File Information
File: 05-lr/acl_arc_1_sum/cleansed_text/xml_by_section/metho/94/p94-1035_metho.xml
Size: 21,567 bytes
Last Modified: 2025-10-06 14:13:55
<?xml version="1.0" standalone="yes"?> <Paper uid="P94-1035"> <Title>An Attributive Logic of Set Descriptions Set Operations</Title> <Section position="4" start_page="0" end_page="255" type="metho"> <SectionTitle> 2 The logic of Set descriptions </SectionTitle> <Paragraph position="0"> In this section we provide the semantics of feature terms augmented with set descriptions and various constraints over set descriptions. We assume an alphabet consisting of x, y, z,... 6 )2 the set of variables; f,g,... E Y: the set of relation symbols; el, c2,... E C the set of constant symbols; A,B,C,... 6 7 ) the set of primitive concept symbols and a,b,... 6 .At the set of atomic symbols. Furthermore, we require that /,T E T'.</Paragraph> <Paragraph position="1"> The syntax of our term language defined by the following BNF definition:</Paragraph> <Paragraph position="3"> where S, T, T1,..., Tn are terms; a is an atom; c is a constant; C is a primitive concept and f is a relation symbol.</Paragraph> <Paragraph position="4"> The interpretation of relation symbols and atoms is provided by an interpretation Z =</4I I > where/41 is an arbitrary non-empty set and I is an interpretation function that maps : 1. every relation symbol f * ~&quot; to a binary relation fl C_/4I x/4I 2. every atom a * .At to an element a I * bl x Notation: * Let if(e) denote the set {e'\[ (e,e') * if} * Let fI(e) T mean fl(e) = 0 Z is required to satisfy the following properties : 1. if al ~ a2 then all # hi2 (distinctness) 2. for any atom a * At and for any relation f * ~&quot; there exists no e * U 1 such that (a, e) * fl (atomicity) For a given interpretation Z an Z-assignment a is a function that maps : 1. every variable x * \]2 to an element a(x) * 141 2. every constant c * C to an element a(c) */41 such that for distinct constants Cl, c2 : a(cl) # a(c2) 3. every primitive concept C * 7 ) to a subset a(C) C /41 such that:</Paragraph> <Paragraph position="6"> The interpretation of terms is provided by a denotation function \[\[.\]\]z,a that given an interpretation Z and an Z-assignment a maps terms to subsets of/41.</Paragraph> <Paragraph position="7"> The function \[.\]\]z,a is defined as follows :</Paragraph> <Paragraph position="9"> The above definitions fix the syntax and semantics of every term.</Paragraph> <Paragraph position="10"> It follows from the above definitions that:</Paragraph> <Paragraph position="12"> Although disjoint union is not a primitive in the logic it can easily be defined by employing set disjointness and set union operations: f: g(x) eJ h(y) =de/ g(x) # h(y) ~q f: g(x) U h(y) Thus disjoint set union is exactly like set union except that it additionally requires the sets denoted by g(x) and h(y) to be disjoint.</Paragraph> <Paragraph position="13"> The set-valued description of the subcategorisation principle can now be stated as given in example (3).</Paragraph> </Section> <Section position="5" start_page="255" end_page="256" type="metho"> <SectionTitle> (3) Subcategorisation Principle SYN,LOC Y \]\] TRS X n \[HL-DTR\[SYN\[LOC\[SUBCAT c-dtrs(X) ~ subcat(Y) </SectionTitle> <Paragraph position="0"> The description in (3) simply states that the subcat value of the H-DTR is the disjoint union of the subcat value of the mother and the values of C-DTRS. Note that the disjoint union operation is the right operation to be specified to split the set into two disjoint subsets.</Paragraph> <Paragraph position="1"> Employing just union operation would not work since</Paragraph> <Paragraph position="3"> it would permit repetition between members of the SUBCAT attribute and C-DTRS attribute.</Paragraph> <Paragraph position="4"> Alternatively, we can assume that N is the only multi-valued relation symbol while both SUBCAT and C-DTRS are single-valued and then employ the intuitively appealing subcategorisation principle given in (4).</Paragraph> </Section> <Section position="6" start_page="256" end_page="256" type="metho"> <SectionTitle> (4) Subcategorisation Principle TRS \[H-DTRISYNILOCISUBCATIN N(X) ~ N(Y) C-DTRS X </SectionTitle> <Paragraph position="0"> With the availability of set operations, multi-valued structures can be incrementally built. For instance, by employing union operations, semantic indices can be incrementally constructed and by employing membership constraints on the set of semantic indices pronoun resolution may be carried out.