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<Paper uid="P93-1028">
  <Title>A LOGICAL SEMANTICS FOR NONMONOTONIC SORTS</Title>
  <Section position="3" start_page="0" end_page="209" type="metho">
    <SectionTitle>
FEATURE SYSTEMS
</SectionTitle>
    <Paragraph position="0"> Unification-based grammar formalisms use formal objects called feature structures to encode linguistic information. We use a variant of the standard definition. Each structure has a sort (drawn from a finite set 8), and a (possibly empty) set of attributes (drawn from a finite set ~).</Paragraph>
    <Paragraph position="1"> Definition1 A feature structure is a tuple</Paragraph>
    <Paragraph position="3"> that gives the edges and their labels, and * (9 : Q ~ S is a sorting function that gives the labels of the nodes.</Paragraph>
    <Paragraph position="4"> This structure must be connected.</Paragraph>
    <Paragraph position="5"> It is not unusual to require that these structures also be acyclic. For some systems O is defined only for sink nodes (PATR-II, for example). Fig. 1 shows a standard textual representation for a FS.</Paragraph>
    <Paragraph position="6"> We sometimes want to refer to substructures of a FS. If .A is a feature structure as described above, we write .A/f for the feature structure rooted at 6(q, f). This feature structure is defined by Q~ c_ Q, the set of nodes that can be reached from 6(r, f). We will use the letter p (possibly subscripted) to represent paths (that is, finite sequences from .T'*). We will also extend ~ to have paths in its second  &lt;subj agr person&gt; isa 3rd &lt;subj agr number&gt; isa singular &lt;subj agr&gt; = &lt;pred agr&gt; &lt;pred actor&gt; = &lt;subj&gt; &lt;pred rep&gt; isa sleep  1. A 2. A</Paragraph>
  </Section>
  <Section position="4" start_page="209" end_page="209" type="metho">
    <SectionTitle>
5. A
6. A
</SectionTitle>
    <Paragraph position="0"> position, with the notion of iterated application of  We will assume that there is a partial order, -~, defined on S. This ordering is such that the greatest lower bound of any two sorts is unique, if it exists. In other words, (S U {_1_}, -q) is a meetsemilattice (where _l_ represents inconsistency or failure). This allows us to define the most general unifier of two sorts as their greatest lower bound, which write as aAsb. We also assume that there is a most general sort, T, called top. The structure (S, -g) is called the sort hierarchy.</Paragraph>
  </Section>
  <Section position="5" start_page="209" end_page="209" type="metho">
    <SectionTitle>
KASPER-ROUNDS LOGIC
</SectionTitle>
    <Paragraph position="0"> (Kasper 1988) provides a logic for describing feature structures. Fig. 2 shows the domain of these logical formulas. We use the standard notion of satisfaction. Let A = (Q, r, 5, O).</Paragraph>
    <Paragraph position="2"> Note that item 3 is different than Kasper's original formulation. Kasper was working with a flat sort hierarchy and a version of FSs that allowed sorts only on sink nodes. The revised version allows for order-sorted hierarchies and internal sorted nodes.</Paragraph>
  </Section>
  <Section position="6" start_page="209" end_page="210" type="metho">
    <SectionTitle>
NONMONOTONIC SORTS
</SectionTitle>
    <Paragraph position="0"> Figure 3 shows a lexical inheritance hierarchy for  strict (isa) and default (default) values for various suffixes. If we ignore the difference between strict and default values, we find that the information specified for the past participle of mahl is inconsistent. The MIDDLE-VERB template gives +en as the suffix, while VERB gives +t. The declaration of the latter as a default tells the system that it should be dropped in favour of the former. The method of nonmonotonic sorts formalizes this notion of separating strict from default information. Definition 2 A nonmonotonic sort is a pair (s, A / where s E S, and A C S such that for each d E A, d-4 s.</Paragraph>
    <Paragraph position="1"> The first element, s, represents the strict information. The default sorts are gathered together in A. We write Af for the set of nonmonotonic sorts.</Paragraph>
    <Paragraph position="2"> Given a pair of nonmonotonic sorts, we can unify  them to get a third NS that represents their combined information.</Paragraph>
