a note on measures of fuzziness applied to nonmonotonic fuzzy propositional logic

11
-. ," ELSEVIER Fuzzy Sets and Systems 67 (1994) 199-209 FUZZY sets and systems A note on measures of fuzziness applied to nonmonotonic fuzzy propositional logic Ian Maung 1 Department of Mathematics, University of Manchester, England, United Kingdom Received 1 April 1993; revised 1 August 1993 Abstract We undertake a mathematical investigation of the problem of inferring a truth value(s) for a propositional sentence, given truth values for some other set of sentences. Interpreting such a set of sentences and truth values as an abstraction of a fuzzy knowledge base, this mathematical problem can be seen as an abstraction of the important problem of inference in fuzzy expert systems. We postulate axioms for nonmonotonic fuzzy propositional logic (inspired by Paris and Vencovska's axioms for ampliative probabilistic inference). We prove simple but new results regarding the consistency of the axioms. We consider one method of inference. Maximum Fuzziness - infer the truth value that is most fuzzy and yet consistent with the knowledge base. We demonstrate that under certain plausible assumptions about the measure of fuzziness, several of the axioms are not sound for Maximum Fuzziness. Finally, we show that computing close approximations to truth values inferred by Maximum Fuzziness is at least as difficult as monotonic deductive propositional logic. Key word~: Measure of fuzziness; Logic; Ampliative reasoning; Inference process 1. Introduction Suppose that K of the form b(Oi)=cq for l~<i~<n, where 0~ is some propositional sentence and c~ is a fuzzy truth value (in the range [0, 1]) for each i. If we interpret the equation 'b(01)= c~' as meaning that 'the expert's degree of belief in 0~ is c~', then 1 Present address: Department of Computing, University of Brighton, Brighton, England BN2 4GJ. Email: im [email protected]. we see that K is an abstraction of the know- ledge base of an idealized fuzzy expert system [2, 7, 32]. Now, given some other sentences of interest such as 0, we would like to infer some truth values, b(O) denoting (under this interpretation) the expert's de- gree of belief in 0. A simple 'practical' example (from the domain of medical expert systems) is that the 01s may be statements relating to the presence of symptoms in a patient and diagnostic rules of the form 'if symptoms then virus present', and 0 may be of the form 'patient has virus'. This formal inference problem is thus a simplified formal abstraction of the important practical problem of inference in 0165-0114/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0165-0114(94)00147-Y

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Page 1: A note on measures of fuzziness applied to nonmonotonic fuzzy propositional logic

• -. , "

E L S E V I E R Fuzzy Sets and Systems 67 (1994) 199-209

FUZZY sets and systems

A note on measures of fuzziness applied to nonmonotonic fuzzy propositional logic

Ian M a u n g 1

Department of Mathematics, University of Manchester, England, United Kingdom

Received 1 April 1993; revised 1 August 1993

Abstract

We undertake a mathematical investigation of the problem of inferring a truth value(s) for a propositional sentence, given truth values for some other set of sentences. Interpreting such a set of sentences and truth values as an abstraction of a fuzzy knowledge base, this mathematical problem can be seen as an abstraction of the important problem of inference in fuzzy expert systems.

We postulate axioms for nonmonotonic fuzzy propositional logic (inspired by Paris and Vencovska's axioms for ampliative probabilistic inference). We prove simple but new results regarding the consistency of the axioms.

We consider one method of inference. Maximum Fuzziness - infer the truth value that is most fuzzy and yet consistent with the knowledge base. We demonstrate that under certain plausible assumptions about the measure of fuzziness, several of the axioms are not sound for Maximum Fuzziness. Finally, we show that computing close approximations to truth values inferred by Maximum Fuzziness is at least as difficult as monotonic deductive propositional logic.

Key word~: Measure of fuzziness; Logic; Ampliative reasoning; Inference process

1. Introduction

Suppose that K of the form

b(Oi)=cq for l ~ < i ~ < n ,

where 0~ is some proposi t ional sentence and c~ is a fuzzy truth value (in the range [0, 1]) for each i. If we interpret the equat ion 'b(01)= c~' as meaning that ' the expert 's degree of belief in 0~ is c~', then

1 Present address: Department of Computing, University of Brighton, Brighton, England BN2 4GJ. Email: im 1 @unix.bton.ac.uk.

we see that K is an abstract ion of the know- ledge base of an idealized fuzzy expert system [2, 7, 32].

Now, given some other sentences of interest such as 0, we would like to infer some truth values, b(O) denot ing (under this interpretation) the expert 's de- gree of belief in 0. A simple 'practical ' example (from the domain of medical expert systems) is that the 01s may be statements relating to the presence of symptoms in a patient and diagnostic rules of the form 'if symptoms then virus present', and 0 may be of the form 'patient has virus'. This formal inference problem is thus a simplified formal abstract ion of the impor tan t practical problem of inference in

0165-0114/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 4 ) 0 0 1 4 7 - Y

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200 I. Maung / Fuzzy Sets and Systems 67 (1994) 199-209

fuzzy expert systems. From now on, when we use terms such as knowledge base and expert system, we are referring to the mathematical abstraction, not the intuitive or practical concept.

