Probabilities , Possibilities , and Fuzzy Sets John A. Drakopoulos Stanford University Department of Computer Science Knowledge Systems Laboratory 701 Welch Road , Palo Alto , CA 94304-0106 January 24 , 1994 Abstract A formal analysis of probabilities , possibilities , and fuzzy sets is presented in this paper . A number of theorems proved show the above measures have equal representational power when their domains are infinite . However , for finite domains , it is proved that probabilities have a higher representational power than both possibilities and fuzzy sets . The cost of this increased power is high computational complexity and reduced computational efficiency . The resulting trade-off of high complexity and representational power versus computational efficiency is discussed under the spectrum of experimental systems and applications . 1 Introduction Probabilities , possibilities , and fuzzy sets are all measures used to formalize and quantify uncertainty . There is an on-going debate regarding the appropriateness of each measure in formalizing uncertainty . Arguments in favor of probabilities can be found in -LSB- 21 , 5 -RSB- while arguments more in favor of possibilities and fuzzy sets are presented in -LSB- 20 , 17 -RSB- . A brief presentation and qualitative comparison of the above measures as well as MYCIN 's certainty factors -LRB- -LSB- 30 -RSB- -RRB- and Dempster-Shafer theory -LRB- -LSB- 8 , 29 -RSB- -RRB- appear in -LSB- 15 -RSB- . However , a quantitative comparison of those measures in terms of both efficiency and expressiveness is not given . Such a comparison is necessary in order to evaluate and characterize those measures . Some attempts to this direction regard a notion of consistency between probabilities and possibilities -LSB- 9 , 7 -RSB- and transformations from probabilities and possibilities to Dempster-Shafer theory -LSB- 18 -RSB- . However , those transformations can not tell us about the exact relationship of probabilities and possibilities . Furthermore , the transformations from possibilities to Dempster-Shafer theory as appear in -LSB- 18 -RSB- require the elements of the universal set to be ordered in descending possibility values . However , such an ordering does not always exist when the universal set is infinite . This limits the applicability of the transformation to finite universal sets . Finally , there has been no study so far of the extensions of the universal sets that are necessary in order to create mappings among probabilities , possibilities , and fuzzy sets . In this paper , we intend to resolve the above issues by presenting a number of theorems that specify some important relationships between probabilities , possibilities , and fuzzy sets . The proofs made here use some formal definitions of the above measures . However , since there is a wide variability in the fuzzy set theory literature in terms of definitions , the results presented here should be construed as applicable only in cases where the definitions used are either equivalent or imply ours . On the other hand , it has been proved in -LSB- 3 , 14 -RSB- that , under very reasonable assumptions , the only truth functional connectives for fuzzy sets that are possible are the usual max?min -LRB- or equivalently sup ? inf which are used here -RRB- . Thus our definitions correspond to the most conventional case which is -LRB- most probably -RRB- most widely used . The rest of this paper is organized along the general scheme of some preliminary definitions followed by a number of theorems and a discussion of their implications to systems , architectures , and applications . In the next section , basic definitions are introduced . These set up the context under which our theorems are presented and consist the foundations of probability , possibility , and fuzzy set theory . In section 3 , we present a number of theorems that relate those theories and compare them from the viewpoint of relative expressiveness i.e. the ability of one to simulate the others . For infi - nite domains , it is proved that all measures have the same expressiveness and as such can be considered to be equivalent in terms of representational power . However , for finite domains , probabilities are proved to be more expressive than both possibilities and fuzzy sets but they have higher computational demands . Possibilities and fuzzy sets can actually simulate probabilities but , in that case , their space requirements are exponential when compared to those of probabilities . The increased complexity of probabilities gives rise to a trade-off which is the subject of section 4 . This is a trade-off of complexity and representational power versus efficiency . Experimental systems and approaches as well as different applications are discussed . Finally , in our concluding section , it is indicated that both probabilistic and possibilistic approaches are worthwhile . A choice should be made individually for each application depending on its demands in terms of accuracy and efficiency . 2 Mathematical Measures , Systems , and System Classes A formal definition of probability , possibility , and fuzzy set theory is given below . We begin with the definition of a sigma-algebra :1 Definition 1 A sigma-algebra B on a set ? is a set of subsets of ? satisfying the following properties : -LRB- a -RRB- ? 2 B -LRB- b -RRB- 8A -LRB- if A 2 B then ? A 2 B -RRB- ; ? A stands for the complement of A -LRB- c -RRB- if A1 ; A2 ; ... is any sequence of sets in B then -LSB- i Ai 2 B Now , we define define a probability system and a probability measure2 as follows : Definition 2 A probability system is a triple , -LRB- ? ; B ; P -RRB- , where ? is an arbitrary set -LRB- the set of all possible outcomes -RRB- , B is a sigma-algebra on ? -LRB- the set of the events of interest -RRB- and P is a real valued function defined for each A 2 B such that : -LRB- a -RRB- 8A 2 B < = P -LRB- A -RRB- < = 1 -LRB- b -RRB- P -LRB- ? -RRB- = 1 -LRB- c -RRB- if A1 ; A2 ; ... is any sequence of pairwise disjoint sets in B then P -LRB- S1i = 1 Ai -RRB- = P1i = 1 P -LRB- Ai -RRB- A function P that satisfies the three conditions above is called a probability measure . The third property is called countable additivity . For a discussion of sigma-algebras and probability measures see -LSB- 26 -RSB- . Similarly , we define a possibility system , and a possibility measure -LRB- -LSB- 10 -RSB- -RRB- : Definition 3 A possibility system is a triple , -LRB- ? ; B ; ? -RRB- , where ? is an arbitrary set -LRB- the set of all possible outcomes -RRB- , B is a sigma-algebra on ? -LRB- the set of the events of interest -RRB- and ? is a real valued function defined for each A 2 B such that : -LRB- a -RRB- ? -LRB- ;-RRB- = -LRB- b -RRB- ? -LRB- ? -RRB- = 1 -LRB- c -RRB- 8I if fAi = i 2 Ig is a set of sets in B then ? -LRB- S i2I Ai -RRB- = supi2I ? -LRB- Ai -RRB- Now the function ? is a possibility measure . -LRB- Note that no restriction is imposed upon the set I which could be uncountable -RRB- .3 Furthermore , we define system classes to be classes of different systems . Hence , we define : Definition 4 A probability system class P ? is the class of all probability systems -LRB- ? ; 2 ? ; P -RRB- . A possibility system class ?? is the class of all possibility systems -LRB- ? ; 2 ? ; ? -RRB- . A measure F is called extensional iff for any A1 ; A2 ; ... the value of F -LRB- S1i = 1 Ai -RRB- can be computed from the values of F -LRB- A1 -RRB- ; F -LRB- A2 -RRB- ; . . . without any direct reference to A1 ; A2 ; ... . Otherwise , F would be called intensional . For example , probability measures are intensional for P -LRB- S1i = 1 Ai -RRB- depends not only on P -LRB- A1 -RRB- ; P -LRB- A2 -RRB- ; ... but also on whether the sets A1 ; A2 ; ... are disjoint or not . On the other hand , possibility measures are extensional for ? -LRB- S1i = 1 Ai -RRB- = sup1i = 1 ? -LRB- Ai -RRB- . The above definition can be generalized to extensional and intensional systems -LSB- 28 -RSB- . The actual trade-off has been claimed to be between computational efficiency and semantic clarity or content -LSB- 28 -RSB- . In the extreme case , one can define an extensional measure that is trivial to compute but carries little or no information . On the other hand , one can define an intensional measure that carries a lot of information but is difficult to compute . However , this is mostly an empirical observation and no mathematical proofs have been given in the literature to establish it . Nevertheless , in this paper , we prove that for the particular intensional and extensional measures studied -LRB- i.e. probabilities , possibilities , and fuzzy sets -RRB- there is a trade-off between computational efficiency and representational power , when the domains of the measures are finite . For infinite domains , possibilities and fuzzy sets have an advantage in terms of computational complexity while no measure has an advantage over the others in terms of representational power as defined in section 3 . Another important difference between probability and possibility measures is that the former implies the formula P -LRB- ? A -RRB- = 1 ? P -LRB- A -RRB- -LRB- 1 -RRB- where ? A stands for the complement of A . A similar formula is not implied by the definition of possibilities . Furthermore , if their definition is extended to include the above formula then our measure ? reduces to binary possibility : 8A -LRB- ? -LRB- A -RRB- = 1 _ ? -LRB- A -RRB- = 0 -RRB- since 1 = ? -LRB- ? -RRB- = ? -LRB- A -LSB- ? A -RRB- = sup -LRB- ? -LRB- A -RRB- ; 1 ? ? -LRB- A -RRB- -RRB- -LRB- 2 -RRB- In 1965 , L.A.Zadeh overcame this difficulty by abandoning traditional set theory and finding shelter in what he called fuzzy set theory -LSB- 34 -RSB- . The basic idea in that theory is the notion of a fuzzy set which is the mathematical concept of set that allows partial membership for elements in its domain . Thus , an element x may partially belong to a fuzzy set A. Each fuzzy set is characterized by its membership function fA -LRB- x -RRB- that states , for any given x in the domain of A , \ how much of x belongs to A '' . Furthermore , Zadeh defined fuzzy set subset-hood based on membership functions : 8A ; B -LRB- A ? B , fA < = fB -RRB- where fA < = fB means that 8x -LRB- fA -LRB- x -RRB- < = fB -LRB- x -RRB- -RRB- . However , it should be noted that the above definition of fuzzy set subsethood is not universally accepted within the fuzzy set community and is often against the meaning of \ if A then B '' . Before , we give a formal definition of fuzzy sets let us first introduce the concept of the base of a sigma-algebra : Definition 5 A set is a base of a sigma-algebra B iff the closure of under set union and set complementation -LRB- wrt the universal set ? of B -RRB- is equal to B -LRB- written as c -LRB- -RRB- = B -RRB- and no proper subset of has this property . Now , the complete definition of a fuzzy set system is as follows : Definition 6 A fuzzy set system is a triple , -LRB- ; D ; f -RRB- , where is a set of all fuzzy sets of interest , D is the domain of the fuzzy sets in , and f is a real valued function defined for each A 2 and each x 2 D such that : -LRB- a -RRB- 8x 2 D f ; -LRB- x -RRB- = ; ; is the empty set in -LRB- b -RRB- 8x 2 D f ? -LRB- x -RRB- = 1 ; ? is the universal set in -LRB- c -RRB- 8I if fAi = i 2 Ig is a set of fuzzy sets in then 8x 2 D fSi2I Ai -LRB- x -RRB- = supi2I fAi -LRB- x -RRB- 8x 2 D fTi2I Ai -LRB- x -RRB- = infi2I fAi -LRB- x -RRB- -LRB- d -RRB- 8A 2 8x 2 D f ? A -LRB- x -RRB- = 1 ? fA -LRB- x -RRB- Now observe that parts -LRB- c -RRB- ; -LRB- d -RRB- in the above definition should not be seen as requirements on the function f but rather as definitions of fuzzy set union , intersection and complementation . For example , the union of a sequence of fuzzy sets A1 ; A2 ; ... is defined to be a fuzzy set whose membership function is equal to sup1i = 1 fAi . Furthermore , we can define the closure c -LRB- -RRB- of recursively as follows 8A 2 A 2 c -LRB- -RRB- 8I if fAi = i 2 Ig is a set of fuzzy sets in c -LRB- -RRB- then S i2I Ai 2 c -LRB- -RRB- Intuitively , c -LRB- -RRB- is the fuzzy analog of sigma-algebras as defined for ordinary sets . Pushing this analogy further , if has the property that for no proper subset of would be c -LRB- -RRB- = c -LRB- -RRB- then will be called a base of c -LRB- -RRB- . Alternatively , we could have defined fuzzy sigma-algebras to be closures of sets of fuzzy sets such as above . Now , a fuzzy set system class F ;D is the class of all fuzzy set systems -LRB- ; D ; f -RRB- . Fuzzy membership functions , seen as measures on fuzzy sets , are extensional measures similar to possibilities . However , in fuzzy set and in particular in fuzzy logic literature , fA -LRB- x -RRB- has often been regarded as a compatibility function i.e. a function that denotes the degree in which the proposition \ x is A '' is true -LRB- see -LSB- 35 -RSB- for more details -RRB- . For example , if A stands for the imprecise concept of being \ young '' then fyoung -LRB- Andrew -RRB- is the degree to which it is true that \ Andrew is young '' . Both interpretations are valid but do not affect the formal definition of fuzzy set systems as given above . However , in author 's opinion viewing fuzzy set membership functions as measures is a mathematically clearer interpretation than seeing them as compatibility functions for imprecise concepts such as \ young '' above . Finally , since A -LSB- ? A ae ? for all purely fuzzy sets A , derivation -LRB- 2 -RRB- above does not hold i.e. fuzzy set membership functions are not necessarily binary as those of ordinary sets . However , it should be noted that , in addition to the classical sup ? inf pair , several algebraic structures on the interval -LSB- 0 ; 1 -RSB- have been defined and studied in the literature and consequently have been used to define fuzzy set connectives -LRB- see -LSB- 2 , 1 , 14 , 32 -RSB- -RRB- . Those alternative definitions relax some of the requirements on the connectives -LRB- mainly distributivity -RRB- so that the definitions are consistent . On the other hand , as proved in -LSB- 3 , 14 -RSB- , under very reasonable assumptions -LRB- mainly distributivity and monotonicity -RRB- the classical sup ? inf pair is not only natural but also the only one possible . Thus , our definition here , which is based on the sup ? inf connectives and is equivalent to Zadeh 's original definition -LRB- -LSB- 34 -RSB- -RRB- , corresponds to the most natural and most widespread definition used in the fuzzy set research community . Nevertheless , it is worth noting that our theorems are intented for the classical connectives for fuzzy sets and do not necessarily apply for other connectives . 