MODELLING AND REASONING VAGUE CONCEPTS concept to the purposes volume - PDF document

MODELLING AND REASONING VAGUE CONCEPTS concept to the purposes volume we attention to concepts to describe it should is a a set instances to it is that all expression can basic labels combining expressions are restricted to the standard logical and implication Hence, the set label expressions identifying concepts can defined as follows: Label Expressions for a correspond to basic labels identifying individuals, such . . possible expressions but not with brown with medium first proposed proposed 101 the treatment of vague concepts in artificial intelligence has fuzzy set volume, we will argue that aspects this theory in turn will to develop modelling and reasoning imprecise concepts. current theories based on fuzzy set semantic, rather than purely axiomatic, perspective operational interpretations fuzzy sets, taking into account their consistency the mathematical calculus fuzzy sets, truth-functional calculus calculus 101, is centred around the extension theoretic operations as union and sets are classical (crisp) sets that allow elements to partial membership. and only and where fuzzy sets this definition definition 11 allowing x to have partial membership XA (x) in A. Fuzzy sets can applied directly to model vague to be objects in the which satisfy 0 it is simply assumed membership in the extension. In other words, the extension of a vague fuzzy set. in order to avoid any we shall also use 0 to denote the extension of an expression expression 101 the extension of a vague fuzzy set set 11. Now given this possible framework You are immediately faced with a difficult a finite basic label infinitely many expressions in recursive definition cannot hope explicitly define a membership function for any but a small subset Fuzzy set overcome this problem providing a mechanism to which can be determined uniquely defining a mapping function for each each 112 -+ [O,l], fv : [O, 112 --+ [O, 11, f+ : [O, 112 --+ [O,1] and f, : [O, 11 -+ [O, I]. The value of xe (x) for any expression 0 can then according to following recursive rules: referred to due to the fact extends the recursive mechanism truth-values of sentences from propositional variables propositional logic. In mental assumption of fuzzy set theory that the functions coincide classical logic operators in (0, 1). Beyond this constraint it is somewhat unclear as to what should the precise combination functions. However, there is a wide consensus consensus should satisfy of axioms: fv (a, 0) = a D2 'da, b, c E [0, 11 if b 5 c then fv (a, b) fv (a, c) is a continuous function function N3 f, is a decreasing function function N4 'da E [O,l] f, (f, (a)) = a Axioms C1-C4 mean that fA is a norm or as defined by by in the context of probabilistic metric spaces. Similarly according to D1-D4 triangular conorm (t-conorm). functions satisfy t-norm and and 101. Other possibilities are, for (0, a and, for disjunction, disjunction, that fA and fv are bounded as follows: where fA is the t-norm defined defined ()0 : otherwise and fv is the drastic Concepts and Fuzzy Sets Interestingly, adding the additional idempotence restricts t-norms t-conorms to Idempotence Axioms Axioms 11 fv (a, a) = a THEOREM 2 fA satisfies CI-C.5 ifand only if fA = min Proof (+) Trivially min satisfies CI-C.5 (+) Assume fA satisfies Cl -C5 For a, b E [O,1] suppose a I b then by axioms Alternatively, for for suppose b 5 a then by axioms CI, C2, C3 and C.5 and therefore fA (a, b) = b = min (a, b) THEOREM 3 fv satisfies Dl-D5 ifand only iffv = max The most common negation again infinitely possibilities including, parameterised functions Somewhat surprisingly, functions essentially as can following theorem due to Trillas Trillas THEOREM 4 Iff, satisfies NI-N4 then ([O,l], f,, )is isomorphic to ([O, 11,l - x, 4 Proof Since f, (1) = 0 and f, (0) = 1 then by continuity follows that some value Now define MODELLING AND REASONING WITH is easy increasing and and hence Similarly, for a possible and its In view strong result assume that on to consider possible relationships between t-norms and t-conorms. constraints relating t-norms and t-conorms come from the impo- classical logic these is the duality relationship that emerges from the assumption that vague satisfy de Law i.e. In the context functional fuzzy set theory this means that: that: fv (a, b) = 1 - fA (1 - a, 1 the table in known t-norms with their associated t-conorm duals: Another obvious choice wish vague preserve is 8 (similarly 8 8). In terms (t-conorms) this leads to the idempotence axiom as we seen from theorem restricts us to t-norms with associated In additional, to constraints based on classical logical equivalences be desirable for fuzzy memberships to be additive in the generates the following equation relating t-norms and t-conorms: 'Ja, b b f~ (a, b) = a + b - fA (a, b) Making the additional assumption that the dual we obtain Frank's Frank's 'Ja, b E [0, I] fA (a, b) - fA (1 - a, 1 - b) = a + b + 1 Frank [3 11 showed that for to satisfy this equation it must correspond to an the following parameterised family family m) In Frank's t-norms the parameter effectively pro- vides a measure the dependence between the membership functions smaller the value the dependence dependence or [46]). 