It is clear that in general we cannot compute the truth value of 0 from K using fuzzy logic alone. This is analogous to exact knowledge bases (which are abstractly just collections of sentences of some logic e.g. predicates or pro- positions), where we cannot infer the truth of a sentence from the truth of other sentences by deductive logic alone. For such cases non- monotonic logics have been developed [11,24]. The most well-known example is negation-by- failure [27] by which we assume that any pro- position is false unless it can be deduced from the knowledge base. It is nonmonotonic because if we later discover that some particular pro- position that could not be deduced from the knowledge base is in fact true, then we will have to withdraw our inferred conclusion that it was false.

When many different values of b(O) are con- sistent with K, we shall need nonmonotonic axioms to infer (ideally unique) values for b(O). In this paper, we shall not treat the case where no value of b(O) is consistent with K (i.e. where K is inconsistent). We seek to specify the inference engine of a fuzzy expert system - for the situation where the knowledge base is incomplete but con- sistent. An interesting alternative approach using an interval interpretation of degree of belief (i.e. a degree of falsity as well as a degree of truth) is given in [1].

We present a formal framework for describing nonmonotonic fuzzy logic, and normative axioms for any nonmonotonic inference process. We also consider measures of fuzziness of fuzzy valuations (fuzzy subsets of sets of proposi- tions) and natural axioms for measures of fuzziness. We introduce the Maximum Fuzziness nonmono- tonic inference process and analyse its sound- ness with regard to the axioms. We discuss the consistency Of the normative axioms in the fuzzy logic framework. We conclude by proving that finding even reasonable estimates of the truth values inferred by Maximum Fuzziness is NP- hard and hence probably infeasible.

2. A framework for nonmonotonic fuzzy logic

Fuzzy logic was first applied to approximate reasoning by Zadeh [34]. Some alternative ap- proaches are given by Mizumoto [17], Raha and Ray [23], Tan et al. [28], Thornber [29] and Trillas and Valverde [31]. Recently there have been some attempts to rework the formal foundations of fuzzy logic [19, 25, 30], but we shall adopt a conventional approach here [4]. Fuzzy Logic [4] is a radically different approach to the modelling of inexact knowledge and belief, compared to probabilistic logic [18], and Dempster-Shafer theory [26]. In particular, different belief values may be assigned to logically equivalent sentences. We interpret belief values as generalized truth values, and the standard logical connectives as generalized truth functions. That is, ^, v, ~,*-~ and -7 are interpreted as functions i, u, imp,/if: [0, 1] z ~ [0, 1] and c: [0, 1] --* [0, 1], respectively. The choice of these func- tions determines the particular fuzzy logic, 2~ °. Formally,

Definition 1. A fuzzy logic, ~ is a set of functions

i ze, u ~, imp z~,/ff~: [0, 1] z ~ [0, 1]

and

ca°: [0, l] - , [0, 1].

The most popular and well-known fuzzy logic [9] is defined as follows.

Definition 2. Standard fuz zy logic, ~o., defined by the functions:

i*(a,b) = min{a,b},

u*(a,b) = max{a,b},

imp*(a,b) = min{1 - a + b, 1},

iff*(a,b) = 1 - l a - bl,

c*(a) = 1 -- a

for any a, b e [0, 1].

Suppose that SL is the set of propositional sen- tences formed from the finite alphabet L of proposi- tional symbols and the connectives A, v , - - , , and --7 by structural recursion in the usual way.

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I. Maung / Fuzzy Sets and Systems 67 (1994) 199-209 201

Let ~ denote domain restriction, and pow(S) be the set of all subsets of a set, S. ~+ is the set of nonnegative reals.

Definition 7. We say that £,e is a continuous f u z zy logic if i xe, u "~, imp -~, iff xe and c ~ are all continuous functions (with respect to the Euclidean metric).

Definition 3. Suppose that ~q' is a fuzzy logic. We call b a £# belieffunction on L if b : SL ~ [0, 1] and satisfies:

(5('1) b(O ^ 49) = i~e(b(O),b(49)),

(~'2) b(O v ~) = u~e(b(O),b(b)),

(~3 ) b(O ~ 0 ) = impSe(b(O),b(49)),

(~4 ) b(O.--~49) = iff-~(b(O),b(49)),

(&a5) b(--qO)= c:e(b(O))

for any sentences 0, 49 e SL.

Let ~f'BL be the set of L~' belief functions on L.

Definition 4. If v : L --* [0, 1], we call v a f u z zy valu- ation on L. Let VL be the set of all such v's. If v e VL and v(p)e{0, 1} for every p e L , we call v a crisp valuation on L. Let TL be the set of crisp valuations on L.

Definition 5. If ~ is a permutat ion on L and v e VL, define ~(v) 6 VL by

~(v)(~(p)) = v(p) for any p eL.

Proposition 1. For every v ~ VL and any ~ , there is some unique b ~ ~ ' B L such that b~,L = v.

Proof. Routine verification using induction on the logical complexity of sentences. For any v, we de- note this b by 5('b ~, abbreviated by b v when the choice of £/' is clear.

Proposition 2. For any Z~' and any b ~ ~q~BL, there is some unique v ~ VL such that b = ~ b v.

Proof. Let v = b~ L. The result follows by induc- tion on the logical complexity of sentences.

Definition 6. For any L, we define dL as the Euclid- ean metric on [0, l] L.

