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<Paper uid="C82-1068">
  <Title>TEST-SCORE SEMANTICS FOR NATURAL LANGUAGES</Title>
  <Section position="3" start_page="425" end_page="425" type="metho">
    <SectionTitle>
TEST-SCORE SEMANTICS FOR NATURAL LANGUAGES 421
</SectionTitle>
    <Paragraph position="0"> bility theory and fuzzy logic provide a more appropriate framework for dealing with natural languages than the traditional logical systems in which there are no gradations for truth, membership and belief, and no tools for coming to grips with vagueness, fuzziness and randomness.</Paragraph>
    <Paragraph position="1"> In what follows, we shall sketch some of the main ideas underlying test-score semantics and illustrate them with simple examples. A more detailed exposition and additional examples may be found in Zadeh (1981).</Paragraph>
  </Section>
  <Section position="4" start_page="425" end_page="425" type="metho">
    <SectionTitle>
BASIC ASPECTS OF TEST-SCORE SEMANTICS
</SectionTitle>
    <Paragraph position="0"> As was stated earlier, the point of departure in test-score semantics is the assumption that any semantic entity may be represented as a system of elastic constraints on a collection of objects or derived objects in a universe of discourse.</Paragraph>
    <Paragraph position="1"> Assuming that each object may be characterized by one or more fuzzy relations, the collection of objects in a universe of discourse may be identified with a collection of relations which constitute a fuzzv relational database or. equivalently, a state description (Carnap (1952)). In this database, then, a derived'object would be characterized by one or more fuzzy relations which are derived from other relations in the database by operations expressed in an appropriate relation-manipulating language. null In more concrete terms, let SE denote a semantic entity, e.g., the proposition p 4 During much of the past decade Pat earned far more than all of his close friends put together, whose meaning we wish to represent. To this end, we must (a) identify the constraints which are implicit or explicit in SE; (b) describe the tests which must be performed to ascertain the degree to which each constraint is satisfied; and (c) specify the manner in which the degrees in question or, equivalently, the partial test scores are to be aggregated to yield an overall test score. In general, the overall test score would be represented as a vector whose components are numbers in the unit interval or, more generally, possibility/probability distributions over this interval.</Paragraph>
    <Paragraph position="2"> Spelled out in greater detail, the process of meaning-representation in test-score semantics involves three distinct phases. In Phase 1, an explanatory database frame or EDF, for short, is constructed. EDF consists of a collection of relational frames each of which specifies the name of a relation, the names of its attributes and their respective domains,with the understanding that the meaning of each relation in EDF is known to the addressee of the meaning-representation process. Thus, the choice of EDF is not unique and is strongly influenced by the desideratum of explanatory effectiveness as well as by the assumption made regarding the knowledge profile of the addressee of the meaning-representation process. For example, in the case of the proposition p 4 During much of the past decade Pat --- -earned far more than all of his close friends put together, the EDF might consist ---7----- ----of the relational frames</Paragraph>
    <Paragraph position="4"> Year, counting backward from the present; MUCH [Proportion; ~1, where in is the degree to which a numerical value of Proportion fits the meaning of much in the context of p; and FAR.MORE [Numberl; Number2; P]. in which p is the degree to which Number1 fits the description far more in relation to NumberP. In effect, the composition of EDF is determined by the information that is needed for an assessment of the compatibility of the given SE with any designated object or, more generally, a specified state of affairs in the universe of discourse.</Paragraph>
    <Paragraph position="5"> In Phase 2, a test procedure is constructed which upon application to an explanatory database -that is, an instantiation of EDF - yields the test scores, 428 L.A. ZADEH Tl,...</Paragraph>
    <Paragraph position="6"> bv the 3Tn, which represent the degrees to which the elastic constraints induced constituents of SE are satisfied. For example, in the case of p', the test ppocedure would yield the test scores for the constraints induced by close friend, @, far more, etc. --In Phase 3, the partial test scores obtained in Phase 2 are aggregated into an overall test score, T, which serves as a measure of the compatibility of SE with ED, the explanatory database. As was stated earlier, the components of T are numbers in the unit interval or, more generally, possibility/probability distributions over this interval. In particular, when the semantic entity is a proposition, p, and the overall test score, T, is a scalar, 'I may be interpreted as the truth of p relative to ED or, equivalently, as the possibility of ED given p. In this interpretation, then, the classical truth-conditional semantics may be viewed as a special case of test-score semantics which results when the constraints induced by p are inelastic and the overall test score is allowed to be only pass or fail --The test procedure which yields the overall test score T is interpreted as the meaning of SE.</Paragraph>
