C.29.2:2 - Problem

Knowing a condition that the answer satisfies does not always tell a practitioner how to get that answer. Writing y = argmin f(x) specifies a selection problem; it supplies a computation only together with a way to represent and search or otherwise solve the admitted problem. Conversely, a running procedure may return a value without establishing that it satisfies the requested condition.

Three failures make this gap expensive:

• A representation identifies cases that require different answers. No later procedure using only that representation can recover the missing distinction without another input.
• An operation such as “solve”, “update” or “sample” hides the very construction that is unavailable. Giving it a name does not make it elementary or obtainable.
• A procedure is assessed under the wrong semantics or cost model. Exact integers, fixed-width words and rounded values have different operations; a single arithmetic instruction may manipulate a growing number of bits.

The repair is to connect the requested answer to an actual construction, then follow the construction’s meaning and costs. It need not start by writing software. A hand calculation, an invariant or a storage count can locate the decisive obstacle first.