Chapter 2, Building Abstractions with Data

Section - 2.1 - Introduction to Data Abstraction

Exercise 2.15

I think Eva Lu Ator is right.

Lets consider the two formulas:

  • $$ \frac 1 { { \frac 1 { I_1 } } + { \frac 1 { I_2 } } } $$
  • $\frac {I_1 I_2} {I_1 + I_2}$.

We know that they are algebraically equivalent but they are not equal by interval arithmetic. We have already seen that the reason they are not same is because in interval arithmetic there is no identity operation i.e. $\frac I I \ne [1,1]$ for any interval $I$ except when lower-bound and upper-bound are equal.

Lets consider why $\frac I I \ne [1, 1]$. All arithmetic operation suggested by Alyssa P. Hacker works by assuming that all the variables in the expression are independent of each other. Thus in expression $\frac I I$, both numerator and denominator are independent of each other by Alyssa arithmetic. Thus if numerator takes a value $v_1 \in I$ then denominator may have some other value $v_2 \in I$ such that $v_1 \ne v_2$. This leads to a result $\frac I I \ne 1$.

We can extend this analysis to all the operations where interval-variable is repeated:

  • $\frac I I$.
  • $I + I$.
  • $I - I$.
  • $I \times I$.

We can even extend it further: if an interval-variable is repeated anywhere in the formulae it will introduce errors because these interval-variables behave independently while we expect them to behave such that the values taken by an interval-variable in all the occurrences in an expression should be equal.

Thus we accept the suggestion from Eva Lu Ator, which means that all interval-variables will occur only once thus there is no chance that any error is caused because of same interval-variables behaving independently in an expression.