</Paragraph> <Paragraph position="1"> The set difference operation f : g(y) - h(z) is not available from the constructs described so far. However, assume that we are given the term x R f : g(y) - h(z) and it is known that hZ(~(z)) C_ gZ(a(y)) for every interpretation 27, (~ such that \[x R f : g(y)- h(z)~ z,~ C/ 0. Then the term x N f : g(y) - h(z) (assuming the obvious interpretation for the set difference operation) is consistent iff the term y \[\] g : f(x) t~ h(z) is consistent. This is so since for setsG, F,H:G-F=HAFCG i\]:f G = F W H. See figure 1 for verification.</Paragraph> </Section> <Section position="7" start_page="256" end_page="258" type="metho"> <SectionTitle> 3 Consistency checking </SectionTitle> <Paragraph position="0"> To employ a term language for knowledge representation tasks or in constraint programming languages the minimal operation that needs to be supported is that of consistency checking of terms.</Paragraph> <Paragraph position="1"> A term T is consistent if there exists an interpretation 2: and an/:-assignment (~ such that \[T\] z'a ~ 0. In order to develop constraint solving algorithms for consistency testing of terms we follow the approaches in \[Smolka, 1992\] \[Hollunder and Nutt, 1990\].</Paragraph> <Paragraph position="2"> A containment constraint is a constraint of the form x = T where x is a variable and T is an term.</Paragraph> <Paragraph position="3"> Constraint simplification rules - I x=yACs (SEquals) x = y A \[x/y\]Cs if x ~ y and x occurs in Cs (SConst) x=~Ay=~ACs x=yAx=~ACs where ~ ranges over a, c.</Paragraph> <Paragraph position="4"> (SFeat) x= f :yAx= F :zZACs x=/:yAy= ACs where F ranges over f, 3f, Vf (SExists) x=gf:yAx=Vf:zAC~ x= f :yAy=zACs (SForallE) x = V__\] : C A x = 9f : y A C~_ x =V/: CAx = 3/: yAy = CAC~ if C ranges over C, -~C, -~a, --c, -~z and Cs Vy =C.</Paragraph> <Paragraph position="5"> In addition, for the purposes of consistency checking we need to introduce disjunctive constraints which are of the form x -- Xl U ... U x,~.</Paragraph> <Paragraph position="6"> We say that an interpretation Z and an/-assignment a satisfies a constraint K written 27, a ~ K if.</Paragraph> <Paragraph position="8"> xi:l <i<n.</Paragraph> <Paragraph position="9"> A constraint system Cs is a conjunction of constraints. null We say that an interpretation 27 and an Z-assignment a satisfy a constraint system Cs iffZ, a satisfies every constraint in Cs.</Paragraph> <Paragraph position="10"> The following lemma demonstrates the usefulness of constraint systems for the purposes of consistency checking.</Paragraph> <Paragraph position="11"> Lemma 1 An term T is consistent iff there exists a variable x, an interpretation Z and an Z-assignment a such that Z, a satisfies the constraint system x = T. Now we are ready to turn our attention to constraint solving rules that will allow us to determine the consistency of a given constraint system.</Paragraph> <Paragraph position="13"> where F ranges over f, Vf</Paragraph> <Paragraph position="15"> if C ranges over C, -~C, -~a, -~c, -~z and there exists xi : 1 < i < n such that Cs ~1 xi = C.</Paragraph> <Paragraph position="17"> We say that a constraint system C8 is basic if none of the decomposition rules (see figure 2) are applicable to c8.</Paragraph> <Paragraph position="18"> The purpose of the decomposition rules is to break down a complex constraint into possibly a number of simpler constraints upon which the constraint simplification rules (see figures 3, 4 and 5 ) can apply by possibly introducing new variables.</Paragraph> <Paragraph position="19"> The first phase of consistency checking of a term T consists of exhaustively applying the decomposition rules to an initial constraint of the form x = T (where x does not occur in T) until no rules are applicable.</Paragraph> <Paragraph position="20"> This transforms any given constraint system into basic form.</Paragraph> <Paragraph position="21"> The constraint simplification rules (see figures 3, 4 and 5 ) either eliminate variable equalities of the form x = y or generate them from existing constraints. However, they do not introduce new variables.