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form is the if-then-else statement. A statement of the form if b then S else T is executed by evaluating the boolean expression b; if b evaluates to true, the statement S is executed, otherwise ('else') the statement T is executed. For example, the statement if x ^ O then y := -x else y := x assigns to y the absolute value of x. The use of 'else' leads to an asymmetry in the branches of a conditional statement. In the evaluation of the absolute value of x, the case x = 0 can be handled by either the assignment y := -x or by the assignment y := x\ it does not matter which is chosen. The asymmetry becomes more pronounced when there are more cases to be considered, as in, for example, then i := i+l else if b[j] <a[i] then j := j+l else ij := i + l , j+l . A modest improvement can be obtained if each branch of a conditional statement is 'guarded' by the condition under which it is executed. An example is the statement

This chapter presented an overview of fuzzy rule based reasoning systems. In the next chapter a specific example of such systems is discussed, namely fuzzy controllers.

Table 3.6 Comparison of the numerical solution with the analytical solution for a 20 A single GaAs quantum well surrounded by Gao.25Alo.75Asbarriers, as a function of the number (N) of points per A in the mesh note the step length 5z 1/N (A)

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the guards being x < 0 and x ^ 0, respectively. The conditional statement is executed by evaluating the guards and then executing any one of the statements whose guard evaluates to true. (If one such guard is found, the subsequent execution of its associated statement need not wait for all the guards to be evaluated.) Note that this makes evaluation of a conditional statement non-deterministic in that it may be the case that both guards evaluate to true. This is illustrated by the example: in the case that x = 0 it is permitted to execute either the assignment

N (A TV ( A 11) Analytical solution Numerical solution Analytical solution Numerical solution E1 (meV) E1 (meV) 2 4 6 8 10 12 270.764969 270.764969 270.764969 270.764969 270.764969 270.764969 270.663106 270.739499 270.753649 270.758601 270.760894 270.762139

10: Sequential Composition and Conditional Statements y := -x or the assignment y := x. But, it does not matter which is executed since both will assign the value 0 to y. It is precisely this non-determinism that is the advantage of the guarded command style of writing conditionals. It saves the programmer from making arbitrary and possibly confusing distinctions. Also, efficiency improvements can be automated in an optimizing compiler. (Freeing the programmer from the task of making detailed efficiency improvements improves reliability and eases program reuse.) The number of guarded commands in a conditional statement is finite but otherwise unbounded. An example with three guards is if a [ i ] < b [ j ]

1. Consider the following rule base:

However, as before, the agreement between the numerical solution and that obtained from the analytical form can be improved by increasing the computational accuracy of the shooting method, i.e. increasing the number of points per A (decreasing 8z). This is highlighted in Table 3.6, which compares the solutions from the two methods for a decreasing step length 6z. When using the form for the variable effective mass shooting equation in equation (3.53), the BenDaniel-Duke boundary conditions are 'hard-wired' in. Thus it is unnecessary to repeat the analysis described in Section 3.4 in order to recover them.

n a[i]>b[j]

The convergence tests of the numerical solution to Schrodinger's equation have provided the confidence to apply it to systems for which the analytical solutions have not (in this book!) been developed or do not exist. The simplest example, of the former would be the symmetric double quantum well of Fig. 3.15. The potential function V(z) required for the numerical solution is simply the (in this case) conduction band edge as given in the figure. Figure 3.16 displays the results of calculations of the lowest two energy states as a function of the central barrier width for a double quantum well with the Al barrier concentration x = 0.2 and a fixed well width lw 60 A. When the wells are separated by a large distance, the interaction between the eigenstates localised within each well is very small and the wells behave as two independent single quantum wells. However, as illustrated in Fig. 3.16, as the central barrier is decreased, the energy levels interact, with one being forced to higher energies and the other to lower energies. This is directly analogous to the formation of a pair of bonding and antibonding orbitals when two hydrogen atoms are brought together to form a hydrogen molecule. In that case, as here, the wave function of the lower (bonding) state is a sum of the wave functions of the separate atoms (wells), and the higher (anti-bonding)

Compare this statement with the equivalent if-then-else statement above to see which you think is clearer. An if-then statement (i.e. a conditional statement without an else clause) is expressed using the skip statement. The statement if b then S is expressed as

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