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where the last condition is true for an open neighborhood containing s = 0. An alternative way to conceptualize this loop is to consider the two branches, 0 R and 0 1, exiting state 0 as branches in parallel. Recall that parallel owgraphs lead to nite mixture distributions. In this case, each time the feedback loop is taken, another term in the mixture is generated so that we have in nite parallel branches. This is an in nite mixture with weights p, p(1 p), p(1 p)2 , . . . . This direct approach of (2.18) is the least elegant of the three feedback loop derivations. Example 2.14: Engineering Application: Hydraulic Pump with Feedback. We return to the full hydraulic pump system of Example 2.8. To solve this owgraph (Figure 2.10) for the MGF of T , MT (s), we set up a balance equation at state 0. A balance equation equates the inputs into a state with the outputs of the state. The idea is based on conservation laws applied to systems and originates from Kirchhoff s laws governing circuit behavior, which are derived from conservation laws applied to circuits. In our case, the input into state 0 is p10 M10 (s). The output from state 0 is M01 (s). This is magni ed by the factor MT (s), which is the MGF of the loop that is fed back into itself. In systems analysis, this factor is called the gain of the system. The balance equation for state 0 in Figure 2.10 is MT (s) = M01 (s)p10 M10 (s)MT (s) + M01 (s)p12 M12 (s). (2.19)

Case adaptation (Continued) neural network, 152, 209 rule-based adaptation, 149 support vector machine, 154 pseudo code, 144 strategies, 137 reinstantiation, 138 substitution, 139 transformation, 142 Case base, 2. See also Case organization client, 167 maintenance, 20. See also Case-base maintenance Case-base competence case density, 108, 191 competence error, 192 coverage, 190 fuzzy integral, 188 group coverage, 190 group density, 190 l-fuzzy measure, 194 link detection, 193 reachability, 190 relative coverage, 192 Case base maintenance completeness, 164 consistency , 163 correctness, 163 fuzzy integral approach, 187 qualitative, 163 quantitative, 165 rough-fuzzy approach, 166 updating cases, 167 Case-based reasoning components, 2 example, 21 features, 2 guidelines of using, 9 learning, 19 life-cycle, 6 retain, 7 retrieve, 7 reuse, 7 revise, 7 merits, 9 methodology or technology, 26 soft case-based reasoning, 27. See also Soft case-based reasoning structure, 3, 4 task-method decomposition, 7 vs. human reasoning, 5 vs. rule-based system, 4 Case coverage, 165

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Speci c construction techniques, like recursive construction, concatenation of small af ne functions, are used for designing certain (n, m, d, x) functions This is an area of active research Recursive Construction Basic idea proposed in the work by Siegenthalor [35] To start with, one can consider an unbalanced function g on n m 1 variables Next, note that the (n m)-variable function h = Xn m + g(X1 , , Xn m 1 ) is balanced Now consider the function f on n variables as Xn + + Xn m+1 + h(X1 , , Xn m ) This is an (n, m, d, x) function We will talk about the values of d, x little later That is, after getting the balanced function h, addition of each new variable increases the order of correlation immunity by 1 Now interpret this construction in the following way.

The Prisoner s Dilemma is famous because the way the police have set up the potential pay-offs means that each criminal has an incentive to confess no matter what the other criminal does For example, concentrate on Ronnie Should he confess or remain silent Well, first examine which option is best for him if his partner in the other interrogation room confesses Looking at the left column of pay-offs, you can see from the upper-left rectangle that if Ronnie confesses while Reggie confesses, Ronnie gets five years On the other hand, the bottom-left rectangle tells you that if Ronnie remains silent while Reggie confesses, Ronnie gets ten years Clearly, the best thing for Ronnie to do if Reggie confesses is also to confess But consider whether Ronnie should confess or remain silent while Reggie remains silent in the other interrogation room.

where d is the cost function between two cities. The tness function of the TSP is f pmk 1 Totalcost pmk C:6

Let hk,i be a k-variable resilient function of order i Just as notation, we consider the unbalanced functions as resilient functions of order 1 and balanced non-CI functions as resilient functions of order 0 It is now clear that Xk+1 + hk,i is always a (k + 1)-variable, (i + 1)-resilient function Let us call this c (complement) operation, since the truth table of hk,i and its complement are concatenated to get the (k + 1)-variable function Extension of this kind of construction has been discussed in the work by Camion et al [7] If i is even, then (1 + Xk+1 )hk,i (X1 , , Xk ) + Xk+1 hk,i (1 + X1 , , 1 + Xk ) is (k + 1)variable, (i + 1)-resilient function.

We call this as r (reverse) operation, since the truth table of hk,i and its reverse string are concatenated to get the (k + 1)-variable function If i is odd, then (1 + Xk+1 )hk,i (X1 , , Xk ) + Xk+1 (1 + hk,i (1 + X1 , , 1 + Xk )) is (k + 1)-variable, (i + 1)-resilient function We call this as rc (reverse and complement) operation, since the truth table of hk,i and its reverse and then complemented string are concatenated to get the (k + 1)-variable function Example of Recursive Construction.

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