    <Paragraph position="3"> Definition 3 The nonmonotonic sort unifier of nonmonotonic sorts (sl,Az) and (s2,As) is the nonmonotonic sort (s, A) where * S ~ 81Ass2, and * A = {dAss I d E Az U A2 A (dAss) -~ s}.</Paragraph>
    <Paragraph position="4">  The nonmonotonic sort unifier is undefined if saAss2 is undefined. We write nzA~n2 for the NS unifier of nl and n2.</Paragraph>
    <Paragraph position="5"> The method strengthens consistent defaults while eliminating redundant and inconsistent ones. It should be clear from this definition that NS unification is both commutative and associative. Thus we may speak of the NS unifier of a set of NSs, without regard to the order those NSs appear. Looking back to our German verbs example, the past participle suffix in VERB is (T, {+t}), while that of MIDDLE-VERB is (+en, {}). The lexical entry for mahl gets their nonmonotonic sort unifier, which is (+en, {}). If +tAs+en had been defined, and equal  to, say, +ten, then the NS unifier of (T, {+t}) and (+en, {}) would have been (+an, {+ten}}.</Paragraph>
    <Paragraph position="6"> Once we have nonmonotonic sorts, we can create nonmonotonically sorted feature structures (NS-FSs) by replacing the function 0 : Q ~ S by a function ~ : Q ~ Af. The nodes of the graph are thus labeled by NSs instead of the usual sorts. NSFSs may be unified by the same procedures as before, only replacing sort unification at the nodes with nonmonotonic sort unification. NSFS unification, written with the symbol rlN, is associative and commutative.</Paragraph>
    <Paragraph position="7"> NSFSs allow us to carry around default sorts, but has so far given us no way to apply them. When we are done collecting information, we will want to return to the original system of FSs, using all and only the applicable defaults. To do that, we introduce the notions of explanation and solution.  Definition 4 A sort t is said to be explained by a nonmonotonic sort (s,A} if there is a D C A such that t = S^s(AsD). If t is a maximally specific  explained sort, lhen ~ is called a solution of n. The solutions for {+en, {)) and {T, {+t}) are +en and +t respectively. The latter NS also explains T. Note that, while D is maximal, it's not necessarily the case that D = A. If we have mutually inconsistent defaults in A, then we will have more than one maximal consistent set of defaults, and thus more than one solution. On the other hand, strict information can eliminate defaults during unification. That means that a particular template can inherit conflicting defaults and still have a unique solution--provided that enough strict information is given to disambiguate.</Paragraph>
    <Paragraph position="8"> NSFS solutions are defined in much the same way as NS solutions.</Paragraph>
    <Paragraph position="9"> Definition 5 A FS (Q,r,~,O) is said to be explained by a NSFS (Q,r, 8, Q) if for each node q E Q we have ~2(q) explains O(q). If.A is a maximally specific explained FS, then A is called a solution. null If we look again at our German verbs example, we can see that the solution we get for mahl is the FS that we want. The inconsistent default suffix +t has been eliminated by the strict +en, and the sole remaining default must be applied.</Paragraph>
    <Paragraph position="10"> For the generic way we have defined feature structures, a NSFS solution can be obtained simply by taking NS solutions at each node. More restricted versions of FSs may require more care. For instance, if sorts are not allowed on internal nodes, then defining an attribute for a node will eliminate any default sorts assigned to that node. Another example where care must be taken is with typed feature structures (Carpenter 1992). Here the application of a default at one node can add strict information at another (possibly making a default at the other node inconsistent). The definition of NSFS solution handles both of these cases (and others) by requiring that the solution be a FS as the original system defines them. In both of these cases, however, the work can be (at least partially) delegated to the unification routine (in the former by Mlowing labels with only defaults to be removed when attributes are defined, and in the latter by propagating type restrictions on strict sorts).</Paragraph>