Proposition 3. Suppose that ~ is a continuous fu z zy logic and that (V,),~N+ ~-- VL. I f V, ~ V as n ~ ~ , then ZPb~"(O) ~ ._~eb~(O) for any 0 ~ SL.

Proof. By induction on the logical complexity of 0.

A fuzzy knowledge base is the fuzzy logic repre- sentation of a set of inexact rules provided by an expert. Formally, we have the following definition.

Definition 8. Suppose that K is a finite set of equations:

b(O~)= ~i ( l~<i~<n),

where 0 ieSL, ~ [ 0 , 1 ] , nePC. Then we call K a f u z zy knowledge base on L. Let -L~'VL(K)= { r e VLI ~bv(Oi) = cti for every 1 ~< i ~< n}.

If £#VL(K) ~ O, then we say that K is £~°-consis- tent. Let ~ K L be the set of all fuzzy knowledge bases that are ~-consistent .

An inference process maps fuzzy knowledge bases to sets of fuzzy valuations. Let L be the set of finite subsets of { p, [ n ~ N}.

Definition 9. Given a particular fuzzy logic, ~9 a, we say that N is an &a-inference process if

N : S f K ~ PV,

where ~q°K-- { ( L , K ) I L ~ L , K ~ C # K L } , P V = UL~L poW(VL) and N ( L , K ) ~_ VL for any L 6 L .

We write N L ( K ) instead of N ( L , K ) . For every K ~ K L and O~SL, define N L ( K ) ( O ) = {bv (O)JveNL(K)} . Then, NL(K)(O) is the set of values of b(O) inferred from K.

Nonmonotonic fuzzy logic is not arbitrary, but governed by natural principles of independence and consistency. We introduce a collection of nor- mative axioms that limit the choice of inference process. The axioms are analogous to those intro- duced by Paris and Vencovska [21, 22] for am- pliative probabilistic logic.

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202 L Maung / Fuzzy Sets and Systems 67 (1994) 199 209

Definition 10. We define a collection of logical inference principles for ~-inference processes, as follows: (1) Existence: For any L EL and any K E~KL, NL(K) ~ O.

Justification: It should be possible to infer truth values from consistent knowledge bases. (2) Compatibility: For any L E L and any K E ~KL, NL(K) ~ ~C~'VL(K).

Justification: The truth values inferred from the knowledge base, K, should be consistent with K. (3) Equivalence: If L E L and K1,KzE~q~KL and ~VL(K1) = ~#VL(K2), then NL(K~)= NL(Kz).

Justification: Only the content of the know- ledge base should determine inferred truth values, not the way in which it is expressed. (4) Uniqueness: For every LE L and any K E~q~KL, OESL, NL(K)(O) is a singleton.

Justification: Inferred beliefs can be adequately represented by a single truth value. (5) Convexity: For every L E L and KEZPKL, OESL, NL(K)(O) is a sub-interval of [0, 1].

Justification: Inferred beliefs can be adequately represented by intervals of truth values. (6) Language Invariance: Suppose that L, L ' e L and L ~ L'. If K EZPKL then NL(K)(O)= NL,(K)(O) for any OESL.

Justification: Inferred truth values should be independent of the overlying language. (7) Relevance: If L I , L 2 E L and L1 n L2 = 0, K1EZPKL1, K2EZtKLz. then NLI ~Lz(K1 u K2)(01) =NL,(K1)(OI) for every 01ESL1. (Note that

K1 w KzE ~£PKL~L~.) Justification: Irrelevant information should not

alter inferred truth values. (8) Obstinacy: For any LEL, if NL(K1) satisfies K2, that is, NL(K1) c ~VL(K2) , then NL(Ka w K2) = NL(K1).

Justification: If K2 is already inferred by Ka, then K2 is providing no new information. So adding K2 to the knowledge base should not alter inferred truth values. (9) Atomicity: Suppose that L , L ' E L and p E L. Suppose that K E &°KL and K ° is K with p replaced everywhere by 0, some sentence in L' (where L, L' are disjoint and 0 is such that {v(O)[vE VL,} = [0, 1]). Let L" = L w L ' \ {p} . Then, for any ~9ESL, NL(K)( O ) = NL,,(K°)(~O°). (Note that K ° e ~L~'KL,,.)

Justification: Re-expressing a proposition as a logical combination of simpler propositions should not alter inferred beliefs. In practice, there is a limit to the depth of analysis of a proposition, but in principle any proposition can be decomposed indefinitely. (10) Open-mindedness: Suppose that L E L, 0 E SL and K E ,~PK L and e > 0. If K w {b(0) = e} is consistent, then NL(K)(O) # {0}.

Justification: If 0 is possible according to K, we should not infer that 0 is impossible.

Contrast this with a fuzzy version of negation- by-failure: If K u {b(0)= 0} is consistent, then NL(K)(O) = {0}. (! 1) Symmetry: Suppose that L E L and that c~ is a permutation on L. If K ESfKL, O ESL, then let c~(K), a(0) be K, 0, respectively, with p re- placed by ~ (p) for any p E L, then NL(a(K))(a(O)) = NL(K)(O).