3 System Relationships An important question is whether probability and possibility measures or systems -LRB- fuzzy or not -RRB- are actually different . A number of theorems presented below prove that they have some strong similarities as well as differences . To study them formally , we need to define the notion of relative expressiveness of one system with respect to another . In the following , we write -LRB- ? ; B ; M -RRB- to indicate a system where ? is an arbitrary set , B is a sigma-algebra on ? , and M is a real valued function defined for each A 2 B. Furthermore , we write 2A to indicate the power set of A , for any set A. Now , we define : Definition 7 A system S = -LRB- ? ; B ; M -RRB- is at most as expressive as a system S0 = -LRB- ? 0 ; B0 ; M -RRB- -LRB- written as S < = S0 -RRB- iff 8A 2 B 9A0 2 B0 M -LRB- A -RRB- = M -LRB- A0 -RRB- : The intuition behind this definition is that if S < = S0 then the system S0 can simulate S by using M -LRB- A0 -RRB- whenever S uses M -LRB- A -RRB- . Of course , this raises the question about the existence of a deterministic algorithm or method that could derive A0 given A ; M , and the system class to which M belongs . Our definition above requires only the establishment of the existence of a set A0 . However , in the theorems proved in this paper that determine expressiveness relationships among measures , constructive proofs are given which describe how to construct not only A0 but also ? 0 and M given A ; M , and the system class of M . Now , similarly as before , we define : Definition 8 A system class C ? is at most as expressive as the system class C0 ? 0 -LRB- written as C ? < = C0 ? 0 -RRB- iff 8M -LRB- ? ; 2 ? ; M -RRB- 2 C ? -RRB- 9M -LRB- ? 0 ; 2 ? 0 ; M -RRB- 2 C0 ? 0 -LRB- ? ; 2 ? ; M -RRB- < = -LRB- ? 0 ; 2 ? 0 ; M -RRB- and ! Therefore , C ? < = C0 ? 0 iff each system in C ? can be simulated by a system in C0 ? 0 . Similarly , for fuzzy set systems we define : Definition 9 A system S = -LRB- ? ; 2 ? ; M -RRB- is at most as expressive as a fuzzy set system S0 = -LRB- ; D ; f -RRB- -LRB- written as S < = S0 -RRB- iff 8A 2 2 ? 9A0 2 c -LRB- -RRB- 9x 2 D M -LRB- A -RRB- = fA0 -LRB- x -RRB- : Furthermore , S0 < = S iff 8A0 2 c -LRB- -RRB- 8x 2 D 9A 2 2 ? M -LRB- A -RRB- = fA0 -LRB- x -RRB- : In the case S < = S0 , the fuzzy set system S0 can simulate the system S while in the case S0 < = S the system S can simulate the fuzzy set system S0 . Now , we define : Definition 10 A system class C ? is at most as expressive as the system class F ;D -LRB- written as C ? < = F ;D -RRB- iff 8M -LRB- ? ; 2 ? ; M -RRB- 2 C ? -RRB- 9f -LRB- ; D ; f -RRB- 2 F ;D -LRB- ? ; 2 ? ; M -RRB- < = -LRB- ; D ; f -RRB- and ! Similarly , F ;D < = C ? iff 8f -LRB- ; D ; f -RRB- 2 F ;D -RRB- 9M -LRB- ? ; 2 ? ; M -RRB- 2 C ? -LRB- ; D ; f -RRB- < = -LRB- ? ; 2 ? ; M -RRB- and ! In the following , we denote the cardinal of a set A as card -LRB- A -RRB- and the cardinal of the set N of natural numbers as @ 0 -LRB- see -LSB- 31 -RSB- for more details on cardinal numbers -RRB- . Now , we can proceed presenting and proving a few second order logic theorems that relate probability , possibility , and fuzzy set systems : Theorem 1 If ? is finite then ?? < = P ? Proof To prove the above theorem we must prove that , given a finite set ? and a possibility system -LRB- ? ; 2 ? ; ? -RRB- , we can construct a probability system -LRB- ? ; 2 ? ; P -RRB- that can simulate it . Therefore , we need to define P so that 8A 2 2 ? 9A0 2 2 ? ? -LRB- A -RRB- = P -LRB- A0 -RRB- To this purpose , since ? is finite , we can reorder its elements and write ? as f ! 1 ; ! 2 ; ... ! ng ; for some n 2 N ; so that ssi < = ssi +1 ; for i = ; 1 ; 2 ; ... ; n ? 1 ; where ssi = ? -LRB- ! i -RRB- ; i = 1 ; 2 ; ... ; n ss0 = Now , we define P -LRB- ! i -RRB- = ssi ? ssi ? 1 ; i = 1 ; 2 ; ... ; n and require P to satisfy the countable additivity property . This completely defines P over 2 ? . Furthermore , we define A0 = f ! 1 ; ! 2 ; ... ; ! sAg where sA = supfi = ! i 2 Ag . Then it is ? -LRB- A -RRB- = sA supi fssi = ! i 2 Ag = sssA = sA X i = 1 ssi ? ssi ? 1 = sA X i = 1 P -LRB- f!ig -RRB- = P -LRB- A0 -RRB- and obviously 8A 2 2 ? = P -LRB- ;-RRB- < = P -LRB- A -RRB- < = P -LRB- ? -RRB- = ? -LRB- ? -RRB- = 1 Therefore , -LRB- ? ; 2 ? ; P -RRB- would be the desired probability system . Q.E.D. Theorem 2 8 ? P ? < = ? 2 ? Proof We would like to prove that 8 ? ; P 9 ? -LRB- ? ; 2 ? ; P -RRB- < = -LRB- 2 ? ; 22 ? ; ? -RRB- Therefore , we have to construct ? , given ? ; P . This is a straightforward construction . Simply define ? as follows : ? -LRB- fA0g -RRB- = P -LRB- A0 -RRB- ; 8A0 2 2 ? ? -LRB- ;-RRB- = ? -LRB- 2 ? -RRB- = 1 ? -LRB- -LSB- 1i = 1Ai -RRB- = 1 sup i = 1 ? -LRB- Ai -RRB- ; 8A1 ; A2 ; ... 2 22 ? This completely defines ? over 22 ? . Furthermore , it is trivial to verify that ? is a possibility measure . Therefore , the possibility system -LRB- 2 ? ; 22 ? ; ? -RRB- is the desired one . Q.E.D. Theorem 3 If ? 0 is finite and card -LRB- ? -RRB- + 1 < card -LRB- 2 ? 0 -RRB- then not -LRB- P ? 0 < = ?? -RRB- Proof We assume that ? 0 is finite and that card -LRB- ? -RRB- + 1 < card -LRB- 2 ? 0 -RRB- . We would like to prove that 9P 8 ? not -LRB- -LRB- ? 0 ; 2 ? 0 ; P -RRB- < = -LRB- ? ; 2 ? ; ? -RRB- -RRB- However , if it were -LRB- ? 0 ; 2 ? 0 ; P -RRB- < = -LRB- ? ; 2 ? ; ? -RRB- then it should be that 8A0 2 2 ? 0 9A 2 2 ? P -LRB- A0 -RRB- = ? -LRB- A -RRB- and so card -LRB- P2 ? 0 -RRB- < = card -LRB- ? 2 ? -RRB- -LRB- 3 -RRB- where PX = fP -LRB- A0 -RRB- = A0 2 Xg , for any X . ? X is defined similarly . Now , to prove the theorem , it is enough to construct P so that the above inequality can not hold for any ? . Since ? 0 is finite we can write it as ? 0 = f ! 01 ; ! 02 ; ... ! 0ng , for some n 2 N . Now we can define P -LRB- f ! 0ig -RRB- = m 1 2i ; i = 1 ; 2 ; ... ; n where m is a constant chosen so that the probabilities above sum up to one -LRB- i.e. m = 2n 2n ? 1 -RRB- . This choice of basic probability values make P to be an 1-1 function in 2 ? 0 since now the probability values of two different sets are equal if and only if those sets contain exactly the same ! 0is . Therefore card -LRB- P2 ? 0 -RRB- = card -LRB- 2 ? 0 -RRB- -LRB- 4 -RRB- Furthermore , measure ? , for each A 2 2 ? , computes a supremum function over the elements of ? contained in A. Therefore , it has to be a function onto ?? = f ? -LRB- f!g -RRB- = ! 2 ? g . The only exception , is when A = ; . In that case , ? -LRB- A -RRB- = 0 , a value which may not belong to ?? . Therefore , card -LRB- ? 2 ? -RRB- = card -LRB- ?? -RRB- + 1 < = card -LRB- ? -RRB- + 1 -LRB- 5 -RRB- Theorem 's hypothesis and equations -LRB- 4 -RRB- , -LRB- 5 -RRB- result in card -LRB- P2 ? 0 -RRB- > card -LRB- ? 2 ? -RRB- -LRB- 6 -RRB- Equations -LRB- 3 -RRB- , -LRB- 6 -RRB- result in the desired contradiction . Q.E.D. Theorem 4 If ? is infinite then P ? = ?? where ?? = P ? means that ?? < = P ? and P ? < = ?? Furthermore , if is infinite or D is uncountably infinite then P ? = F ;D where P ? = F ;D means that P ? < = F ;D and F ;D < = P ? Proof First we observe that if a possibility system -LRB- ? ; 2 ? ; ? -RRB- could be found such that its range would cover the whole interval -LSB- 0 ; 1 -RSB- -LRB- i.e -LSB- 0 ; 1 -RSB- ? ? 