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<Paper uid="E95-1023">
  <Title>Deterministic Consistency Checking of LP Constraints</Title>
  <Section position="3" start_page="165" end_page="166" type="metho">
    <SectionTitle>
2 Outline of an alternative
</SectionTitle>
    <Paragraph position="0"> approach To motivate our approach we start with an example on scrambling in German subordinate clauses.  (4) dab er einen Mann in der Strafle lanfen that he a man in the street walking sah.</Paragraph>
    <Paragraph position="1"> saw.</Paragraph>
    <Paragraph position="2"> that he saw a man walking in the street. (5) dab er in der Strafle einen Mann laufen sah. (6) daft einen Mann er in der Stral3e lanfen sah. (7) daft einen Mann in der Stral\]e er lanfen sah. (8) daft in der Stral\]e er einen Mann laufen sah. (9) daft in der Strai\]e einen Mann er laufen sah.  The above data can be captured precisely if we can state that sah requires-both its verbal argument laufen and its NP argument er to precede it. Similarly, laufen would require both its arguments einen Mann and in der Strafle to precede it. This is illustrated schematically in (10) below. (10) {er} {sah} / { einen mann, in der strasse} { laufen } Our idea is to employ a specification such as the one given in (11) which is a partial specification of the lexical entry for the verb sah. The specification can be thought of as a formal specification of the intuitive description given in (12).</Paragraph>
    <Paragraph position="4"> For space reasons, our treatment is necessarily somewhat superficial since we do not take into account other interacting phenomena such as fronting or extraposition.</Paragraph>
    <Paragraph position="5"> The definition in (11) does not make specific assumption about whether a context-free backbone is employed or not. However, if a CFG backbone is employed then we assume that the value of the subcat attribute is treated as an unordered sequence (i.e. a set) as defined in (11).</Paragraph>
    <Paragraph position="7"> The essential idea is to use set-valued descriptions to model word-order domains. In paxticulax subset constraints (Manandhar, 1994) are employed to construct larger domains from smaller ones. Thus in example (11) the domain of the verb is constructed by including the domains of the subcategorised arguments (enforced by the constraints dora :D NPdomf3dom :D ViDom). Note that in this example the verb itself is not part of its own domain. The binary constraint Vi &lt; V enforces precedence ordering between the signs Vi and V. The constraint V~dom &lt; do,~ {V} ensures that every element of the set ViDom precedes the sign V. In other words, the set ViDom is in the domain precedence relation with the singleton {V}.</Paragraph>
    <Paragraph position="8"> However there are strong constraints on ordering in the middle field. For instance, when pronomial complements are involved then not all permutations are acceptable. Examples such as (13) are considered ungrammatical.</Paragraph>
    <Paragraph position="9"> (13) *dab in der Strafle ihn er laufen sah.</Paragraph>
    <Paragraph position="10"> According to Uszkoreit (Uszkoreit, 1985), ordering of arguments in the middle field is governed by the following set of LP constraints given in (14) which axe to be interpreted disjunctively.</Paragraph>
    <Paragraph position="12"> The LP constraint in (14) states that for every pair of constituents in the middle field at least one of the conditions should apply otherwise the sentence is considered ungrammatical. A related but more elaborate LP rule mechanism is considered in (Steinberger, 1994).</Paragraph>
    <Paragraph position="13">  To approximate this complex LP constraint employing the kind of logical machinery described in this paper, we can use a description such as the one given in (15). The definition given in (15) extends the description given in (11).</Paragraph>
    <Paragraph position="14">  The definition in (15) can be understood as follows. The feature constraint syn : dora : MF coinstantiates the middle field domain to the variable MF. To keep the example simple, we assume that the whole domain is in the middle field and we ignore fronting or extraposition. A more complex condition would be needed to handle these. The rest of the definition in (15) ensures that for every pair of elements x and y such that x and y are both members of MF and x precedes y at least one of the LP constraints hold. If every LP constraint is violated then an inconsistency results. The constraints in (15) is a weaker representation  of the disjunctive specification given in (16). (16) Sx~y if (x e MF A y e MF A x &lt; y) then</Paragraph>
