Specification of our search problem

We are now ready to formulate the specification of our search problem:

Given an array $a$ and integer $x$, find an integer $i$ such that

1. if there is no $j$ such that $a[j]$ is $x$, then $i$ is $-1$,
2. otherwise, $a[i]$ is $x$.

The first clause says that if $x$ doesn't occur in the array then $i$ should be $-1$, and the second says that if it does then $i$ should be a position where it occurs. If there is more than one position where $x$ occurs, then this specification allows you to return any of them -- this would be the case e.g. if $a$ were $[17,13,17]$ and $x$ were $17$. Thus, the specification is ambiguous. Hence different algorithms with different behaviours can satisfy the same specification -- for example, one algorithm may return the smallest position at which $x$ occurs and an another the largest. There is nothing wrong with ambiguous specifications. In fact, they do occur in practice often.