2 ? -RRB- then it should be 8P -LRB- ? ; 2 ? ; P -RRB- < = -LRB- ? ; 2 ? ; ? -RRB- and so it would be P ? < = ?? . To see that , observe that it should be P -LRB- A0 -RRB- 2 -LSB- 0 ; 1 -RSB- ? ? 2 ? -LRB- for each A0 2 2 ? -RRB- and so there would be A 2 2 ? such that P -LRB- A0 -RRB- = ? -LRB- A -RRB- . In addition , if -LSB- 0 ; 1 -RSB- ? P2 ? then , in the same way , it would be 8 ? -LRB- ? ; 2 ? ; ? -RRB- < = -LRB- ? ; 2 ? ; P -RRB- and so , it would be P ? = ?? . Thus , all we have to prove is that both probability and possibility systems can cover the whole interval -LSB- 0 ; 1 -RSB- when ? is infinite . Now , consider the set IJ = f -LRB- i ; j -RRB- = i ; j 2 N and i < = jg . Obviously card -LRB- IJ -RRB- = @ 0 < = card -LRB- ? -RRB- -LRB- since ? is infinite -RRB- . Therefore , there is an injection g from IJ to ? mapping -LRB- i ; j -RRB- 2 IJ to g -LRB- i ; j -RRB- 2 ? . In addition , for each ! 2 ? define : ? -LRB- f!g -RRB- = -LRB- ij if 9 -LRB- i ; j -RRB- 2 IJ g -LRB- i ; j -RRB- = ! otherwise -RRB- P -LRB- f!g -RRB- = -LRB- 12i if 9i 2 N g -LRB- i ; i -RRB- = ! otherwise -RRB- We shall now prove that the above two measures both cover the interval -LSB- 0 ; 1 -RSB- . For each x 2 -LSB- 0 ; 1 -RSB- define Ax = f ! 2 ? = ? -LRB- ! -RRB- < = xg The obviously ? -LRB- Ax -RRB- = supf ? -LRB- ! -RRB- = ? -LRB- ! -RRB- < = xg = supf i j = -LRB- i ; j -RRB- 2 IJ and i j < = xg = x Since , 8x 2 -LSB- 0 ; 1 -RSB- Ax 2 2 ? ; it would be -LSB- 0 ; 1 -RSB- ? ? 2 ? -LRB- i.e. ? covers the whole interval -LSB- 0 ; 1 -RSB- -RRB- .4 Similarly , for each x 2 -LSB- 0 ; 1 -RSB- define Bx = f ! 2 ? = 9i 2 N ! = g -LRB- i ; i -RRB- and yi -LRB- x -RRB- > = 1 2i g where y0 -LRB- x -RRB- = x yi +1 -LRB- x -RRB- = yi -LRB- x -RRB- ? 1 = 2i if yi -LRB- x -RRB- > = 1 = 2i yi +1 -LRB- x -RRB- = yi -LRB- x -RRB- if yi -LRB- x -RRB- < 1 = 2i Now , it can be easily shown that fyig is a decreasing sequence such that 8i 2 N < = yi < = x = 2i +1 Therefore , it is limi ! 1 yi = 0 . Furthermore , it can be easily verified that yi ? yi +1 = -LRB- 1 = 2i if 9i 2 N g -LRB- i ; i -RRB- 2 Bx otherwise -RRB- Thus P -LRB- Bx -RRB- = X ! 2Bx P -LRB- f!g -RRB- = X 9i2N g -LRB- i ; i -RRB- 2Bx 1 2i = X i2N yi ? yi +1 = y0 ? lim i ! 1 yi = x Since , 8x 2 -LSB- 0 ; 1 -RSB- Bx 2 2 ? ; it would be -LSB- 0 ; 1 -RSB- ? P2 ? -LRB- i.e. P covers the whole interval -LSB- 0 ; 1 -RSB- -RRB- .5 Similarly , for the second part of the proof , we have to show that a system in F ;D can cover the whole interval -LSB- 0 ; 1 -RSB- when it would be either infinite or D uncountably infinite . If is infinite then it must be card -LRB- -RRB- > = @ 0 . Thus there should be an injection h from IJ to beta and we can define 8d 2 D 8 -LRB- i ; j -RRB- 2 IJ fh -LRB- i ; j -RRB- -LRB- d -RRB- = i j If we define 8x 2 -LSB- 0 ; 1 -RSB- Cx = fh -LRB- i ; j -RRB- = ij < = xg then it would be 8d 2 D fCx -LRB- d -RRB- = x. Therefore , f would cover the whole interval -LSB- 0 ; 1 -RSB- . On the other hand , if D is uncountable then there is a bijection u from as subset D0 of D to the interval -LSB- 0 ; 1 -RSB- . Now , if we define 8d 2 D fA -LRB- d -RRB- = u -LRB- d -RRB- , where A is a fuzzy set in , then it would be fA -LRB- D0 -RRB- = -LSB- 0 ; 1 -RSB- and so f would cover the whole interval -LSB- 0 ; 1 -RSB- . Q.E.D. . The previous few theorems showed that probabilities are more expressive -LRB- or equivalently have more representational power -RRB- than possibilities , for finite domains ? . However , as the last theorem showed , their differences in expressiveness disappear when the domain is infinite . Similar relationships hold between probabilities and fuzzy sets as proved in the last theorem and will be shown in the next few ones . Nevertheless , it should be noted that the case of finite domains is of particular interest in Computer Science where only a finite subset of the rational numbers is representable in our finite machines . Furthermore , the case of finite domains does not \ overflow '' the range of the above measures allowing for interesting relationships among them to be formed . It is those relationships that reveal the differences between the two mathematical operators -LRB- P and sup -RRB- that define the fundamental operations in each system class examined in this paper . The next few theorems formally describe expressiveness relationship between fuzzy sets and either probabilities or possibilities . Theorem 5 8 ? ; P ? < = F ; 2 ? -LRB- assuming card -LRB- -RRB- > = 3 -RRB- . 8 ? ; D ?? < = F ;D -LRB- assuming D is non-empty and card -LRB- -RRB- > = card -LRB- ? -RRB- -RRB- . Proof For the first part of the proof , we would like to prove that if card -LRB- -RRB- > = 3 then 8 ? ; P 9f -LRB- ? ; 2 ? ; P -RRB- < = -LRB- ; 2 ? ; f -RRB- i.e. we would like to construct f , given ; ? ; P , so that 8A 2 2 ? 9A0 2 c -LRB- -RRB- 9x 2 2 ? P -LRB- A -RRB- = fA0 -LRB- x -RRB- We choose A0 to be a fuzzy set with domain 2 ? so that 8x 2 2 ? fA0 -LRB- x -RRB- = P -LRB- x -RRB- This is possible , since card -LRB- -RRB- > = 3 and so contains at least one fuzzy set other than the empty or universal fuzzy set . Now , extend the definition of f to the rest of c -LRB- -RRB- according to the properties of fuzzy membership functions -LRB- -LRB- a -RRB- ; -LRB- b -RRB- ; -LRB- c -RRB- ; -LRB- d -RRB- in definition -LRB- 6 -RRB- -RRB- . This extension is possible since f is completely defined for A0 while f 's values are irrelevant to our proof for all other fuzzy sets in -LRB- except of course the empty set ; and the universal fuzzy set ? 0 where f should be equal to or 1 , respectively , for each x 2 2 ? -RRB- . Therefore -LRB- ; 2 ? ; f -RRB- is the desired fuzzy set system . For the second part , we have to construct ; f , given D ; ? ; ? , so that if D is non-empty and card -LRB- -RRB- > = card -LRB- ? -RRB- then 8A 2 2 ? 9A0 2 c -LRB- -RRB- 9x 2 D ? -LRB- A -RRB- = fA0 -LRB- x -RRB- To this purpose , consider the base ? = ff!g = ! 2 ? g of the sigma-algebra 2 ? and make contain a purely fuzzy set A0 for each element A of ? . This is possible since card -LRB- ? -RRB- = card -LRB- ? -RRB- < = card -LRB- -RRB- Furthermore , since D is non-empty -LRB- i.e. there exist at least one x 2 D -RRB- , we can define 8A 2 ? fA0 -LRB- x -RRB- = ? -LRB- A -RRB- Since A0 2 ? c -LRB- -RRB- , the fuzzy set system -LRB- ; D ; f -RRB- is the desired one .6 Q.E.D. Theorem 6 If card -LRB- ? -RRB- > = card -LRB- -RRB- ? card -LRB- D -RRB- and card -LRB- -RRB- < @ 0 then F ;D < = P ? Proof Let -LRB- ; D ; f -RRB- 2 F ;D . Then , for each x 2 D , -LRB- ; fxg ; f -RRB- is equivalent to a possibility system -LRB- ? x ; 2 ? x ; ? x -RRB- i.e. -LRB- ; fxg ; f -RRB- < = -LRB- ? x ; 2 ? x ; ? x -RRB- -LRB- 7 -RRB- and -LRB- ? x ; 2 ? x ; ? x -RRB- < = -LRB- ; fxg ; f -RRB- where the elements of c -LRB- -RRB- are in a bijective relation with the elements of 2 ? x . Hence , it must be card -LRB- c -LRB- -RRB- -RRB- < = card -LRB- 2 ? x -RRB- . Therefore , the smallest cardinality of ? x that would be guarranteed not violate the above relation is card -LRB- -RRB- . In other words , ? x can be chosen to be finite . Then by theorem 1 we get ?? x < = P?x and so -LRB- ? x ; 2 ? x ; ? x -RRB- < = -LRB- ? x ; 2 ? x ; Px -RRB- -LRB- 8 -RRB- Then by -LRB- 7 -RRB- , -LRB- 8 -RRB- and the transisitivity of \ < = '' , we get : -LRB- ; fxg ; f -RRB- < = -LRB- ? x ; 2 ? x ; Px -RRB- Now , by choosing the sets ? x to be disjoint for different x 's and taking the union over all x 2 D in the last formula above we get : -LRB- ; D ; f -RRB- < = -LRB- ? ; 2 ? ; P -RRB- where ? = -LSB- x2D?x P = extension of all Px over 2 ? Obviously , card -LRB- ? -RRB- = card -LRB- -LSB- x2D?x -RRB- > = card -LRB- -RRB- ? card -LRB- D -RRB- Q.E.D. Theorem 7 If ? ; ; D are finite and both card -LRB- -RRB- and card -LRB- D -RRB- are polynomially related to card -LRB- ? -RRB- then not -LRB- P ? < = F ;D -RRB- Proof We would like to prove that 9P 8f not -LRB- -LRB- ? ; 2 ? ; P -RRB- < = -LRB- ; D ; f -RRB- -RRB- We assume to the contrary that -LRB- ? ; 2 ? ; P -RRB- < = -LRB- ; D ; f -RRB- . Then it must be card -LRB- P2 ? -RRB- < = 2 ? card -LRB- -RRB- ? card -LRB- D -RRB- -LRB- the factor of 2 in the formula above is because each fuzzy set in contributes at most card -LRB- D -RRB- values by itself and at most card -LRB- D -RRB- values through its complement -RRB- . However , card -LRB- P2 ? -RRB- can be as large as 2card -LRB- ? -RRB- -LRB- see the proof of theorem 3 -RRB- . Therefore it must be 2card -LRB- ? -RRB- < = 2 ? card -LRB- -RRB- ? card -LRB- D -RRB- This and the hypothesis that both card -LRB- -RRB- and card -LRB- D -RRB- are polynomially related to card -LRB- ? -RRB- result in the desired contradiction . Q.E.D. Figure 1 represents our results graphically . The nodes in the graph represent system classes and the links represent \ at most as expressive as '' relations . The labels on some of the arcs indicate special conditions under which the corresponding \ < = '' relations hold . The meanings of the labels are as follows : Label Meaning > = 3 card -LRB- -RRB- > = 3 D > = 1 card -LRB- D -RRB- > = 1 , > = ? card -LRB- -RRB- > = card -LRB- ? -RRB- :D < = ? card -LRB- -RRB- ? card -LRB- D -RRB- < = card -LRB- ? -RRB- We assume that the sets ? ; ; D are finite . The dashed line is a link that has not been proved explicitly in any of the previous theorems but can be derived easily from the following : card -LRB- ? -RRB- < = card -LRB- ? 0 -RRB- -RRB- C ? < = C ? 0 4 Discussion of Different Measures of Un - certainty Theorems 1 and 3 of the previous section showed that , for finite domains , possibilities are less expressive than probabilities . Another way to express that is that probabilities can simulate possibilities without any extra space requirements while the opposite is not true , in general . Theorem 2 reveals that when extra space requirements can be greatly tolerated then probabilities can be simulated by possibilities , too . However , as theorems 1 and 3 showed , for any given finite space , probabilities have strictly greater representational power than possibilities . Thus , probabilities , when compared to possibilities , can represent more mappings from any given finite domain to the interval -LSB- 0 ; 1 -RSB- and as such can be regarded as being \ denser '' than possibilities in terms of representable mappings . Unfortunately , the space requirements to simulate probabilities by possibilities are excessive since the size of 2 ? is exponentially larger than the size of ? . Therefore , in practice , we may regard that probabilities can not be simulated by possibilities when the domain is finite . Theorems 5 , 6 , and 7 reveal that a similar relation exists between probabilities and fuzzy sets , in finite spaces . Probabilties can simualte fuzzy sets without any extra space requirements : they both require O -LRB- card -LRB- -RRB- ? card -LRB- D -RRB- -RRB- space . However , fuzzy serts can not simulate probabilities without exponentially greater requirements -LRB- theorem 7 -RRB- . The second part of theorem 5 shows that fuzzy sets can simulate possibilities without extra space requirements . This comes at no surprise for fuzzy sets have been defined in a possibilistic way . It must be stressed , however , that those relationships hold for finite domains . When domains become infinite , all measures studied here are capable of covering the whole -LSB- 0 ; 1 -RSB- interval and as such are representationally equivalent to each other -LRB- i.e. any measure representable in one of them is representable in the other without any extension or extra requirements on the domain -RRB- . Of course , this can be seen as an \ overflowing '' effect , in the sense that the above measures overflow -LRB- i.e. cover -RRB- their range when their domains become appropriately large . Furthermore , studying them in such an \ overflowing '' state would not give us insight on their intrinsic representational capabilities . Thus , it is the author 's view that finite domains are more appropriate for comparing the corresponding system classes in term of the mappings they can represent . The flip side of the representational advantage of probabilities , in finite domains , is their efficiency disadvantage . Possibilistic measures such as possibilities and fuzzy sets are easier to compute than probabilities since the former are extensional while the later are intensional measures . This theoretical advantage of possibilities is very noticeable in practice where one can easily compute the possibility value of a compound logical expression or relation among events or entities by simply applying the definition . On the other hand , using probabilities , such a task is either very difficult or impossible . For example , evaluating a Bayesian Network -LSB- 4 , 27 -RSB- , is computationally an NP-hard problem -LRB- -LSB- 6 -RSB- -RRB- . Although there are simple and efficient algorithms for the restricted class of singly connected networks -LRB- -LSB- 23 -RSB- -RRB- , in general , exact algorithms are very complicated and often very inefficient . This led to approximate evaluations of the conditional probabilities in a Bayesian network -LRB- -LSB- 4 , 28 -RSB- -RRB- , a solution which introduces an error which in turn reduces the information probabilities carry . Furthermore , the structure of the network imply independence assumptions which do not always capture all the possible situations and as such act as a new source of error and information loss . Therefore , it appears that probabilities often carry too much information imposing approximate or inefficient solutions . Probabilistic approaches are not limited to Bayesian networks . Probabilistic logic -LSB- 24 -RSB- is another example . In general , Bayesian statistics have been used as a statistical theory of evidence , uncertainty , and inference -LRB- -LSB- 13 , 22 -RSB- -RRB- . Furthermore , a number of probabilistic techniques have been applied for pattern recognition and classification . These include -LRB- but are not limited to -RRB- Bayes classifier and learning , Parzen windows , hidden Markov models , knearest neighbor , and stochastic approximation methods . A good overview of them appers in -LSB- 12 -RSB- . Despite the wide range of applications and techniques of probabilities , problems like the ones mentioned above as well as some empirical evidence that people are very poor probability estimators -LRB- see -LSB- 16 -RSB- -RRB- led researcher to find alternative solutions and explore new measures . Approaches that do not exacly fit in the probability landscape include MYCIN 's certainty factors -LSB- 30 -RSB- and Dempster-Shafer theory -LSB- 8 , 29 -RSB- . Certainty factors require strong independence conditions to hold among their primitive units -LRB- rules -RRB- in order to give accurate results . Therefore , they have problems similar to those probabilities had though to a lesser extent . On the other hand , the information they carry is not as precise and semantically clear as that of probabilities . In short , certainty factors stand between probabilities and possibilities both in terms of information content and computational efficiency . Dempster-Shafer theory has been shown to be a broad theory that subsumes both probabilities and possibilities . When the focal elements -LRB- e.g. elements of non-zero basic probability density or assignment m -RRB- are sigletons or nested then Dempster-Shafer theory collapses to probability or possibility theory , respectively -LSB- 17 , 18 -RSB- . However , the case of possibilities , as appears in -LSB- 17 , 18 -RSB- , applies only for finite universal sets for their elements must be ordered in descending possibility values . Such an ordering is not always possible for infinite sets . Of course , a transformation similar to the one used in theorems 1 , 4 can be used to reduce possibilities to Dempster-Shafer theory in the general case . However , the opposite transform requires exponentially larger spaces , when the domain is finite . This is due to theorem 3 and the fact that Dempster-Shafer theory subsumes probability theory . In short , the structure of the focal elements -LRB- singletons or nested -RRB- will determine whether the defined measures will be probability or possibility measures -LRB- respectively -RRB- . However , they can not be both at the same time . On the other hand , if the focal elements are neither sigletons nor nested then Dempster-Shafer theory is neither probability nor possibility theory . In that case , similar to certainty factors , Dempster-Shafer theory stands in between probability and possibility theory . Computational efficiency is sacrificed if the entity primitive units -LRB- sets of propositions -RRB- are large . Accuracy is reduced unless strong independence conditions hold among the functional primitive units -LRB- belief functions -RRB- . Possibilistic measures , which are the most efficient of all to compute for they require no special applicability conditions , do have the least representational power of all , when domains are finite . Therefore , in that case , their clear computational advantage should not come at a surprise . On the other hand , for infinite domains , possibilistic measures maintain their computational advantage at no apparent cost for their representational power is then equal to that of probabilities . Applications of possibilities are mainly within the spectrum of fuzzy set applications . Fuzzy sets and logic have been used in representation and approximate reasoning -LSB- 33 -RSB- , pattern recognition -LSB- 11 , 25 -RSB- , operations research -LSB- 11 -RSB- , and modeling uncertainty and control -LSB- 11 -RSB- . Applications include decision support systems , expert systems , natural language processing -LRB- -LSB- 33 -RSB- -RRB- , database management , linear programming , robotics , vision -LRB- -LSB- 11 -RSB- -RRB- , clustering , classification , image analysis -LRB- -LSB- 11 , 25 -RSB- -RRB- , and speech recognition -LRB- -LSB- 25 -RSB- -RRB- . 5 Conclusion The choice between probabilities and possibilities or fuzzy sets is not easy . For infinite spaces , it appears that all system classes examined here are equivalent with respect to the measures they can represent . The disadvantage in this case is with the probabilities which have very high computational cost . On the other hand , for finite spaces , probabilities should give higher flexibility -LRB- for they can represent more mappings -RRB- at the a cost of higher computational complexity . That complexity can be that high that would make it impossible for the actual computations to be numerically accurate or computationally tractable -LRB- assuming of course that non-deterministic polynomial space is not identical to deterministic polynomial space -LRB- i.e. NP < > P -RRB- -RRB- . Over-simplifying a probabilistic model so that its computations can be easily performed may reduce the information conveyed to such a level that possibilities would be preferable not only for their simplicity and efficiency advantage but also for their information content . Depending on the nature of each individual application or situation under consideration and its computational demands in terms of accuracy , simplicity , and efficiency , one can choose the best measure to use in his model or system . The existence of possibilistic and probabilistic measures and their already proved irreducible differences should not be seen as a limiting constraint but rather as a resourcefull option . Acknowledgements This research was partially supported by NASA grant NAG2-581 -LRB- ARPA Order # 8607 -RRB- and Teknowledge Federal Systems , Inc. contract 71715-1 -LRB- ARPA contract DAAA21-92-C-0028 -RRB- . References -LSB- 1 -RSB- C. Alsina and E. Trillas . On almost distributive Lukasiewicz triplets . Fuzzy Sets and Systems , 92:175 -LCB- 178 , 1983 . -LSB- 2 -RSB- C. Alsina , E. Trillas , and L. Valverde . On some logical connectives for fuzzy set theory . Journal of Mathematical Analysis and Applications , 93 -LRB- 1 -RRB- :15 -LCB- 26 , 1983 . -LSB- 3 -RSB- R. Bellman and M. Giertz . On the analytic formalization of the theory of fuzzy sets . Information Sciences , 5:149 -LCB- 156 , 1973 . -LSB- 4 -RSB- E. Charniak . Bayesian networks without tears . AI Magazine , 12 -LRB- 4 -RRB- :50 -LCB- 63 , 1991 . -LSB- 5 -RSB- P. Cheeseman . Probabilistic versus fuzzy reasoning . In L.N. Kanal and J.F. Lemmer , editors , Uncertainty in Artificial Intelligence , pages 85 -LCB- 102 . 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This definition is equivalent to the definition of possibility measures given by Zadeh in -LSB- 36 -RSB- for ordinary sets . In addition , Zadeh extends his definition to apply for fuzzy sets . He defines , ? -LRB- A -RRB- = sup ! 2 ? finf -LRB- fF -LRB- ! -RRB- ; fA -LRB- ! -RRB- -RRB- where F , A are fuzzy subsets of ? . As it can be trivially seen that definition implies that ? -LRB- S i2I Ai -RRB- = supi2I ? -LRB- Ai -RRB- and ? -LRB- ;-RRB- = 0 . Furthermore , if sup ! 2 ? fF -LRB- ! -RRB- = 1 then ? -LRB- ? -RRB- = 1 . Therefore , upon the condition that sup ! 2 ? fF -LRB- ! -RRB- = 1 ; Zadeh 's extension of possibility measures to fuzzy sets implies definition 3 . In that case the theorems presented here apply to the possibility measures of fuzzy sets , too . 4 . Now , givenA ; ? , it would be A0 = A ? -LRB- A -RRB- . 5 . Now , givenA ; P , it would be A0 = BP -LRB- A -RRB- . 6 . The definition of f in the rest of D is irrelevant to this proof . For example , f may be zero there . PW PW 2WP b , DF ,2 WFb b . D < = W b > = WD > = 1 b > = 3 Figure 1 : Expressiveness relations among probabilities , possibilities , and fuzzy sets .

