NOT.1:5 - Archetypal Grounding

NOT.1:5.1 - Same components, different composition

A team is designing a diagram notation for constructing pipelines of functions. Its first sketch shows the input, output and an unordered collection of function names. The current operation is to determine the result and then exchange two functions without losing their connections.

Use the supplied functions f(z) = z + 1 and g(z) = 2z, with input 3. Applying f then g gives g(f(3)) = 8. Applying g then f gives f(g(3)) = 7. Both cases have the same named components. The unordered sketch therefore loses a difference that the operation needs.

The designer tries directed connections, with the rule that a connection passes the output of one function to the input of the next:

3 -> f -> g -> result     gives 8
3 -> g -> f -> result     gives 7

Now the reader can recover the composition order and derive each result from the supplied functions. Exchanging the two functions changes their connections; merely dragging a box on the page must either preserve those connections or expose that it changed them. The design has acquired both an expression requirement and a question about the editing operation.

If the diagram is only an inventory of available functions, the original unordered sketch can remain adequate. If later work needs branching or shared intermediate results, try those operations before deciding how their connections and identities will be expressed. Success with this linear case leaves those questions open.

NOT.1:5.2 - Same starting times, different durations

A notation for a sequence of signals lists their starting times in units of a shared pulse. Two sequences both start signals at 0 and 2. In sequence A each signal lasts one unit; in sequence B each lasts two. The intervals are half-open: the signal is active at its start and inactive at its end.

For the question “when should each signal begin?”, the list 0, 2 suffices. For “is a signal active at time 1.5?”, it does not: the answer is no for A and yes for B. The omitted duration now changes the answer.

Try pairs of (start, duration). Sequence A becomes (0, 1), (2, 1); B becomes (0, 2), (2, 2). The reader adds duration to start and compares 1.5 with the resulting interval. This exposes the needed difference with one simple convention. A timeline with interval bars might make repeated overlap questions easier; the pair notation is easier to transmit as plain text. Their comparative value depends on the operation and access.

If a performer’s action determines the duration later, retain that unresolved value and identify which questions remain answerable. Suppose the first signal’s duration is still to be chosen between 1 and 2. Write (0, d) with d in {1, 2}, not yet chosen. In either permitted case the signal is active at 0.5; at 1.5 the answer remains unresolved. Choosing d = 2 makes the latter answer yes. The starting-time question remains answerable throughout. This continuation adds a requirement to preserve an unfinished choice, with no default that prescribes behavior the work has left open. Temporal and embodied notation design in NOT.8 develops such cases beyond this elementary timing example.