    <Paragraph position="16"> The description in (16) non-deterministicaily requires that at least one of the LP constraints hold.</Paragraph>
    <Paragraph position="17"> On the other hand, the description in (15) waits until either one of the LP constraints is satisfied (in which case it succeeds) or all the LP constraints are violated (in which case it fails). Thus the description in (15) can be solved deterministically. null Thus (15) should rule out the ungrammatical example in (13) if the assumptions regarding focus are made as in (17).</Paragraph>
    <Paragraph position="18">  (17) *dab in der Strafle ihn er laufen sah.</Paragraph>
    <Paragraph position="19"> pprn:- focus:th:theme pprn: + tw. agent  Note that it is not necessary to know whether the PP in der Strafle is focussed to rule out (17) since the fact that the pronoun ihn is focus:- is enough to trigger the inconsistency.</Paragraph>
  </Section>
  <Section position="4" start_page="166" end_page="167" type="metho">
    <SectionTitle>
3 Some generic LP constraints
</SectionTitle>
    <Paragraph position="0"> As suggested by the example in (11), in general we would want support within typed feature formalisms for at least the following kinds of LP constraints. null  1. Sign1 &lt; Signs 2. Doml &lt; dom Dom~ (Dotal and Dom~ are set-valued) 3. Doml is included in Dom~ The constraint Sign1 &lt; Sign~ states that Sign1 precedes Signs. The constraint Dom~ &lt; dora Dom2 states that every element of the set described by Doml precedes every element of the set described by Dom=. Constraints such as Doml is included  in Dora2 essentially builds larger domains from smaller ones and can be thought of as achieving the same effect as Reape's domain union operation. Note crucially that within our approach the specification of precedence constraints (such as Sign1 &lt; Sign~ and Dom~ &lt; ~om Dom2) is independent of the domain building constraint (i.e. the constraint Doml is included in Dom=). This we believe is a generaiisation of Reape's approach. Other constraints such as the following involving immediate precedence and first element of a domain are of lesser importance. However, these could be of the form:  1. Sign1 immediately-precedes Sign= 2. First daughter of Dom~ is Sign1 To be able to state descriptions such as in (15), we also want to introduce guarded (or conditional) LP constraints such the following: 1. if Sign~ is NP\[acc\] A Sign2 is NP\[dat\] then Sign, &lt; Sign= ( Guards on Feature constraints) 2. if Sign~ &lt; Sign2 then ......</Paragraph>
    <Paragraph position="1"> ( Guards on precedence constraints) 3. 3x3y (/fx:NP\[acc\] E Dom A</Paragraph>
    <Paragraph position="3"> Guarded constraints can be thought of as conditional constraints whose execution depends on the presence of other constraints. The condition part  G of a guarded constraint if G then S else T is known as a guard. The consequent S is executed if the current set of constraints entail the guard G. The consequent T is executed if the current set of constraints disentail the guard G. If the current set of constraints neither entail nor disentail G then the execution of the whole guarded constraint is blocked until more information is available. null The application of guarded constraints within computational linguistics has not been well explored. However, the Horn extended feature structures described in (Hegner, 1991) can be thought of as adding guards to feature structures. On the other hand, within logic programming guarded logic programming languages have a longer history originating with committed-choice languages (Ueda, 1985) and popularised by the concurrent constraint programming paradigm due to Saraswat (Saraswat and Rinard, 1990) (Saraswat, 1993).</Paragraph>
    <Paragraph position="4"> For space reasons, we do not cover the logic of guarded feature constraints, guards on set membership constraints and guards o.n precedence constraints. Guarded feature constraints have been extensively studied in (Ait-Kaci et al., 1992) (Smolka and Treinen, 1994) (Ait-Kaci and Podelski, 1994).</Paragraph>
  </Section>
  <Section position="5" start_page="167" end_page="168" type="metho">