Justification: o~(K) and 0~(0) are just renamed versions of K, 0, respectively. Inferred beliefs should be invariant under renaming. (12) Continuity: Suppose that Kin, KEoL#KL for every meN. Let K,. = {b(Oi)= 0~}m)[1 ~< i ~ n} for each m e N and K be {b(Oi)=c~ill<_i<~n} where e}m~ ~i as m ~ o c for each i. Then, NL(K,,)(O)~NL(K)(O) as m ~ o e , for every OESL.

Justification: Microscopic changes in the knowledge base should not result in macroscopic changes in inferred beliefs.

3. Measures of fuzziness

The uncertainty of a fuzzy valuation is called its fuzziness. The concept of fuzziness is normally ex- pressed as a pre-order relation, ~L on the set VL for every L. The idea is that vl ~< L V2 means that v2 is at least as fuzzy as vl. We define Vl <LV2 to mean that vl <~LV2 but v2 ~ L v l . The following axioms are generally accepted as reasonable (see E5, 9]): (F1) (a) IfVl,VZe VLand [vl(p) -- ½1 /> [v2(p) -- ½l

for every pEL then v~ ~<Lv2. (b) If, in addition, [ v l ( q ) - ½l > Iv 2 (q ) - ½l

for some qEL, then Vl <Lv2.

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L Maung / Fuzzy Sets and Systems 67 (1994) 199-209 203

(F2) Suppose that v* ~ VL is defined by v~(p) - ½ for every p eL. Then, v <L v* for any

vL\{v*}. (F3) (a) If vl e TL and v2 e VL then vl <~ z V2.

(b) If, in addition, v2 ¢ TL then vl <Lv2. Note that (F2) and (F3) follow from (F1). In

addition, we introduce the following axioms: (F4) If • is a permutation on L and v ~ VL, then

v <<.L~(V) and ~(v)~< L V. (The fuzziness of a valuation should be invariant under renam- ing of propositions.)

(F5) Suppose that V,v'eVL where L = L~ ~ L2 and L1, L2 are disjoint. (a) Ifv ~ L1 <~L,V' I L1 and v $ L 2 ~L2 vt J~ L2

then v ~ L V'. (b) If v + Lt <~ L, V' ~ Ll and v + Lz < L~ V' j, L2

then v < L V'.

Definition 11. Suppose that F:V--,~ + (where V = { (L, v)[ L ~ L, v ~ VL }). We denote F(L,. ): VL ~+ by FL for any L e L. Define a pre-order relation, ~< FL, on VL, for any L ~ L, by

U 1 ~ F L U 2 if and only if FL(Vl)~< FL(U2).

If ~<FL satisfies axioms (F1)-(F5) for any L ~ L, then we call F a measure of fuzziness. We say that F is continuous if FL is continuous for every L e L and that F is normalized if Fz(v*)= ILl for any L~L.

Note that F should be continuous since micro- scopic differences in valuations should not result in macroscopic differences in their fuzzinesses. Many continuous, normalized measures of fuzzi- ness have been proposed in the literature, for example

FL(V) = -- {p~L V(p) Iogz v(P)

-t- ~ (1 -- v (p) ) log2(1 -- v(p))} pe L

was hypothesized by DeLuca and Termini (see [3]). In addition, see [5, 8, 9, 33] for alternative suggestions.

4. M a x i m u m Fuzziness Inference Process

Definition 12. Suppose that 5e is a fuzzy logic. We define an L~'-inference process, MF, corresponding to a measure of fuzziness, F by

MFL(K) = arg max ve ~/"Vt. ( K )

{FL(V)} for any K ~ ~KL ,

for each L 6 L

i.e. the set of all those ve~t'VL(K) such that FL(v) is maximized.

The Maximum Fuzziness Inference Process is motivated by the more general 'principle of min- imum information' for ampliative reasoning [9]:

'Use all the information available, but no more'.

In fuzzy logic, the fuzziness of a valuation measures the amount of uncertainty associated with it. Since an increase in information results in a reduction in uncertainty, and vice versa, we can view the nega- tive fuzziness of a valuation as a measure of its information content.

Given K, we should assume precisely the in- formation contained in K; no more and no less. Hence, we should infer from K the least informative (i.e. most fuzzy) valuations that are still consistent with K. These valuations form the set MFL(K).

We now restrict attention to those continuous fuzzy logics, £,e such that

i~(1,0) = ia~(0, 1) = 0 and i'~(1, 1) (il) i~°(0,0) = ~-- l ,

(i2) i-~(a,b) = i'~(b,a) for any a,b~[O, 1], (i3) i - ~ ( a , b ) = O o n l y i f a = O o r b = O , 04) i'~(a,b) = 1 only if a = 1 and b = 1, (ul) ua"(1, 1) = u'~(1,O) = u-~(O, 1) -- 1 and u~(O,O)

~-~ O,

(u2) u'~(a,b) = u~(b,a) for any a, be[O, 1], (u3) u-~(a,b)= 1 only if a = 1 o r b = 1, (u4) u'~(a,b) = 0 only i f a = 0 and b = O, (cl) c~(1) = O, c~(O) = 1, (c2) c'Z(a) - 0 only i fa = l, and c'~(a) = 1 only if

a ~ - O .