2.2 Functionality and As described in the previous section fuzzy logic logic 101 is truth-functional, a property which significantly reduces both the complexity and storage re- the calculus. Truth-functionality a rather strong assumption that significantly reduces the number standard Boolean prop- erties that can the calculus. For instance, Dubois and Prade MODELLING AND REASONING VAGUE CONCEPTS CONCEPTS effectively showed no non-trivial truth-functional calculus can idempotence together excluded middle. and Prade Prade Ifx is truth-functional both idempotence the law middle then that the only idempotent t-conorm LE Vx by the law then the then by negation axiom axiom somewhat controversially proved a related result Zadeh's orig- calculus. Elkan's result on the membership values this calculus is satisfy a particular re-expressions of logical implication. Clearly and Prade Prade in that it only concerns particular choice t-norm, t-conorm and negation function. function. Let fA (a, b) = min(a, b), fv (a, b) = max(a, b) and f, (a) = 1 - a. For this calculus controversy associated with this theorem theorem that such a result that previous practical ap- fuzzy logic somehow paradoxical. with Elkan's logic is it assumes a priori concepts should a specific standard logical equivalence, namely No justification this law in of vague except that a particular implication. Given such an and assuming above reduction theorem follows trivially. their reply Elkan, Dubois etal. etal. claim that has confused notions of uncertainty, they should satisfy standard Boolean equivalences while degrees truth need not. one sense this point that Elkan seems to be confusing modelling object domain with modelling the concepts in the the other hand we not agree that you can separate these two domains. is to convey information in fuzzy set linking of fuzzy membership functions with possibility distributions [115]). It does not seem reasonable that questions process should those relating vague concepts. two concepts L1 and the information convey about and the relationship this information two separate assertions Furthermore, it state that uncertainty and be) truth-functional. Rather, we claim that to develop an operational semantics for and investigate what Indeed this emphasis semantic approach forms the basis of main object Elkan's work. problem of equivalences must should be investigated within context of a particular semantics. not helpful select such an equivalence largely arbitrarily then proceed had been resolved. of operation semantics for that we section of this and throughout volume. However, for the moment we shall take a different perspective on the result of and Prade (and to a degree that it provides an insight into what assumption truth-functionality actually is. suggest that truth-functionality a special of a somewhat weaker formalizing the following property: Functionality there exists determined only from the membership values of basic labels). underlying intuition that meaning of from the component concepts while, sequel, avoiding problems highlighted and Prade and of Elkan. The measure is function function lIn -t [O,1] such that Vx E fl ve (x) = fe (VL~ (4,. . a, VL, (4) The following example shows that functional measures triviality result result EXAMPLE 9 Functional but Non-Truth Functional denote the is a probability measure then that REASONING WITH independent for all functional measure. Howevel; since defined by probability measure Clearly, howevel; be determined the function for any be evaluated recursively as a linear combinations . . . Clearly then and the and hence are not the restrictions Prude's theorem theorem 2.3 Operational Semantics for Membership Membership Walley proposes a number measure should to provide effective means of modelling uncertainty in intelligent the following interpretability requirement: 'the measure should have clear interpretation that sufficiently defined assessment, to understand the conclusions the system and as a action, and combining and updating measures' Concepts and Thus according semantics should not only provide of understanding the numerical levels of uncertainty propositions but must underlying cal- culus. In