</Paragraph> <Paragraph position="22"> The constraint simplification rules given in figure 3 are the analog of the feature simplification rules provided in \[Smolka, 1991\]. The main difference being that our simplification rules have been modified to deal with relation symbols as opposed to just feature symbols.</Paragraph> <Paragraph position="23"> The constraint simplification rules given in figure 4 simplify constraints involving set descriptions when they interact with other constraints such as feature constraints - rule (SSetF), singleton sets - rule (SSet), duplicate elements in a set - rule (SDup), universally quantified constraint - rule (SForall), another set description - rule (SSetSet). Rule (SDis) on the other hand simplifies disjunctive constraints. Amongst all simplification rules - II the constraint simplification rules in figures 3 and 4 only rule (SDis) is non-deterministic and creates a n-ary choice point.</Paragraph> <Paragraph position="24"> Rules (SSet) and (SDup) are redundant as completeness (see section below) is not affected by these rules. However these rules result in a simpler normal form.</Paragraph> <Paragraph position="25"> The following syntactic notion of entailment is employed to render a slightly compact presentation of the constraint solving rules for dealing with set operations given in figure 5.</Paragraph> <Paragraph position="26"> A constraint system Cs syntactically entails the (conjunction of) constraint(s) C/ if Cs F C/ is derivable from the following deduction rules: 1. C/AC8 FC/ 2. C~Fx=x 3. CsFx=y >CsFy=x 4. CsFx=yACsFy=z >CsFx=z 5. Cs F x = -~y > C~ F y = -~x 6. CsFx=f:y >CsFx=3f:y 7. CsFx=f:y >CsFx=Vf:y 8. CsFx=I:{...,xi,...} >C~Fz=3I:zi Note that the above definitions are an incomplete list of deduction rules. However C~ I- C/ implies C~ ~ C/ where ~ is the semantic entailment relation defined as for predicate logic.</Paragraph> <Paragraph position="27"> We write C8 t/C/ if it is not the case that C~ I- C/. The constraint simplification rules given in figure 5 deal with constraints involving set operations. Rule (C_) propagates g-values of y into I-values of x in the presence of the constraint x = f :_D g(y). Rule</Paragraph> <Paragraph position="29"> constraint x = f :_D g(y) (correspondingly x = f :D h(z)) in the presence of the constraint x = f : g(y) U h(z). Also in the presence of x = f : g(y) U h(z) rule (UDown) non-deterministically propagates an I-value of x to either an g-value of y or an h-value of z (if neither already holds). The notation y = 3g : xi \] z =</Paragraph> <Paragraph position="31"> tes an f-value of x both as a g-value of y and h-value of z in the presence of the constraint x = f : g(y) n h(z).</Paragraph> <Paragraph position="32"> Finally, rule (nUp) propagates a common g-value of y and h-value of z as an f-value of x in the presence of the constraint x = f : g(y) n h(z).</Paragraph> </Section> <Section position="8" start_page="258" end_page="259" type="metho"> <SectionTitle> 4 Invariance, Completeness and </SectionTitle> <Paragraph position="0"> Termination In this section we establish the main results of this paper - namely that our consistency checking procedure for set descriptions and set operations is invariant, complete and terminating. In other words, we have a decision procedure for determining the consistency of terms in our extended feature logic.</Paragraph> <Paragraph position="1"> For the purpose of showing invariance of our rules we distinguish between deterministic and non-deterministic rules. Amongst all our rules only rule (SDis) given in figure 4 and rule (UDown) are non-deterministic while all the other rules are deterministic. null Theorem 2 (Invariance) 1. If a decomposition rule transforms Cs to C~s then Cs is consistent iff C~ is consistent.</Paragraph> <Paragraph position="2"> 2. Let Z,a be any interpretation, assignment pair and let Cs be any constraint system.</Paragraph> <Paragraph position="3"> * If a deterministic simplification rule transforms Cs to C' s then: iff p c&quot; * If a non-deterministic simplification rule applies to Cs then there is at least one non-deterministic choice which transforms Cs to C' s such that: z,a p iffz, apc; A constraint system Cs is in normal form if no rules are applicable to Cs.