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O

artificial intelligence
fuzzy sets
possibility
probability
relations
uncertainty


Memorandum COSOR 95-13 , 1995 , Eindhoven University of Technology Routing Control of a Motorway Network - A Summary ? H.J.C. Huijberts Dept. of Math . and Comp . Sc . Eindhoven Univ. of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands hjch@win.tue.nl J.H. van Schuppen CWI P.O. Box 94079 1090 GB Amsterdam The Netherlands J.H.van.Schuppen@cwi.nl Abstract The problem of setting route directives in a motorway network is formulated as a dynamic game problem . After restricting attention to the Nash equilibrium concept , a search procedure for such an equilibrium is proposed . 1 Introduction The purpose of this paper is to provide a short account of an investigation into routing control . The authors were asked to participate in the DRIVE II Project DYNA . The aim of this project is to develop , produce , and test a real-time prediction algorithm for traffic flow on a motorway network . The 1994 extension of the project also defines as aim traffic control . The task of the CWI team was to perform research on a control algorithm . Only a proposal for a control algorithm had to be developed , no simulations were required . In many economically advanced countries the traffic flow in peak hours on particular stretches of the motorway -LRB- freeway -RRB- network exceeds the available capacity . Because governments do not want to increase network capacity much more there is an interest in the application of communication and control to improve the utilization of the existing network . Government policies in Europe , the U.S.A. , and in Japan stimulate this approach . In this investigation attention is limited to routing control of a motorway network . Routing control is defined as a control effort that seeks to influence the route choice of users of a network . Routing control is a form of traffic control and a tool of traffic management . The contribution of the investigation includes the definition of a control system for the use of route directives of a motorway network . The routing problem is formulated as a dynamic game problem for which attention is ? This paper is to appear in the proceedings of the 1995 European Control Conference , Rome , Italy restricted to a Nash equilibrium . A search procedure for this Nash equilibrium is proposed that makes use of the prediction algorithm developed by others in the project . Further research on routing control is required . The algorithm must be tested in simulation and subsequently be tested on the road . At the theoretical level the existence and the search procedure for the Nash equilibrium need attention . The final report of the investigation is published elsewhere , see -LSB- 3 -RSB- . Other investigations into routing control For an introduction to routing control see -LSB- 1 -RSB- . M. Papageorgiou -LSB- 5 -RSB- has proposed to use optimal control theory to synthesize a control law that meets the control objectives . As part of the DRIVE Project QUO VADIS several routing control laws have been proposed by M. Papageorgiou and others . These control laws are in the process of being evaluated by simulation or by road testing , see -LSB- 4 , 6 -RSB- . Experience with routing control An experiment with routing control was performed near Amsterdam , The Netherlands . On November 14 , 1991 a variable message sign was put in use called the RIA-sign for Route choice Information Amsterdam . The sign is located on motorway A8 between Zaandam and the Amsterdam ringway A10 in the direction of Amsterdam . The sign provides information about traffic conditions on the ringway , in particular , on the length of traffic jams at the Coentunnel , in the western link , and at the Zeeburgertunnel , in the northern link , and on possible obstructions in these tunnels . On the basis of this actual state information car drivers can make an en-route choice by either turning left to the A10 Noord or right to the A10 West . A study has been made of the effectiveness of the RIA - sign by interviewing a panel of drivers . The overall conclusion is that the sign is valued positively by drivers and has a positive effect on traffic flow . After the introduction of the sign the panel members were less frequently caught up in queues than before . The average travel time of the panel decreased by eight percent . The duration and severity of congestion decreased from period to period . Five to seven percent of all trips were influenced by the sign . 2 Modeling of traffic flow in a motorway network 2.1 Model for traffic flow without control The model for traffic flow in a motorway network with control to be mentioned below is based on a similar model without control . Terminology on a motorway network follows . Consider a motorway network on which a control and signalling system is installed . Such a system is assumed to have a detector station about every 500 m. of the motorway . At a detector station information is collected on the traffic flow . Detector stations are also assumed present at onramps , at off-ramps , and at motorway intersections . A model of a motorway network consists of a graph in which the nodes correspond to locations of detector stations . A section of the network is an ordered pair of nodes such that there is a motorway connection between these nodes that does not pass another node . An origin-destination pair -LRB- OD -RRB- is an ordered tuple of nodes where the first node corresponds to a location with an on-ramp and the second to a location with an off-ramp . A route of an OD pair is a path from the associated origin to the associated destination . A path is specified by a chain of sections . In general an OD pair may have several routes . A link is a chain of sections between two adjacent intersections of a motorway network . A summary follows of a state space model for traffic flow in a motorway network . For motorway traffic flow a macroscopic model of H.J. Payne will be used that has been modified by S.A. Smulders . The state variables of this model are traffic density and average speed per section for all sections . Because the purpose of this model is routing , the traffic density per section must be distinguished per OD pair and per route . Denote the discrete time axis by T . Denote the traffic density of OD pair -LRB- i ; j -RRB- , of route k , and of section -LRB- l ; m -RRB- by x -LRB- -LRB- i ; j -RRB- ; k ; -LRB- l ; m -RRB- ; :-RRB- : T ! R + in veh/km . lane . The density in section -LRB- l ; m -RRB- may then be calculated by x -LRB- -LRB- l ; m -RRB- ; t -RRB- = X x -LRB- -LRB- i ; j -RRB- ; k ; -LRB- l ; m -RRB- ; t -RRB- ; where the sum is over all OD pairs and all routes that use the section -LRB- l ; m -RRB- . The state vector x contains the traffic densities defined above and the average speed per section in all sections . The state transition equations are deduced from Payne 's model and from the structure of the network . The resulting state space model is described by the discrete-time dynamic system x -LRB- t + 1 -RRB- = f -LRB- x -LRB- t -RRB- ; > = -LRB- t -RRB- -RRB- ; x -LRB- t0 -RRB- = x0 : -LRB- 1 -RRB- In this model the vector > = specifies the entry flows at origins . 2.2 Control measures Attention in this summary is restricted to those control measures that have a direct impact on the en-route choice of network users . 1 . Radio . Transmission of messages on regular radio channels . The messages contain almost exclusively information on exceptional circumstances , such as congestion and unforeseen lane closures . 2 . RDS-TMC . The Radio Data System -LRB- RDS -RRB- has recently come on the market as part of a car radio . Via a Travel Message Channel -LRB- TMC -RRB- information is communicated to the car radio and stored there . The user of a car equipped with RDS may by the push of a button start RDS to sound an audio message in his or her language with traffic information . 3 . DRIP . A Dynamisch Route Information Paneel -LRB- DRIP ; Dynamic route information board -RRB- is an example of a variable messages sign -LRB- VMS -RRB- . An example of a DRIP is the RIA sign discussed in Section 1 . That sign displays the length of the traffic jams in the western and the northern link of the ring around Amsterdam . 4 . VDS . Variable Direction Signs -LRB- VDS -RRB- are currently not installed in The Netherlands . In principle DRIP 's can be used as VDS 's . 2.3 Routing information Routing information provided by control measures may be distinguished into the following forms : 1 . Actual information . Information reflecting the state of traffic in the motorway network at the time the information is sent to the communication channel or displayed . Most current control measures mentioned operate in this mode . 2 . Predictions . For example , prediction of travel times for one or more routes . 3 . Route directives . A network user is provided a route directive . A note on terminology follows . A route directive is information that points travellers to a particular route . A route directive may be provided by a variable direction sign , a DRIP , a radio message , or RDS . A route directive is considered as an advice , travellers may decide to follow or to ignore the directive . Which type of routing information is to be preferred for routing control ? Travel time predictions are considered with reservations . Car drivers may compare the travel time prediction with the experienced travel time . Because the predictions have a large uncertainty there will always be a difference between these travel times . Therefore drivers may lose confidence in the predictions . This will then affect the reliability of routing control . In conclusion , the routing information of actual state information and of travel directions are preferred over travel time predictions . The advantages of travel directives over actual state information are as of yet not clear . 