    <SectionTitle>
4 A feature logic with LP
</SectionTitle>
    <Paragraph position="0"> constraints In this section we provide formal definitions for the syntax and semantics of an extended feature logic that directly supports linear precedence constraints as logical primitives. The logic described in this paper is a further development of the one described in (Manandhar, 1993).</Paragraph>
    <Paragraph position="1"> The syntax of the constraint language is defined by the following BNF definitions.</Paragraph>
    <Paragraph position="2"> Syntax Let ~ be the set of relation symbols and let 79 be the set of irreflexive relation symbols. We shall require that :7- and 79 are disjoint.</Paragraph>
    <Paragraph position="4"> where f E .7- and p E 79 The constraint x = f : y specifies that y is the only f-value of x. The constraint x = 3f : y states that y is one of the f-values of x.</Paragraph>
    <Paragraph position="5"> The constraint x = 3p + : y just says that x is related to y via the transitive closure of p. The precedence constraint such as Sign1 precedes Sign= is intended to be captured by the constraint Sign1 = 3p + :Sign= where p denotes the (user chosen) immediate precedence relation.</Paragraph>
    <Paragraph position="6"> Similarly, x = 31o* : y states that x is related to y via the transitive, reflexive closure of p. This constraint is similar to the constraint x = 3p + : y except that it permits x and y to be equal.</Paragraph>
    <Paragraph position="7"> The constraints f(x) : p+ : g(y) and f(x) : p* : g(y) are intended to enforce precedence between two word-ordering domains. The constraint f(x) : p+ : g(y) states that every f-value of x precedes (i.e. is in the p+ relation with) every g-value of y. The constraint f(x) : p* : g(y) is analogous.</Paragraph>
    <Paragraph position="8"> The constraint x = If p 1\]y states that y is the first daughter amongst the f-values of x (i.e. is in the p* relation with every f-value of x).</Paragraph>
    <Paragraph position="9"> Since our language supports both feature constraints and set-membership constraints the conventional semantics for feature logic (Smolka, 1992) needs to be extended. The essential difference being that we interpret every feature/relation as a binary relation on the domain of interpretation. Feature constraints then require that they behave functionally on the variable upon which the constraint is expressed.</Paragraph>
    <Paragraph position="10"> A precise semantics of our constraint language is given next.</Paragraph>
    <Paragraph position="11"> Semantics An interpretation structure 27 =&lt;//z, .I &gt; is a structure such that:  * ///is an arbitrary non-empty set * .i is an interpretation function which maps: - every relation f E ~- to a binary relation: /I _c///x U I - every relation p E 79 to a binary relation:  pi C //i X //I with the added condition that (pZ)+ is irreflexive A variable assignment ~ is a function c~:12 ~//J.</Paragraph>
    <Paragraph position="12"> We shall write fI(e) to mean the set: if(e) = {e' e//I I (e,e') E fi} We say that an interpretation 27 and a variable assignment a satisfies a constraint C/ written Z, a C/ if the following conditions are satisfied:</Paragraph>
    <Paragraph position="14"> Given the above semantics, it turns out that the first-daughter constraint can be defined in terms of other constraints in the logic. Let f_p_l be a distinct relation symbol then we can equivalently define the first-daughter constraint by:</Paragraph>
    <Paragraph position="16"> The translation states that y (which is the f_p_lvalue of x) precedes or is equal to every f-value of x and y is a f-value of x. For this to work, we require that the feature symbol f_p_l appears only in the translation of the constraint x = \[f p 1\]y.</Paragraph>
    <Section position="1" start_page="168" end_page="168" type="sub_section">
      <SectionTitle>
4.1 Two Restrictions
</SectionTitle>
      <Paragraph position="0"> The logic we have described comes with 2 limitations which at first glance appears to be somewhat severe, namely:  This is so because it turns out that adding both functionM precedence and atoms in general leads to a non-deterministic constraint solving procedure. To illustrate this, consider the following constraints: x= f :yAy=aAx=3f* :z where a is assumed to be an atom.</Paragraph>