</Paragraph> <Paragraph position="4"> Let succ(x, f) denote the set: succ(x, f) = {y I c8 x = 3f : y} A constraint system Cs in normal form contains a clash if there exists a variable x in C8 such that any of the following conditions are satisfied : 1. C~Fx=al andC~Fx=a2suchthatal ~a2 2. Cs F x = cl and Cs F x = c2 such thatcl ~c2 3. Cs F x = S and Cs F x = -,S where S ranges over x, a, c, C.</Paragraph> <Paragraph position="5"> 4. CsFx=3f:yandCsFx=a 5. C~ F f(x) C/ g(y) and succ(x, f) n succ(y, g) 7~ 6. C~ F x = f: {xz,...,xn}= and Isucc(x,f) I < n If Cs does not contain a clash then C~ is called clashfree. null The constraint solving process can terminate as soon as a clash-free constraint system in normal form is found or alternatively all the choice points are exhausted. null The purpose of the clash definition is highlighted in the completeness theorem given below.</Paragraph> <Paragraph position="6"> For a constraint system Cs in normal form an equivalence relation ~_ on variables occurring in Cs is defined as follows: x-~ y ifC~ F x = y For a variable x we represent its equivalence class by Theorem 3 (Completeness) A constraint system Cs in normal form is consistent iff Cs is clash-free. Proof Sketch: For the first part, let C~ be a constraint system containing a clash then it is clear from the definition of clash that there is no interpretation Z and Z-assignment a which satisfies Cs.</Paragraph> <Paragraph position="7"> Let C~ be a clash-free constraint system in normal form.</Paragraph> <Paragraph position="8"> We shall construct an interpretation 7~ =< L/R, .R > and a variable assignment a such that T~, a ~ Cs. Let U R = V U ,4t UC.</Paragraph> <Paragraph position="9"> The assignment function a is defined as follows: 1. For every variable x in )2 (a) if C8 }- x = a then ~(x) = a (b) if the previous condition does not apply then ~(x) = choose(Ix\]) where choose(\[x\]) denotes a unique representative (chosen arbitrarily) from the equivalence class \[x\].</Paragraph> <Paragraph position="10"> 2. For every constant c in C: (a) if Cs F x = c then a(c) = (~(x) (b) if c is a constant such that the previous condition does not apply then (~(c) -- c 3. For every primitive concept C in P:</Paragraph> <Paragraph position="12"> It can be shown by a case by case analysis that for every constraint K in C~: 7~,a~ K.</Paragraph> <Paragraph position="13"> Hence we have the theorem.</Paragraph> <Paragraph position="14"> Theorem 4 (Termination) The consistency checking procedure terminates in a fi- null nite number of steps.</Paragraph> <Paragraph position="15"> Proof Sketch: Termination is obvious if we observe the following properties: 1. Since decomposition rules breakdown terms into smaller ones these rules must terminate. 2. None of the simplification rules introduce new variables and hence there is an upper bound on the number of variables.</Paragraph> <Paragraph position="16"> 3. Every simplification rule does either of the following: (a) reduces the 'effective' number of variables. A variable x is considered to be ineffective if it occurs only once in Cs within the constraint x = y such that rule (SEquals) does not apply. A variable that is not ineffective is considered to be effective.</Paragraph> <Paragraph position="17"> (b) adds a constraint of the form x = C where C ranges over y, a, c, C, -~y, -~a, -~c, -~C which means there is an upper bound on the number of constraints of the form x = C that the simplification rules can add. This is so since the number of variables, atoms, constants and primitive concepts are bounded for every constraint system in basic form.</Paragraph> <Paragraph position="18"> (c) increases the size of succ(x,f). But the size of succ(x, f) is bounded by the number of variables in Cs which remains constant during the application of the simplification rules. Hence our constraint solving rules cannot indefinitely increase the size of succ(x, f).</Paragraph> </Section> <Section position="9" start_page="259" end_page="259" type="metho"> <SectionTitle> 5 NP-completeness </SectionTitle> <Paragraph position="0"> In this section, we show that consistency checking of terms within the logic described in this paper is NP-complete. This result holds even if the terms involving set operations are excluded. We prove this result by providing a polynomial time translation of the well-known NP-complete problem of determining the satisfiability of propositional formulas \[Garey and Johnson, 1979\].