    <Paragraph position="11"> What is done in other systems in one step has been here broken into two steps--gathering information and taking a solution. It is important that the second step be carried out appropriately, since it re-introduces the nonmonotonicity that we've taken out of the first step. For a lexicon, templates exist in order to organize information about words. Thus it is appropriate to take the solution of a lexical entry (which corresponds to a word) but not of a higher template (which does not). If the lexicon were queried for the lexical entry for mahl, then, it would collect the information from all appropriate templates using NSFS unification, and return the solution of that NSFS as the result.</Paragraph>
  </Section>
  <Section position="7" start_page="210" end_page="211" type="metho">
    <SectionTitle>
DEFAULT LOGIC
</SectionTitle>
    <Paragraph position="0"> The semantics for nonmonotonic sorts is motivated by default logic (Reiter 1980). What we want a default sort to mean is: &amp;quot;if it is consistent for this node to have that sort, then it does.&amp;quot; But where Reiter based his DL on a first order language, we want to base ours on Kasper-P~ounds logic. This will require some minor alterations to lZeiter's formalism. null A default theory is a pair (D, W) where D is a set of default inferences and W is a set of sentences from the underlying logic. The default inferences are triples, written in the form ~:Mp Each of the greek letters here represents a wff from the logic. The meaning of the default inference is that if ~ is believed and it is consistent to assume t5, then 7 can be believed.</Paragraph>
    <Paragraph position="1"> Given a default theory (D, W), we are interested in knowing what can we believe. Such a set of beliefs, cMled an extension, is a closure of W under the usual rules of inference combined with the default rules of inference given in D. An extension E is a minimal closed set containing W and such that if c~ :M fl/7 is a default, and if ~ E E and consistent with E then 7 E E (that is, if we believe ~x and fl is consistent with what we believe, then we also believe 7).</Paragraph>
    <Paragraph position="2"> l~eiter can test a formula for consistency by testing for the absence of its negation. Since Kasper-Rounds logic does not have negation, we will not be able to do that. Fortunately, we have do have our  own natural notion of consistency--a set of formulas is consistent if it is satisfiable. Testing a set of Kasper-Rounds formulas for consistency thus simply reduces to finding a satisfier for that set. Formally, we encode our logic as an information  system (Scott 1982). An information system (IS) is a triple (A, C, b) where A is a countable set of &amp;quot;atoms,&amp;quot; Cis a class of finite subsets of A, and t- is a binary relation between subsets of A and elements of A. A set X is said to be consistent if every finite  subset of X is an element of C. A set G is closed if for every X _C G such that X l- a, we have a E G. Following thestyle used for information systems, we will write G for the closure of G.</Paragraph>
    <Paragraph position="3"> In our case, A is the wffs of SFML (except FALSE), and C is the class of satisfiable sets. The entailment relation encodes the semantics of the particular unification system we are using. That is, we have</Paragraph>
    <Paragraph position="5"> represents the transitivity of path equations.</Paragraph>
  </Section>
  <Section position="8" start_page="211" end_page="211" type="metho">
    <SectionTitle>
DEFAULT KASPER-ROUNDS
LOGIC
</SectionTitle>
    <Paragraph position="0"> In the previous section we described the generic form of default logic. We will not need the full generality to describe default sorts. We will restrict our attention to closed precondition-free normal defaults. That is, all of our defaults will be of the form: :M~ We will write D E as an abbreviation for this default inference. Here fl stands for a generic wff from the base language. Even this is more general than we truly need, since we are really only interested in default sorts. Nevertheless, we will prove things in the more general form.</Paragraph>
    <Paragraph position="1"> Note that our default inferences are closed and normal. This means that we will always have an extension and that the extension(s) will be consistent if and only if W is consistent. These follow from our equivalents of Reiter's theorem 3.1 and corollaries 2.2 and 2.3.</Paragraph>