      <Paragraph position="1"> The above constraints state that y is the f-value of x and y is the atom a and z is related to x by the reflexive-transitive closure of f.</Paragraph>
      <Paragraph position="2"> Determining consistency of such constraints in general involves solving for the following disjunctive choices of constraints.</Paragraph>
      <Paragraph position="4"> if x = 3f : y (\[ Cs where G ranges over g, 3g  However for practical reasons we want to eliminate any form of backtracking since this is very likely to be expensive for implemented systems. On the other hand, we certainly cannot prohibit atoms since they are crucially required in grammar specification. But disallowing functional precedence is less problematic from a grammar development perspective.</Paragraph>
    </Section>
    <Section position="2" start_page="168" end_page="168" type="sub_section">
      <SectionTitle>
4.2 Imposing the restriction
</SectionTitle>
      <Paragraph position="0"> We note that precedence can be restricted to non-atomic types such as HPSG signs without compromising the grammar in any way. We then need to ensure that precedence constraints never have to consider atoms as their values. This can be easily achieved within current typed feature formglisms by employing appropriateness conditions (Carpenter, 1992).</Paragraph>
      <Paragraph position="1"> An appropriateness condition just states that a given feature (in our case a relation) can only be defined on certain (appropriate) types. The assumption we make is that precedence is specified in such a way that is appropriate only for non-atomic types. This restriction can be imposed by the system (i.e. a typed feature formalism) itself.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="168" end_page="169" type="metho">
    <SectionTitle>
5 Constraint Solving
</SectionTitle>
    <Paragraph position="0"> We are now ready to consider consistency checking rules for our constraint language. To simplify the presentation we have split up the rules into two groups given in figure 1 and figure 2.</Paragraph>
    <Paragraph position="1"> The constraint solving rules given in figure 1 deal with constraints involving features, setmemberships, subset and first daughter. Rules (Equals) and (Feat) are the usual feature logic rules (Smolka, 1992) that deal with equality and features. By \[x/y\]Cs we mean replacing every occurrence of x with y in Cs. Rule (FeatEx-</Paragraph>
    <Paragraph position="3"> where R ranges over p+, p*</Paragraph>
    <Paragraph position="5"> G ranges over g, 3g).</Paragraph>
    <Paragraph position="6"> The constraint solving rules given in figure 2 deal with constraints involving the precedes and the precedes or equal to relations and domain precedence. Rule (TransConj) eliminates the weaker constraint x = 3p*:y when both x=2p*:y Ax=3p +:y hold. Rule (TransClos) effectively computes the transitive closure of the precedence relation one-step at a time. Rule (Cycle) detects cyclic relations that are consistent, namely, when x precedes or equals y and vice versa then x = y is asserted. Finally rule (DomPrec) propagates constraints involving domain precedence.</Paragraph>
    <Paragraph position="7"> We say that a set of constraints are in normal form if no constraint solving rules are applicable to it. We say that a set of constraints in normal form contains a clash if it contains constraints of the form:</Paragraph>
    <Paragraph position="9"> In the following sections we show that our constraint solving rules are sound and every clash-free constraint system in normal form is consistent. null</Paragraph>
    <Section position="1" start_page="169" end_page="169" type="sub_section">
      <SectionTitle>
5.1 Soundness, Completeness and
Termination
</SectionTitle>
      <Paragraph position="0"> Theorem 1 (Soundness) Let Z,o~ be any interpretation, assignment pair and let Cs be any set of constraints. If a constraint solving rule transforms Cs to Crs then: z,a ~ C, iffz, a ~ C', Proof Sketch: The soundness claim can be verified by checking that every rule indeed preserves the interpretation of every variable and every relation symbol.</Paragraph>
      <Paragraph position="1"> Let succ(x,f) and succ(x,p) and denote the sets:</Paragraph>
      <Paragraph position="3"> tem Cs in normal form is consistent iff Cs is clash-flee.</Paragraph>