2.4 Modeling of the control system The control measures that have an effect on the en-route choices are : VDS , DRIP , radio , RDS , and route guidance systems . The route information received by drivers via these information channels will have an effect on : -LRB- 1 -RRB- The exit of the motorway network to be used for the trip . -LRB- 2 -RRB- The route to be followed . It is known that the reactions of car drivers to en-route messages is mixed . Some car drivers will not see or will not notice the message , some who do receive it do not know how to interpret the message , etc. . Experience with routing information and variable direction signs is needed in combination with interviews of road users for modeling of drivers reactions . It is practical if route directives are not adjusted continuously but only at regularly spaced times and held fixed in between these times . One may think of updating the route directives every 5 minutes . The effect of route directives on the traffic flow is modelled by inputs and by splitting fractions . At each intersection of a motorway where the possible routes of an OD pair branch the splitting fraction describes which part of the flow follows each route . The effect of an input on the splitting fraction at an intersection reflects the compliance of drivers to the route directives . Let u -LRB- -LRB- i ; j -RRB- ; k ; m ; :-RRB- : T ! -LSB- 0 ; 1 -RSB- ; represent the route directives for OD pair -LRB- i ; j -RRB- , for route k , at intersection m. Let the vector u contain all routing fractions . The control system for the state space model is then described by the recursion x -LRB- t + 1 -RRB- = f1 -LRB- x -LRB- t -RRB- ; u -LRB- t -RRB- ; > = -LRB- t -RRB- -RRB- ; x -LRB- t0 -RRB- = x0 : -LRB- 2 -RRB- 3 Routing control 3.1 Problem formulation and approach The routing control problem may be formulated as having one decision-maker or many decision-makers . In the formulation of one decision-maker , the road operator , who is responsible to the government for the network , selects the control objectives and decides on routing . In the formulation of multi decision-making a decision-maker corresponds to the flow of an OD pair . Although in principle the road users make the route choice individually , the choice will be similar for all users travelling on the same OD pair asuming that they are provided with the same routing information and assuming rational behaviour . Because of this distinction one must work with the decision criteria of user optimality and of system optimality . System optimality refers to one decision-maker , the road operator , and user optimality refers to multi decision-makers , one decision-maker for each OD pair . Control objectives of the flow for a particular OD pair are : travel time , travel distance , travel costs , road surface -LRB- quality -RRB- , safety , congestion and stress , ease of driving , environmental factors , pleasant scenery . For the investigation on which this paper reports travel time will be taken as the cost function . Let G denote the class of control laws and for OD pair -LRB- i ; j -RRB- , G -LRB- i ; j -RRB- the corresponding class of control laws . Elements of G and G -LRB- i ; j -RRB- are denoted by g and g -LRB- i ; j -RRB- respectively . Let J -LRB- -LRB- i ; j -RRB- ; g -RRB- represent the cost , say the total travel time , over a specified time horizon for the traffic flow of OD pair -LRB- i ; j -RRB- when control law g is used . The cost is to be minimized . The authors have decided to narrow down the dynamic user optimality concept to the Nash equilibrium concept . Definition 3.1 For the routing problem formulated above a control law g ? 2 G is said to be a Nash equilibrium if for all OD pairs -LRB- i ; j -RRB- 2 OD J -LRB- -LRB- i ; j -RRB- ; g ? -RRB- < = J -LRB- -LRB- i ; j -RRB- ; g -RRB- ; where g -LRB- k ; l -RRB- = ae g ? -LRB- k ; l -RRB- ; 8 -LRB- k ; l -RRB- 2 OD ; -LRB- k ; l -RRB- < > -LRB- i ; j -RRB- ; g1 -LRB- k ; l -RRB- ; if -LRB- k ; l -RRB- = -LRB- i ; j -RRB- ; and g1 -LRB- i ; j -RRB- 2 G -LRB- i ; j -RRB- is arbitrary . Thus a control law is a Nash equilibrium if , for each OD pair , the cost increases when for only that OD pair another control law than the optimal one is used . The restriction to a Nash equilibrium seems quite reasonable considering the facts that in any reasonable size network there are in general many OD flows and that drivers on different OD flows do not coordinate their decisions . Open questions are whether a Nash equilibrium exists , and , if so , whether it is unique . In general a Nash equilibrium , if it exists , is non unique . Problem 3.2 Consider the routing control problem for route directives with the Nash equilibrium concept formulated above . Determine a control law g ? 2 G that is a Nash equilibrium . In this investigation control synthesis will be based on a version of the separation principle . According to this principle a control law is synthesized by combining a predictor or filter with a control law based on state feedback . The control law consists of a predictor and a controller . The predictor has been developed in the project by other teams . It provides estimates of intensities and average speeds in links of the motorway network and predictions of the intensities at entry points of the network . It also can provide predictions of intensities and average speeds in links of the network . The controller uses the predictions of the state variables and produces the input , the variable direction settings . What remains to be done is to synthesize the controller , a control law based on state estimates and predictions of arrival intensities . 3.2 Routing control law If a Nash equilibrium exists , how can it be determined ? For a dynamic game problem one can derive a set of dynamic programming equations that characterize a Nash equilibrium , see -LSB- 2 , Section 6.6.2 Th . 6 , Section 6.5.2 Th . A-6 -RSB- . Considering the size of the state space of the system , for the Amsterdam network of the order of 60 ? 103 , this approach is practically not feasible . Therefore a realistic approximation procedure for the Nash equilibrium should be considered . Principle of routing control law The definition of a Nash equilibrium suggests a search procedure to determine such an equilibrium . It consists of an optimization per OD pair and a search procedure over OD pairs . Principle 3.3 Routing control law . 1 . Select and order a set of origin-destination network nodes pairs . 2 . Perform for all OD pairs selected in step 1 an online input design based on a finite-horizon look ahead procedure using travel time predictions produced by the predictor . 3 . Repeat the steps 1 and 2 until for no OD pair a further improvement can be made . 4 . Fix the route directives for the next discretization interval . 5 . Repeat the first four steps of this procedure in the subsequent intervals . The principle formulated above leads to a control law that can be computed with the predictor and is therefore feasible in practice . No claim of optimality for this control law is made . 4 Concluding remarks The proposed control law needs further investigation , testing by simulation , and testing on the road . It will take several more years of development work before it comes to a road test . A future investigation into routing control may address the following points : 1 . At the theoretical level there are the question of existence of a Nash equilibrium and the characterization of the set of such equilibria for the routing control problem . 2 . The search procedure over selected OD pairs . A theoretical question is whether the search procedure terminates in a finite number of steps if there exists a Nash equilibrium . A practical question is whether a few steps of the procedure provides an acceptable solution to the routing control problem and , if so , how these steps should be performed . Acknowledgements The authors thank M. Ben-Akiva for his stimulation of the CWI team on the routing control problem during the lifetime of the project . They thank J.C.C. de Ruiter and S.A. Smulders of the Rijkswaterstaat team for information on the routing control problem . References -LSB- 1 -RSB- W. Balz and H. Zackor . Freeway network control . In M. Papageorgiou , editor , Concise encyclopedia of traffic and transportation systems , pages 156 -LCB- 162 . Pergamon Press , Oxford , 1991 . -LSB- 2 -RSB- T. Basar and G.J. Olsder . Dynamic noncooperative game theory . Academic Press , New York , 1982 . -LSB- 3 -RSB- H.J.C. Huijberts and J.H. van Schuppen . Routing control of a motorway network . Report BS-R95xy , CWI , Amsterdam , 1995 . -LSB- 4 -RSB- J.M. Morin , P. Ricquez , and J.F. Gabard . Aid-todecision for motorway network management : The expert system OPERA . In ERTICO , editor , Proceedings of the First World Congress on Applications of Transport Telematics and Intelligent Vehicle-Highway Systems , pages 1388 -LCB- 1395 , Boston , 1994 . Artech House . -LSB- 5 -RSB- M. Papageorgiou . Dynamic modeling , assignment , and route guidance in traffic networks . Transpn . Res . - B , 24B :471 -LCB- 495 , 1990 . -LSB- 6 -RSB- M. Papageorgiou , P. Gower , A. Messmer , and J.M. Morin . Control strategies for variable message signs . In ERTICO , editor , Proceedings of the First World Congress on Applications of Transport Telematics and Intelligent Vehicle-Highway Systems , pages 1229 -LCB- 1236 , Boston , 1994 . Artech House .

O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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dynamic game
motorway traffic
route directives
routing control