote transitions from a common origin, so the expressions containing non-local paths can be converted to the equivalence class notation, using the schema 11 :... :In :<p> = \[<11 .... ,In >,<p >\].</Paragraph> <Paragraph position="5"> Consider the passive voice alternative of the description of Figure 2, shown here in Figure 5. This description is also represented by the first formula of Figure 6. The formulas to the right in Figure 6 are formed by 1. applying path expansion, 2. converting the attributes containing non-local path values to formulas representing equivalence classes of paths.</Paragraph> <Paragraph position="6"> By following this procedure, the entire functional description of Figure 2 can be represented by the logical formula given in Figure 7.</Paragraph> <Paragraph position="8"> equivalence classes to simplify the result. Consider unifTing X with the description Y = actor : case : nominative.</Paragraph> <Paragraph position="9"> The commutative law (10) makes it possible to unify Y with any of the conjuncts of X. If we unify Y with the disjunction which contains the vo/ce attributes, we can use the distributive law (16) to unify Y with both disjuncts. When Y is unified with the term containing \[< adjunct obj >, < actor >\], the equivalence (22) specifies that we can add the term adjunct : obj : case : nominative.</Paragraph> <Paragraph position="10"> This term is incompatible with the term adjunct : obj : case : objective, and by applying the equivalences (6, 4, 1, and 2) we can transform the entire disjunct to TOP. Equivalence (8) specifies that this disjunction can be eliminated. Thus, we are able to use the path equivalences during unification to reduce the number of disjunctions in a formula without expanding to DNF.</Paragraph> <Paragraph position="11"> Note that path expansion does not require an expansion to full DNF, since disjunctions are not multiplied. While the DNF expansion of a formula may be exponentially larger than the original, the path expansion is at most quadratically larger. The size of the formula with paths expanded is at most n x p, where n is the length of the original formula, and p is the length of the longest path. Since p is generally much less than n the size of the path expansion is usually not a very large quadratic.</Paragraph> </Section> <Section position="6" start_page="263" end_page="264" type="sub_section"> <SectionTitle> 5.5 Value Disjunction and General Disjunction </SectionTitle> <Paragraph position="0"> The path expansion procedure illustrated in Figure 6 can also be used to transform formulas containing value disjucntion into formulas containing general disjunction. For the reasons given above, value disjunctions which contain non-local path expressions must be converted into general disjunctions for further simplification.</Paragraph> <Paragraph position="1"> While it is possible to convert value disjunctions into general disjunctions, it is not always possible to convert general disjunctions into value disjunctions. For example, the first disjunction in the formula of Figure 7 cannot be converted into a value disjunction. The left side of equivalence (9) requires both disjuncts to begin with a common label prefix. The terms of these two disjuncts contain several different prefixes (voice, actor, subj, goat, and adjunct), so they cannot be combined into a common value.</Paragraph> <Paragraph position="2"> Before the equivalences of section 4 were formulated, the first author attempted to implement a facility to represent disjunctive feature structures with non-local paths using only value disjunction.</Paragraph> <Paragraph position="3"> It seemed that the unification algorithm would be simpler if it had to deal with disjuncti+ns only in the context of attribute values, rather than in more general contexts. While it w~ possible to write down grammatical definitions using only value disjunction, it was very difficult to achieve a correct unification algorithm, because each non-local path was much like an unknown variable. The logical calculus presented here clearly demonstrates that a representation of general disjunction provides a more direct method to determine the values for non-local paths.</Paragraph> </Section> </Section> <Section position="7" start_page="264" end_page="264" type="metho"> <SectionTitle> 6 Implementation </SectionTitle> <Paragraph position="0"> The calculus described here is currently being implemented as a program which selectively applies the equivalences of Figure 4 to simplify formulas. A strategy (or algorithm) for simplifying formulas corresponds to choosing a particular order in which to apply the equivalences whenever more than one equivalence matches the form of the formula. The program will make it possible to test and evaluate different strategies, with the correctness of any such strategy following directly from the correctness of the calculus. While this program is primarily of theoretical interest, it might yield useful improvements to current methods for processing feature structures.</Paragraph> <Paragraph position="1"> The original motivation for developing this treatment of feature structures came from work on an experimental parser based on Nigel \[9\], a large systemic grammar of English. The parser is being developed at the USC/Information Sciences Institute by extending the PATR-II system of SRI International. The systemic grammar has been translated into the notation of Functional Unification Grammar, as described in \[6\]. Because this grammar contains a large number (several hundred) of disjunctions, it has been necessary to extend the unification procedure so that it handles disjunctive values containing non-local paths without expansion to DNF. We now think that this implementation of a relatively large grammar can be made more tractable by applying some of the transformations to feature descriptions which have been suggested by the logical calculus.</Paragraph> </Section> class="xml-element"></Paper>