    <Paragraph position="2"> Let's consider now how we would represent the information in Fig. 3 in terms of Kasper-Rounds default logic. The strict statements become normal KR formulas in W. For instance, the information for MIDDLE-VERBs (not counting the inheritance information) is represented as follows: ({}, {past : participle: suffix: +en)) The information for VERB will clearly involve some defaults. In particular, we have two paths leading to default sorts. We interpret these statements as saying that the path exists, and that it has the value indicated by default. Thus we represent the VERB template as:</Paragraph>
    <Paragraph position="4"> past : participle : suffix : -I-, past : participle : prefix : ge+ } Inheritance is done simply by pair-wise set union of ancestors in the hierarchy. Since the entry for mahl contains no local information, the full description for it is simply the union of the two sets above.</Paragraph>
    <Paragraph position="6"> past : participle : suffix : T, past : participle : prefix : ge+, past : participle : suffix : +en} We can then find an extension for that default theory and take the most general satisfier for that formula. It is easy to see that the only extension for raahl is the closure of: past : tense : suffix : +te, past : participle : suffix : +en, past : participle : prefix : ge+ The default suffix +t is not applicable for the past participle due to the presence of +en. The suffix +re is applicable and so appears in the extension.</Paragraph>
  </Section>
  <Section position="9" start_page="211" end_page="212" type="metho">
    <SectionTitle>
DKRL AND NONMONOTONIC
SORTS
</SectionTitle>
    <Paragraph position="0"> In the previous section we defined how to get the right answers from a system using default sorts. In this section we will show that the method of non-monotonic sorts gives us the same answers. First we formalize the relation between NSFSs and default logic.</Paragraph>
    <Paragraph position="1"> Definition 6 Let 79 = (Q, r, 5, ~) be a nonmonotonically sorted feature structure. The default the-</Paragraph>
    <Paragraph position="3"> The default part of DT(79) encodes the default sorts, while the strict part encodes the path equations and strict sorts.</Paragraph>
    <Paragraph position="4"> Theorem 1 The FS .4 is a solution for the NSFS 7) if and only if {C/1.4~C/} is an extension of DT(79).</Paragraph>
    <Paragraph position="5">  Because we are dealing with closed normal default theories, we can form extensions simply by taking maximal consistent sets of defaults. This, of course, is also how we form solutions, so the the solution of a NSFS is an extension of its default theory. We now need to show that NSFS unification behaves properly. That is, we must show that non-monotonic sort unification doesn't create or destroy extensions. We will write (D1, W1)=zx(D2, I4/2) to indicate that (O1, W1) and (D2, W2) have the same set of extensions. We will do this by combining a number of intermediate results.</Paragraph>
    <Paragraph position="6"> Theorem 2 Let (D, W) be a closed normal default theory.</Paragraph>
    <Paragraph position="7">  1. /fc~ A/3 C/* 7, then (D, W to {4 ^/3})=a(D, W to {7})2. /f W U {/3} is inconsistent, then (D t^ {DE} , W)=A(D, W).</Paragraph>
    <Paragraph position="8"> 3. IfW ~-/3, then (D U {DE} , W)=A(D, W).</Paragraph>
    <Paragraph position="9"> 4. IfW~-~ anda^/3C/:~7, then (D tO {DE} , W)=A(D tO {D.y}, W).</Paragraph>
    <Paragraph position="10">  The formulas ~ and /3 represent the (path prefixed) sorts to be unified, and 7 their (path prefixed) greatest lower bound. The first part deals with strict sort unification, and is a simple consequence of the fact that (D, W) has the same extensions as (D, W). The next two deal with inconsistent and redundant default sorts. They are similar to theorems proved in (Delgrande and Jackson 1991): inconsistent defaults are never applicable; while necessary ones are always applicable. The last part allows for strengthening of default sorts. It follows from the previous three. Together they show that nonmonotonic unification preserves the information present in the NSFSs being unified. Theorem 3 Let 791 and 792 be NSFSs. Then DT(79Z RN792)=zx DT(791) to DT(792) (using pair-wise set union).</Paragraph>
  </Section>
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