Justification [9]: (cl), (ul) and (il) are all justified by the

requirement of consistency with crisp logic. (i2)

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204 L Maung / Fuzzy Sets and Systems 67 (1994) 199-209

and (u2) are justified by the requirements that:

b(O ix 49) = b(49 ^ O) and b(O v 49) = b(49 v O)

for any 0, 49 t SL.1

(i3) and (u3) seem restrictive but can be derived from the natural requirements of monotonicity and idempotence: (i5) i~e(A,B) >~ i~e(a,b) for any a,b,A,B~[O, 1]

such that A ~> a and B ~> b, (u5) u'~(A,B) >~ u'~(a,b) for any a,b,A, Bt[O, 1]

such that A ~> a and B ~> b, 06) i'~(a,a) = a for any a t [ 0 , 1], (u6) u-~(a,a) = a for any a t [ 0 , 1] (where (i6) and (u6) are necessary if we assume that b(0/x 0) = b(O v O) = b(O) for any 0 e SL).

We note that standard fuzzy logic, ~q~* is con- tinuous and satisfies all of the above conditions. In this section, we consider which of the twelve prin- ciples introduced in Definition 10 are satisfied by the Maximum Fuzziness Inference Process.

Proposition 4. I f F is a continuous normalized measure of fuzziness and ~ is a continuous fuzzy lo9ic, then MF satisfies Existence, Compatibility, Equivalence, Language Invariance, Relevance, Ob- stinacy and Symmetry.

Proof. Given any L e L and any K t ~ K L , it fol- lows from Proposition 3 that ~ V L ( K ) is a closed set, and hence is compact. Since F~ is continuous, it attains its supremum on ~VL(K) and so MF~(K) ~ O. So M F satisfies Existence and hence Obstinacy. This follows since if L ~ L and K ~, K2 ~ ~C~KL, and MFL(K 1 ) satisfies K E, that is, 0 ~ MFL(K1) ~- &aVL(K2), then MF~(K1) ~_ £ f V L ( K 1 ) c~ ~ ' VL(K2) = ~-q~VL(Klw K2) --~ £~'VL(K~), and so

max {FL(V)} ~< max {FL(V)} v e M F L ( K 1 ) v~c~VL(KI) ¢-~ ~C~VL(K2)

~< max {FL(v)}. V~.LeVL(K D

Rules like this may seem compelling but lose some of their sting when one notices that [20] there is no continuous fuzzy logic such that b(O)= b(~b) whenever 0 and ~b are logically equivalent; there are, however, many noncont inuous ones.

Hence, maxw~vL(Kl~ ~ .~VL(K2){FL(V)} = maxv~ev,,(K~ { FL(V) }.

So, MFL(K1 w K 2 ) = MFL(K1), and so M F satisfies Obstinacy. Routine verification shows that M F satisfies Compatibility and Equivalence. For any c~ (a permutation on L), K t ~ K L , v t m F L ( e ( K ) ) : FL(V) >>- FL(V') for any v ' t ZeVL(e(K)). Now, v ' t S f V z ( e ( K ) ) if and only if c~- I (v ' ) t &aVL(~- l e(K)) = 5fVL(K). Also,

FL(U-I(v)) = FL(V) by (F4)

>~ FL(V') for any v' eZ~'VL(e(K))

~> FL(Ct ~(v')) by (F4).

It follows that ~ I ( v ) tMFL(K) . Hence, ct 1MFL(e(K)) ~_ MFL(K). Since this holds for any ~, K, it follows that ~MFL(e le(K)) ~_ MFL(e(K)). Hence, M F L ( K ) = e - I M F L ( ~ ( K ) ) . So, for any OESL,

MFL(K)(O) = (~-1 MFL(o~(K)))(O)

= MFL(~(K))(~(O))

and we have shown that M F satisfies Symmetry. We now show that M F satisfies Relevance (and

hence Language Invariance also). Suppose that L 1 and L 2 a r e disjoint and that K 1 E ~ K L , ,

K 2 E S f l K L 2 . Let L = L 1 u L2 . Let K = K l W K 2.

For any vtZeVL(K) , let v~ = v + L~ for i = 1, 2. Then v~ZPVL,(K~) for i = 1,2. We claim that v * t MFL(K) if and only if v* ~ MFL,(Ki) for i = 1, 2. We now prove the contrapositive of the 'only if' part of this claim. Given v* ¢ MFL~(K~), there is some Vl¢MFL, (K1) such that v* <FLU1. Define v'~ VL as follows:

v ' ( p ) = v l ( p ) i f p ~ L 1 ,

= v*(p) if p t L z .

Then, v'¢£PVL(K) and v* < ' v~v ~ L1 and v* <~FL~ V* = V'+ L2. Hence, by (F5)(b), v* <F~V' and so v* (~ MFL(K), as required. The 'if' part of this claim also follows from (F5).

Hence, M F z ( K ) = { v t ~,¢VL(K) [ v i t MFL~(Ki) for i = 1,2}. So MFL(K) (p l ) = MFL,(K1)(Pa) for any Pl e L l and so MFL(K)(01) = MFz,(K1)(Oa) for any 01 ~SL1 by induction on the logical com- plexity of 0~. So, MF satisfies Relevance and the proof is complete.

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I. Maung / Fuzzy Sets and Systems 67 (1994) 199-209 205

Proposition 5. I f F is a measure of fuzziness, then M F does not satisfy Uniqueness, Convexity, Atomic- ity, Continuity or Open-mindedness.