</Paragraph> <Paragraph position="1"> Theorem 5 (NP-Completeness) Determining consistency of terms is NP-Complete.</Paragraph> <Paragraph position="2"> Proof: Let C/ be any given propositional formula for which consistency is to be determined. We split our translation into two intuitive parts : truth assignment denoted by A(C/) and evaluation denoted by r(C/).</Paragraph> <Paragraph position="3"> Let a, b,... be the set of propositional variables occurring in C/. We translate every propositional variable a by a variable xa in our logic. Let f be some relation symbol. Let true, false be two atoms.</Paragraph> <Paragraph position="4"> Furthermore, let xl, x2,.., be a finite set of variables distinct from the ones introduced above.</Paragraph> <Paragraph position="5"> We define the translation function A(C/) by: A(C/) = f: {true, false}n The above description forces each of the variable Xa,Xb,... and each of the variables xl,x2,.., to be either equivalent to true or false.</Paragraph> <Paragraph position="6"> We define the evaluation function T(C/) by:</Paragraph> <Paragraph position="8"> where xi 6 {xl,x2,...} is a new variable r(~S) = xi n 3f : (r(S) n ~z~) where xi 6 {xl,x2,...} is a new variable Intuitively speaking T can be understood as follows. Evaluation of a propositional variable is just its value; evaluating a conjunction amounts to evaluating each of the conjuncts; evaluating a disjunction amounts to evaluating either of the disjuncts and finally evaluating a negation involves choosing something other than the value of the term.</Paragraph> <Paragraph position="9"> Determining satisfiability of C/ then amounts to determining the consistency of the following term:</Paragraph> <Paragraph position="11"> Note that the term truenT(C/) forces the value of T(C/) to be true. This translation demonstrates that determining consistency of terms is NP-hard.</Paragraph> <Paragraph position="12"> On the other hand, every deterministic completion of our constraint solving rules terminate in polynomial time since they do not generate new variables and the number of new constraints are polynomially bounded.</Paragraph> <Paragraph position="13"> This means determining consistency of terms is NPeasy. Hence, we conclude that determining consistency of terms is NP-complete.</Paragraph> </Section> <Section position="10" start_page="259" end_page="260" type="metho"> <SectionTitle> 6 Translation to Sch6nfinkel-Bernays </SectionTitle> <Paragraph position="0"> class The Schhnfinkel-Bernays class (see \[Lewis, 1980\]) consists of function-free first-order formulae which have In this section we show that the attributive logic developed in this paper can be encoded within the SchSnfinkel-Bernays subclass of first-order formulae by extending the approach developed in \[Johnson, 1991\]. However formulae such as V f : (3 f : (Vf : T)) which involve an embedded existential quantification cannot be translated into the SchSnfinkel-Bernays class. This means that an unrestricted variant of our logic which does not restrict the universal role quantification cannot be expressed within the SchSnfinkel-Bernays class. In order to put things more concretely, we provide a translation of every construct in our logic into the SchSnfinkel-Bernays class.</Paragraph> <Paragraph position="1"> Let T be any extended feature term. Let x be a variable free in T. Then T is consistent iff the formula (x = T) 6 is consistent where 6 is a translation function from our extended feature logic into the SchSnfinkel-Bernays class. Here we provide only the essential definitions of 6:</Paragraph> <Paragraph position="3"> These translation rules essentially mimic the decomposition rules given in figure 2.</Paragraph> <Paragraph position="4"> Furthermore for every atom a and every feature f in T we need the following axiom: * Vax(-~f(a, x)) For every distinct atoms a, b in T we need the axiom: *a#b Taking into account the NP-completeness result established earlier this translation identifies a NP-complete subclass of formulae within the SchSnfinkel-Bernays class which is suited for NL applications.</Paragraph> </Section> class="xml-element"></Paper>