fuzzy logic logic this means than any interpretation functions should be consistent with truth-functionality. to be the case to investigate new combining imprecise In [25] Dubois and Prade suggest three possible based on the measure similarity between elements and totypes of the concept, this section we and discuss their consistency with functionality assumption the risk associated with making assertion involving vague concepts concepts )2.3.1 Prototype Semantics A direct between membership functions similarity measures been proposed a number including Ruspini [89] and Dubois Dubois [28]. The basic idea of this semantics a follows: assumed that there a prototypical instances that they satisfy denote this set of similarity which elements domain can Typically this to be a function function 11 satisfying the following for the concept then defined to be a subjective similarity between an element and the closest element from Clearly then and hence hence according to prototype semantics normalised membership functions. the type membership functions that with prototype semantics. Clearly this can problem of deciding what relationships hold between more fundamental concepts and the prototypes the component the relationships MODELLING AND REASONING WITH VAGUE CONCEPTS For the it would assume that Clearly, a prototypical not tall cannot also a prototypical would not example, someone who not prototypically not tall. of concepts might naively assume that the prototypes the intersection seen that: might wonder whether a typical person would be basis of objection to prototype theory calculus) raised Osherson etal. etal. For example, they that when the concepts then a much more prototypical of it is Interestingly when viewed at membership function level this suggests that conjunctive combination of membership it is since we would intuitively expect to have a higher membership in the extension of in either of disjunctions of concepts does seem rather For example, the prototypical might reasonably to be composed of we obtain obtain argue against the use of prototypes to model disjunctions using a counter example wealth, liquidity argument presented presented assumes that wealth corresponds to liquidity or investment, however, while there certainly a relationship between these it is not at clear that is a disjunctive one. with the following similarity measure The membership functions for tall, medium, not tall and not determined as (M, F)) MODELLING AND REASONING Clearly then person since both in What seems clear from the above discussion and example that membership similarity measures almost certainly not truth-functional and probably not even functional. For instance, specific. In the meanings is the however, that we take the prototypes corresponding to prototypes of the disjuncts calculus will problem with similarity pally related functionality of emergent calculus. notion of similarity itself. some concepts it may similarity between and prototypes. For example, in the the concept might reasonably measure the similarity between indi- in terms monotonically decreasing the Euclidean between their heights. For other concepts, however, the exact nature underlying similarity much harder consider the concept identify a itself a difficult task, we quantify between a variety different plants and the elements of this prototype set? It identify an easily measurable attribute of plants that could measure the the degree hard to define the degree membership itself! was proposed Giles in series of papers papers and [35]. In this semantics the fundamental fundamental )is that the membership Xe (x) quantifies the level of and Fuzzy assert that 0'. Formally, denote the will pay an 1 unit is false. Given this gamble we may then suppose that a risk value value I] with the assertion 8,. Such values are likely to be subjective probabilities and vary between agents, however, if (8,) will be willing to assert will certainly unwilling to make such an then define this being subjective probability that then interpreted means that will pay your 1 unit they will assert In this willing to assert false, since in this will be repaid by Your risk when asserting equivalent to means that the choice In this the risk of asserting the maximum risks associated with asserting which of two options will choose. Hence, given between membership and risk means that assert either the choice is made by You. this case, since the choice of which the two statements choose the statement minimum associated should correspond to Hence, the corresponding membership will satisfy: this semantics captures the set calculus proposed by by 101, there is something odd about the truth-functional way associated with asserting a compound expression For instance,