£P-inference process satisfying all of the principles: Existence, Compatibility, Equivalence, Convexity and Symmetry.

Proof. The following counterexamples demon- strate this result.

Convexity: Let L = { p , q } and let K be b(p v q ) = 1. Then, MFL(K) = { V l , V 2 } , where vl(p) = 1, vl(q) = ½ and v2(p) = ½, v2(q) = 1. So, M F L ( K ) ( p ) = {½,1} = MFL(K)(q). (By this ex- ample, M F does not satisfy Uniqueness).

Atomicity: Suppose that L = {p,q,r} and K is b(p ^ (q v r)) = 0. Suppose that L' = {Pl,P2,P3} and 0 = ( P l v (P2 v P3)). So L " = L u L ' \ { p } = {p~,p2,P3,q,r}. So MFc,(K °) = {v'} where v'(pi) = ½ for i = 1,2,3 and v'(q) = v'(r) = 0, whereas

MFL(K) = {v} where v(p) = O, v(q) = v(r) = ½. Hence, MFu(K°)(p °) = {bV'(p~ v (P2 v P3))} ~ {0} (by (u4)) whereas MFL(K)(p) --{0}.

Open-mindedness: Suppose that L and K are as in the counterexample for Atomicity. Then K w { b ( p ) = ½} is consistent and yet MFL(K)(p)

= { 0 } . Continuity: (assuming (i5)) Suppose that L =

{q,r} and let Km be v(q ^ r) = O,v(r) = 1/(m + 1), K be v(q ^ r) = O,v(r) = 0. If we assume (i5) then K= is consistent for each m, and MFL(K=)(q) = 0

0, whereas MFL(K)(q) = ½.

We note the following open problem. Open problem: For which Ks does M F give

the same answer for every choice of measure of fuzziness?

Proof. Let L = {p,q}. Let K be {b(p/x q)=O, b(p v q ) = 1}. Then, K is consistent, since if v'(p) = 1 and v'(q) = 0 then bV" (p /x q) = i(v'(p), v'(q)) = i(1,0) = 0 by (il). By (ul), b~'(p v q) = 1. So v' ~ZPVL(K).

Suppose that N is an ~-inference process satisfy- ing Existence, Compatibility, Equivalence and Symmetry. Since N satisfies Existence and Com- patibility, NL(K)(p ^ q) = {0} so i(v(p), v(q)) = 0 for any VENL(K) and hence, by (i3), v ( p ) = 0 or v(q) = 0. Similarly, by (u3), v(p) = 1 or v(q) = 1. If v ( p ) ¢ O then v ( q ) = O and so v ( p ) = 1 and so NL(K)(p)~_{O, 1}. Also, if OeNL(K)(p) then 1 eNL(K)(q). But now let a be the permutation of L swapping p and q. Then, ~ V L ( K ) = ZeVL(a(K)) by (i2) and (u2), and:

NL(K)(p) = NL(~(K))(p) since N satisfies Equivalence

= NL(~(K))(~(q))

= NL(K)(q) since N satisfies Symmetry.

So, if O~NL(K)(p) then I ~ N L ( K ) ( p ) also. Sim- ilarly, if l eNL(K) (p ) then OeNL(K)(p) also. Hence, {0,1} __ NL(K)(p) by Existence and so NL(K) (p )= {0,1}, and so N does not satisfy Convexity.

5. Consistency of the axioms

So, using standard fuzzy logic, Ae*, and the intuitively plausible Maximum Fuzziness Infer- ence Process, leads to a rejection of the extremely natural reasoning principles: Convexity, Atomicity, Continuity and Open-mindedness. We now show that for a very wide range of fuzzy logics, £P, the principles of Existence, Compatibility, Equivalence, Convexity and Symmetry are mutually inconsistent.

6. lnfeasibility of maximum fuzziness

We now consider the complexity of Maximum Fuzziness Inference Processes. We assume that £P is a continuous fuzzy logic, satisfying the condi- tions introduced in Section 4.

Definition 13. Define d* : [0, 1] x pow([O, 1]) by

d*(x ,X) = i n f { l x - y l l y e X }

for any x e [0, 1], X _c [0, 1].

Theorem 1. Suppose that 5~ is a fuzzy logic satisfy- ing (il), (i2), (i3), and (ul), (u2), (u3). Then, there is no

Definition 14. If N is an Ae-inference process, define A P P R OX (N ,e ) to be the problem: Given

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206 1. Maung / Fuzzy Sets and Systems 67 (1994) 199-209

L e L , K ~ P K L and OeSL, compute v(O) such that

d*(v(O), NL(K)(O)) <~ e.

Let CNFL = {0 ~ SL[O is in conjunctive normal form}.

Lemma 1. Suppose that O eCNFL and v e VL. I f 0¢ SAT, then b~(--nO) > O.

Proof. Suppose that O~SL and 0 = (01 /x ( 0 2 . . .

