C.29.3:4.5 - Compare the routes over the claimed scope

Work a small input through both routes with its meanings intact. Then establish why the required relation holds over the input range or histories for which the result will be used.

For a finite case, a complete case analysis can suffice. For an invariant, show it in the initial state and show how every permitted operation preserves it. For an approximate result, propagate the relevant losses to the quantity used by the receiver. For a sampler or statistical estimator, compare the relevant distributions or estimation properties. Use the mathematical, numerical or empirical method appropriate to that claim.

For example, a required binary sampler gives output 1 with probability 0.5. Suppose the device model gives probability p between 0.49 and 0.51 under the intended preparation. The absolute discrepancy in that probability is at most 0.01, so it meets a tolerance of 0.02. Comparing two individual draws cannot establish or refute that distributional result. Estimating p from a finite run requires the sampling assumptions and uncertainty calculation appropriate to those observations.

When the requested result contains several random components, compare their joint behavior. Suppose the target is a pair of independent fair bits. Preparing one fair bit B and returning (B,B) gives the right distribution at each output separately. But the target assigns probability 1/2 to unequal outputs; this system assigns probability 0. Using two independent draws gives each of the four pairs probability 1/4 and restores the requested result. The realization must provide that joint sampling behavior. When its observations are dependent, use the actual joint law or change how the second component is obtained.

Choose changed cases from consequential features of the arrangement: an endpoint of the command range, a different initial state, an overlapping action or a possible indication error. These cases help locate failures. A sampled success supports the tested case; the broader argument supplies the broader conclusion.

Separate three possible outcomes:

• The interpreted execution supplies the required result under the stated conditions.
• It supplies a weaker result, such as an interval or occupancy bound, that may still answer the receiving question.
• A connection fails or remains unknown, and its location determines the next construction, observation or restriction.