      <Paragraph position="4"> Proof Sketch: For the first part, let Cs be a constraint system containing a clash then it is clear from the definition of clash that there is no interpretation E and variable assignment a which satisfies Cs.</Paragraph>
      <Paragraph position="5"> Let Cs be a clash-free constraint system in normal form.</Paragraph>
      <Paragraph position="6"> We shall construct an interpretation 7C/ =&lt; L/n, .n &gt; and a variable assignment a such that TO, a ~ Cs.</Paragraph>
      <Paragraph position="7"> Let L/R = Y.</Paragraph>
      <Paragraph position="8"> The assignment function a is defined as follows:</Paragraph>
      <Paragraph position="10"> Note that for constraints in normal form: if x = y 6 C8 then either x is identical to y or x occurs just once in C~ (in the constraint x = y). Otherwise Rule (Equals) is applicable.</Paragraph>
      <Paragraph position="11"> The interpretation function .R is defined as follows: null</Paragraph>
      <Paragraph position="13"> It can be shown by a case by case analysis that for every constraint K in C,: ~,a~K.</Paragraph>
      <Paragraph position="14"> Hence we have the theorem.</Paragraph>
      <Paragraph position="15">  Theorem 3 (Termination) The consistency checking procedure terminates in a finite number of steps.</Paragraph>
      <Paragraph position="16"> Proof Sketch: The termination claim can be easily verified if we first exclude rules (Subset), (TransClos) and (DomPrec) from consideration. Then for the remainder of the rules termination is obvious since these rules only simplify existing constraints. For these rules: 1. Rule (Subset) increases the size of succ(x, f) but since none of our rules introduces new variables this is terminating.</Paragraph>
      <Paragraph position="17"> 2. Rules (TransClos) and (DomPrec) asserts a relation R between pairs of variables x, y. However, none of these rules apply once x = 3p + : y is known. Furthermore, if x = 3p + : y is known it is never simplified to the weaker x = 3p* : y. This means that these rules converge.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="169" end_page="171" type="metho">
    <SectionTitle>
6 Linearisation of precedence
</SectionTitle>
    <Paragraph position="0"> ordered DAGs The models generated by the completeness theorem interpret (the map of) every precedence relation p as a directed acyclic graph (DAG) as depicted in figure 3. However sentences in natural languages are always totally ordered (i.e. they are strings of words). This then raises the question: Is it possible to generate linearised models? For the logic that we have described this is always possible. We only provide a graphical argument given in figure 3 to illustrate that this is indeed possible.</Paragraph>
    <Paragraph position="1"> The question that arises is then: What happens when we add immediate prece-</Paragraph>
    <Section position="1" start_page="169" end_page="171" type="sub_section">
      <SectionTitle>
6.1 Problem with immediate precedence
</SectionTitle>
      <Paragraph position="0"> However if we add immediate precedence to our logic then it is not clear whether we can guarantee linearisable models. This is highlighted in figure 4.</Paragraph>
      <Paragraph position="1"> As illustrated in this figure consistency checking of constraints involving both linear precedence and immediate precedence with a semantics that requires linearised models is not trivial. So we do not explore this scenario in this paper.</Paragraph>
      <Paragraph position="2"> However, it is possible to add immediate precedence and extend the constraint solving rules described in this paper in such a way that it is sound and complete with respect to the current semantics described in this paper (which does not insist on linearised models).</Paragraph>
      <Paragraph position="3"> 7 Handling immediate precedence In this section, we provide additional constraint solving rules for handling immediate precedence.</Paragraph>
      <Paragraph position="4"> The basic idea is to treat immediate precedence as a functional relation whose inverse too is functional. null In effect what we add to our logic is both precedence as a feature and a new constraint for representing the inverse functional precedence.</Paragraph>
      <Paragraph position="5"> This is summarised by:  The additional rules given in figure below are all that is needed to handle immediate precedence.</Paragraph>
    </Section>
  </Section>
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