^ (0,_ 1 /x 0,) ... ) where the Oi's are disjunctions of literals. Suppose that vsVL and b~(--nO)= O. Then, by (c2), b~(O) = 1. Then, by (i4), b~(O~) = 1 for every 1 ~< l~< n. Fix I. Suppose that 0t is (c~ ~) v (. . . v c~Z))...) for this 1. Then b~(c} ~)) = 1 for some 1 ~<i~< k, by (u3). Define t eTL by t(p) = 0 if v(p) < 1 (and t(p) = 1 otherwise). Then, bt(c} °) = 1 by (cl) and (c2), and so bt(Ol)= 1 by (ul). Since this holds for every 1 <~ l <% n, bt(O) = 1 by (i 1). Hence, 0 e SA T, a contradiction. Hence, for any ve VL, b~(--nO) > 0, as required.

Theorem 2. Suppose that F is a continuous, nor- malized measure of fuzziness and that M F is the corresponding Maximum Fuzziness 5('-Inference Process. Then, for any 0 <~ e < ¼, A P P R O X ( M F , e) is NP-hard.

Proof. Assume that 0 ~< ~ < ¼. We show that the NP-complete problem, CNFSA T (testing the satis- fiability of propositional sentences in CNF) is linear-time reducible to APPROX(N,e) . The NP- completeness of C N F S A T is demonstrated in [6]. Given O~CNF, let L1 = { p ] p appears in 0}. Let m = ]LlI,L1 = {pl . . . . . Pro}. Let n = m + 1 and L2 = {ql . . . . . q,} where the qi's are disjoint from L~. Let L = L1 u L 2 . Define K to be the fuzzy knowledge base consisting of the single equation:

b(--nO/x 4)) = O,

where ~b = (q~ v (q2 v ... v (q,-1 v q , ) . . . ) . Then, if O¢SAT, by Lemma 1, bV(--nO)>0 for every v~ VL. Since 0 # MFL(K) ~_ ~ V L ( K ) (by Existence), MFL(K)(O)= {0} and so MFL(K)(q~) = {0} for every 1 ~< i ~< n.

Suppose now that O~SAT. Let K1 be the know- ledge base consisting of the single rule: b(--n 0) = 0. Then, --10 is not a tautology and so there is some t e TL, such that bt(--nO) -- 0, and hence K1 is con- sistent. So by Existence, MFL, (K1)v e O. Choose some Vl ~MFLI(K1). Suppose that v E ~ V L ( K ) and that bV(-70)>O. Then, v(ql )=O for l~<i~<n. Define v'eVl~ by v'(qi)=½ for l ~ i ~ < n and v '$ LI = vl. Then, v' e ~ V L ( K ) .

Suppose that c(:L--. L by c~(pj) = qj, c~(qj) = pj for 1 ~<j ~< m and ~(q , )= q,. Then, by (F3)(a), ~(v)+ L1 ~<Llv' J, L1 since ~(v) J, L1ETLI. Also, by (F2), ~(v) + L2 <L2v' ~ L2 since (~(v))(q,) = v(q,) = 0. Hence, by (F5)(b), ~(v) < Lv' and so v < Lv' by

(F4). So, if veMFL(K) , then bY(--nO)= 0, and it follows easily from (F2) and (F5)(b) that v ~ Lz = v'~ L2 for any vEMFL(K). Hence, MFL(K)(qi) = {½} for every l~<i~<n. So, if v (q l )>¼ then M F r ( K ) ( q l ) = {½} and so OeSAT. Whereas, if v(ql) < ¼ then MFL(K)(ql ) = {0} and so 0 ¢ SAT. So, given 0 ~< e < ¼ and 0~CNF, we have con- structed K, L linear in the length of 0 such that computing MFL(K)(q l ) to accuracy within e is sufficient to decide if 0 is satisfiable. So C N F S A T is linear-time reducible to A P P R O X (MF, e), and by the NP-completeness of CNFSA T, A P P R O X ( M F , e) is therefore NP-hard.

7. Discussion

In this section, we discuss some alternative fuzzy logics to delimit the scope of the above results. Consider a fuzzy logic, A ° such that

i(a,b) = max{0,a + b - 1},

u(a,b) = min{1,a + b},

imp(a,b) = min{1 - a + b, 1},

c (a )= 1 - - a

for any a, b ~ [0, 1]. Note that 5¢ does not satisfy idempotence ((i6),

(u6)) or (i3), (u3). So Theorem 1 does not apply to A a. But A ° does satisfy (il), (i2), (ul), (u2), (cl), (c2), (i4), (u4) and the monotonicity conditions, (i5), (uS).

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1. Maung / Fuzzy Sets and Systems 67 (1994) 199-209 207

Proposition 6. I f FL(v) = , ~ 1 / 2 -- dL(V,V*), then MFL (K) is a singleton if (i) each Oi in K has at most one connective OR,

(ii) each Oi in K is a clause and, for each i, 0 < ~ i < 1.

Proposition 7. &PVL(K) is a finite union of convex sets.

Proof. Induct ion on the structural complexi ty of the proposi t ions , Oi appear ing in K.

