C ompos ition and inferenc e.6 Properties of composition 121

We have noted that the commonly used sup-min composition is a special case of the sup-T composition where the min operation is used to represent the T-norm. Similarly, in the sup-dot composition, the product operation is used as the T-norm. Hence, it is appropriate to discuss the properties of sup-T composition, since the properties of other specific compositions will follow from the general results.

2.10.6.1 Sup-T composition

Consider two fuzzy relations (fuzzy sets) P and R represented by the two-variable membership functions µP(x, z) and µR(z, y). The sup-T composition Po R of P and R is given by its membership function

µPoR(x, y) = [µP(x, z) TµR(z, y)] (2.50)

Clearly this definition can be generalized to relations having more than two variables.

2.10.6.2 Inf-S composition

Consider two fuzzy relations (fuzzy sets) P and R represented by the two-variable membership functions µP(x, z) and µR(z, y). The inf-S composition P⊗ R of P and R is given by its membership function

µP⊗R(x, y) = [µP(x, z) SµR(z, y)] (2.51)

where inf represents the infemum operation (global minimum, searched piecewise continuously in the defined space), and S represents the S-norm, as usual (see Table 2.2). Again, this definition can be generalized to relations having more than two variables.

The inf-S composition is the complement of the sup-T composition. It may be interpreted as performing the following two steps:

(1) Blend the data P with the knowledge base R by reinforcing the validity (membership) of the knowledge in the overlapping region (Z) and project-ing to the non-overlappproject-ing region (X × Y), using the S-norm (OR operation).

(2) Scan the common region (Z) and compress its validity (membership) to the global minimum in that region, leaving behind knowledge in the non-overlapping region (X × Y).

The inf-max composition is a special case of the inf-S composition.

2.10.6.3 Commutativity

The sup-T composition is commutative; thus

P o R = R o P (2.52)

infz∈Z

sup

z∈Z

2 Fu n dament als of fuzzy logic sy s tems

122 This holds because the T-norm operation in equation (2.50) is commutative (aTb= bTa).

Similarly, the inf-S composition is commutative; thus

P ⊗ R = R ⊗ P (2.53)

This holds because the S-norm operation in equation (2.51) is commutative (aSb= bSa).

2.10.6.4 Associativity

The sup-T composition is associative; thus

P o (Q o R) = (P o Q) o R (2.54)

This may be proved for the sup-min composition as follows. Let the member-ship functions of P, Q, and R be µP(x, y), µQ(y, z) and µR(z, w), respectively.

This provides sufficient generality for the present proof. Now consider two sides of equation (2.54). The membership functions are:

LHS= min[µP(x, y), min(µQ(y, z), µR(z, w))]

RHS= min[ min(µP(x, y), µQ(y, z)), µR(z, w)]

If sup and min are taken in orthogonal directions (i.e., different subspaces or with respect to different variables) then these two operations commute.

Hence, in the RHS expression, the first min (carried out over (z, w)) will com-mute over the sup carried out over y. Hence

RHS= min[min(µP(x, y), µQ(y, z)), µR(z, w)]

Next, since sup commutes with itself and since min of a set of elements can be carried out any two at a time, we have

RHS= min[(µP(x, y), min(µQ(y, z)), µR(z, w))]

In this expression, the second sup is over z, and the first min is over (x, y).

Hence, they commute. We have

RHS= min[µP(x, y), min(µQ(y, z), µR(z, w))]

This result is identical to the LHS.

Note: A simpler proof can be given for the case of discrete membership functions, in view of the matrix multiplication analogy. Specifically, first we observe that the matrix multiplication is associative. Next we observe that matrix multiplication involves product (analogous to min) and addition

supz

supy

supz

supy

supy

supz

supy

supz

supz

supy

2.10 C ompos ition and inferenc e

(analogous to max). Finally, we note that product is associative and so is min; 123 addition is associative and so is max; and product is distributive over addi-tion, and min is distributive over max. This completes the proof.

The inf-S composition is associative; thus

P ⊗ (Q ⊗ R) = (P ⊗ Q) ⊗ R (2.55)

This may be proved for the inf-max composition by following a similar pro-cedure to what was given for the sup-min composition.

2.10.6.5 Distributivity

The sup-T composition is distributive over union (∪). Specifically,

(P ∪ Q) o R = (P o R) ∪ (Q o R) (2.56)

In general, the sup-T composition does not distribute over intersection (∩).

Specifically,

(P ∩ Q) o R ⊂ (P o R) ∩ (Q o R) (2.57)

Graphical proofs may be provided for these, using special cases.

2.10.6.6 DeMorgan’s Laws

Since inf-S composition is the complement of the sup-T composition, DeMorgan’s Laws hold; specifically,

-= { ⊗ | (2.58a)

== { o | (2.58b)

where the over-bar denotes the negation operation for a relation.

2.10.6.7 Inclusion

The inclusion property states that if R1⊂ R2then

Po R1⊂ P o R2 (2.59)

This property is particularly useful in knowledge-based decision-making with incomplete knowledge or data. Specifically, since R1and R2 may represent two knowledge bases or two sets of data, from (2.59) we conclude that a more complete knowledge base provides a more complete inference, or a more complete set of data provides a more complete inference. Furthermore, (2.59) may be used to show the equivalence between the decisions made by using composition (i.e., CRI) and those made by first determining the decisions corresponding to the individual rules and then aggregating these individual decisions (i.e., individual rule-based inference). This topic will be further discussed in Chapter 3.

2 Fu n dament als of fuzzy logic sy s tems

124 Example 2.26

Suppose that a fuzzy rule base (relation) P (x, y) is used to make an inference B( y) from a context A( x ). Also, suppose that another fuzzy rule base Q ( y, z) is used to make an inference C (z) from context B( y). From this information, derive a suitable rule base R ( x, z) that may be used to make an inference C ( z ) from context A( x ). You must justify your derivation.

The relation from a possible “pre-admission grade” ( x_i) to “admission to a group of universities” ( y_j) for a particular student is given by the discrete fuzzy relation:

P( x_i, y_j) =

The relation from a possible “admission to a university” ( yi) to possible “com-pletion time” ( zk) of degree is given by the discrete relation:

Q( y_j, z_k) =

Determine a suitable relation R ( x_i, z_k) that relates possible pre-admission grade ( x_i) to possible completion time ( z_k) of degree. Use both max-min com-position and max-dot comcom-positions.

which, in view of the associativity property of the composition operation (o) can be written:

C( z)= A(x) o [P(x, y) o Q(y, z)]

= A(x) o R(x, z)

where the equivalent rule base is R ( x, z ) = P(x, y) o Q(y, z)

2.10 C ompos ition and inferenc e