    <Paragraph position="7"> To illustrate the phases in question, we shall consider a few simple examples (a) SE 4 Ellen resides in a'small city near Oslo.</Paragraph>
    <Paragraph position="8"> In this case, EDF is assumed to comprise the following relational</Paragraph>
    <Paragraph position="10"> In RESIDENCE, City.Name is the name of the city in which Name resides; in POPULA-TION, Population is the number of residents in City.Name; in SMALL, P is the degree to which a city with a population equal to the value of Population is small; and in NEAR, P is the degree to which City.Namel is near City.NameZ.</Paragraph>
    <Paragraph position="11"> The test procedure which leads to the overall test score T -- and thus represents the meaning of SE - is described below. In this procedure, Steps 1 and 2 involve the determination of the value of an attribute given the values of other attributes; Steps 3 and 4 involve the testing of constraints; and Step 5 involves an aggregation of the partial test scores into the overall test score T.</Paragraph>
    <Paragraph position="12">  1. Find the name of the residence of Ellen: RE! c,ty RameRESIDENCEIName=Ellen] which means that the value of Name is set to Ellen and the value of City.Name is read, yielding RE, the residence of Ellen.</Paragraph>
    <Paragraph position="13"> 2. Find the population of the residence of Ellen: n PRE = Population PDPULATION[City.Name=RE] 3. Test the constraint induced by SMALL:  r,guSMALLIPopulation=RE] where ~~ denotes the resulting test score.</Paragraph>
    <Paragraph position="14">  4. Test the constraint induced by NEAR: T2=uNEAR[City.Name=Oslo; City.Name2=RE] 5. Aggregate ~~ and T2:</Paragraph>
    <Paragraph position="16"> where A stands for min in infix position, and T is the overall test score. This 'TEST-SCORESEMANTICSFORNATURALLANGUAGES 429 mode of aggregation implies that, in SE, the denotation of conjunction is taken to  be the Cartesian product of the denotations of the conjuncts (Zadeh (1981)). (b) SEA During much of the past decade Pat earned far more than all of  his close friends put together.</Paragraph>
    <Paragraph position="17"> In this case, we shall employ the EDF described earlier, that is:</Paragraph>
    <Paragraph position="19"> The test procedure comprises the following steps:  1. Find the fuzzy set of Pat's friends:</Paragraph>
    <Paragraph position="21"> in which the left subscript Namelxu signifies that the relation FRIEND [NameP=Pat] is projected on the domain of the attributes Name1 and u, yielding the fuzzy set of friends of Pat.</Paragraph>
  </Section>
  <Section position="5" start_page="425" end_page="425" type="metho">
    <SectionTitle>
2. Intensify FP to account for the modifier.*:
CFP 4 FP2
</SectionTitle>
    <Paragraph position="0"> in which FP* denotes the fuzzy set which results from squaring the grade of membership of each component of FP. The assumption underlying this step is that the fuzzy set of close friends of Pat may be derived from that of friends of Pat by intensification. null 3. Find the fuzzy multiset of incomes of close friends of Pat in year Year; , i=l,...,lO: ICFP. 4 , AmountINCOMEIName = CFP; Year=Yeari] In stipulating that the right-hand member be treated as a fuzzy multiset, we imply that the identical elements should not be combined, as they would be in the case of a fuzzy set. With this understanding, ICFPi will be of the general form</Paragraph>
    <Paragraph position="2"> where el,..., e m are the incomes of Name ..., Name 1' m, respectively, in Year., andsl,...,,m 6 are the grades of membership of Name,;.-, Namem in the fuzzy se t of close friends of Pat.</Paragraph>
    <Paragraph position="3">  4. Find the total income of close friends of Pat in Yeari , i=l;.., 10: TICFPi =6,el+...+ 6 e mm which represents a weighted arithmetic in Yeari.</Paragraph>
    <Paragraph position="4"> 5. Find Pat's income in Yeari: IPi 4AmountINCOME[Name=Pat; 6. Test the constraint induced sum of the incomes of close friends of Pat Year=Yeari].</Paragraph>
    <Paragraph position="5"> by FAR.MORE: ri$pFAR.MOREINumberl=IPi; Nurnber2= TICFPi] 7. Find the sigma-count (Zadeh (1981)) of years during which Pat's income was far.greater than the total income of all of his close friends: c iq'.; , 430 L.A. ZADEH a. Test the constraint induced by MUCH:  where T represents the overall test score. The two examples described above are intended merely to provide a rough outline of the meaning-representation process in test-score semantics. A more detailed exposition of some of the related issues may be found in Zadeh (1978) and Zadeh (1981) 'Research supported in part by the NSF Grants MCS79-06543 and IST-801896.</Paragraph>
  </Section>
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