Proof. No te that FL is a cont inuous measure of fuzziness, and fur thermore FL is strictly concave on [0, 1] L. Fo r each of (i) and (ii), we can show that K is equivalent to a convex set of constraints on v and hence FL has a unique m a x i m u m on the convex set £PVL(K). (i) is immedia te f rom the definition of L~ a. Fo r par t (ii), suppose that 0i is (Pl ^ "'" /x P,)--*(ql v ... v q,,) and that ~i is 0 < c ~ < 1. Then, b ( (p l /x ... A p , ) - + ( q l v ... v qm)) = c~ implies that

1 - b ( p i /x ... / x p , ) + b(qi v ... v q,,)=o~. (*)

Note that b(p l ^ -.. ^ p , ) > 0 and b(ql v ... v qm) < 1 since e < 1. So b(pi A ... ^ p , ) = Z7-1 b(pi)

?n -- ( n - - 1) and b(ql v ... vq , , ) = Zj=~ b(qj). Sub- sti tuting in ( • ) gives

• b ( p i ) - ~ b ( q j ) = n - o ~ . i = 1 j - I

So M F satisfies Uniqueness, when K is restricted. No te that bo th condit ions (i) and (ii) above fail for s tandard fuzzy logic. An appropr ia t e counter- example for (i) appears in the p roof of Theo rem 1. For (ii), consider K, the fuzzy H o r n clause know- ledge base:

b(p3 ~ q) = 0.7,

b(p3) = 0.5,

b(pl A P2 ~ q) = 0.8.

Then, M F L ( K ) = { v l , v 2 } where v l ( p l ) = 0 . 4 , v l (P2) = 0.5, v l ( P 3 ) = 0.5, v l ( q ) = 0.2 and v2(pl ) = 0.5, v2(P2) = 0.4, Vz(p3 ) = 0.5, Vz(q) = 0.2. So

M F does not even satisfy Convexi ty for these re- strictions of K.

The logic £~' also satisfies the following p roper ty for general fuzzy knowledge bases, K.

A simple corol lary is that MFL(K) is always a finite set for FL strictly concave. However , we conjecture that MFL(K) is not a singleton in gen- eral, and hence M F does not satisfy Convexity. No te also that M F does not satisfy Continuity. To see this, note that if K, . is v(q v r) = 1 -- 1/(2m), v(r) = 1 - ( l / m ) and K is v(q v r ) = 1, v ( r ) = 1, then MFL(Km)(q) = 1/(2m) --* 0 as m --* o¢ whereas MFL(K)(q) = ½.

We next consider a ra ther extreme and unnatura l collection of fuzzy logics; those for which each of i, u, imp, iff, and c are linear. In this case, ~ V L ( K ) is convex for any K ~ £PKL, and so if FL is strictly concave, MFL(K) is unique and so M F satisfies Uniqueness. Indeed, M F satisfies all of the ax ioms of Definit ion 10, except, in general, Atomici ty and Open-mindedness . If we define each of i, u, imp, iff, c as constant functions with value ½, then MFL(K) = v* for every K ~ ~q~KL, and satis- fies all of the principles stated, including Atomici ty and Open-mindedness . However , note that there is no linear fuzzy logic satisfying (il) and (ul) and so we have given up agreement with crisp logic.

8. Conclusions and future directions

We have formula ted an ax iomat ic app roach to n o n m o n o t o n i c fuzzy propos i t ional logic, and evaluated the axioms for consistency and sound- ness with respect to inference based on M a x i m u m Fuzziness. The results presented apply to a wide range of fuzzy logics and measures of fuzziness but are by no means exhaustive. It is an interesting open p rob lem to discover if there is any natural fuzzy logic for which the set of ax ioms is consistent, and if there is any measure of fuzziness for which the M a x i m u m Fuzziness Inference Process satisfies all of the axioms. Given the unique character iza- t ion theorems of M a x i m u m En t ropy [21, 15] and M i n i m u m Specificity [14] for n o n m o n o t o n i c prob- abilistic and possibilistic logic, respectively, it seems

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208 L Maung / Fuzzy Sets and Systems 67 (1994) 199-209

at least possible that a similar characterization can be found for a particular choice of fuzzy logic and measure of fuzziness.

In exact reasoning, nonmonotonic logics such as negation-by-failure can lead to inconsistencies e.g. given --q(p ^ q) and p v q, we would infer both --qp and --7 q which is inconsistent (compare with Theorem 1). However, for particular kinds of exact knowledge bases (e.g. logic programs [10], consisting exclusively of Horn clauses), it is known that negation-by-failure never leads to inconsistencies [27]. It would be interesting to see if we can find similar restrictions on fuzzy logic knowledge bases that ensure that Maximum Fuzziness is well-behaved with respect to the principles. Some simple results in this direc- tion appear in Section 7.

Another problem is that the results presented only concern the case where K is consistent. In practice, the K abstracted from the knowledge base 'mined' from an expert may be inconsistent, and the theory gives no guidance for this case.

A final theoretical issue is the question of extending the scope of the work to fuzzy predicate logic.

It is our hope that the theoretical study initiated here will play the same foundational role for fuzzy expert systems as the theory of deduc- tive and nonmonotonic predicate logics does for certain rule-based and logic-based (e.g. PROLOG) expert systems. However, this investigation has no pretensions of having immediate practical significance for fuzzy expert systems technology, though an empirical study to determine the validity of the axioms and abstrac- tions assumed here might reveal some practical consequences, and is an interesting topic for further research.

Acknowledgements

I would like to thank my Ph.D. supervisor, Jeff Paris, for interesting and lively discussions which have helped to shape the work presented here. I also acknowledge financial support from the Science and Engineering Research Council of Great Britain.

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