NOT.2:5 - Archetypal Grounding

NOT.2:5.1 - A sum with a local index and an external parameter

A team is designing a compact notation for repeated addition. Its proposed expression is:

sum i=1..2 of i + p

The notation has not specified where the summation body ends. At external parameter p = 10, two readings give different results: (1 + 10) + (2 + 10) = 23, or (1 + 2) + 10 = 13.

Choose a constructor sum(index, lower, upper, body). It introduces the index only within body; bounds use the surrounding context. Choose add(left, right) for addition. These rules express the first reading as:

sum(i, 1, 2, add(i, p))

The occurrence of i in add(i, p) reaches the local index declaration. The occurrence of p has no local declaration and uses the external value 10. Substituting the two index values gives 11 and 12, then 23. The second reading has a different construction:

add(sum(i, 1, 2, i), p)

Its outer operation adds the external parameter once and gives 13. The notation now makes the two operand structures recoverable.

Naming also matters. The expression sum(p, 1, 2, add(p, p)) is a valid expression under the chosen lexical rule, but both occurrences in its body refer to the index. It gives 6 and no longer uses the external parameter. To rename the index while preserving that parameter, choose a name such as j that does not capture it: sum(j, 1, 2, add(j, p)) still gives 23. This example identifies the binding condition; NOT.4 supplies the general transformation method.

NOT.2:5.2 - Two ports of the same type with different roles

A diagram notation describes a ratio operation. It has two numeric inputs, numerator and denominator, and a numeric output. The supplied rule divides the numerator by a nonzero denominator. Source A supplies 6 and source B supplies 3.

The first drawing joins both sources to an unlabeled box. Knowing that both connections carry numbers leaves their operand positions undecided. Choose named input ports and preserve their identities when the box is moved or redrawn:

A: 6 -> ratio.numerator
B: 3 -> ratio.denominator
ratio.result -> answer

The result is 2. Reversing the two input connections is another well-formed construction with result 0.5. Port names make that change visible even though the input types remain the same.

Now enclose the ratio in a reusable component. Its exposed inputs refer to those two ports, and its output refers to ratio.result. Renaming the internal ratio box or changing its position can preserve those boundary connections. Swapping the connections changes the expressed operation. A rule that permits the former does not thereby permit the latter.

The same design question arises whenever compatible participants occupy different roles in a relation or operation. More elaborate diagram calculi can give the permitted connections and drawing equivalences mathematical definitions.

NOT.2:5.3 - Scope in a structured sequence

Suppose sequence(A, B) means perform A and then B, and repeat(n, S) means repeat S n times. Then repeat(3, sequence(A, B)) contains three performances of each action. sequence(repeat(3, A), B) contains three of A and one of B. A written delimiter, spoken grouping cue or learned gesture can carry this boundary if the receiver can reliably use its convention. The temporal realization and effort of perceiving or performing it are further questions for NOT.8.

NOT.2:5.4 - Resolve a movable label at the time of its use

Three physical cups have permanent numbers 1, 2 and 3, volumes 20, 10 and 0 ml, and capacity 50 ml each. Removable tags A, B and C initially mark cups 1, 2 and 3 respectively. An instruction H says to transfer 5 ml from A to B. Each letter refers to the cup bearing that tag when the transfer is performed.

Write H; swap-tags(B, C); H, where the swap moves only tags. The first H gives volumes (15, 15, 0) by permanent cup number. After the swap, B marks cup 3. The second H therefore gives (10, 15, 5). Both transfers have sufficient source volume and destination capacity.

The abbreviation H retains a lookup to perform, not cup numbers fixed when H was defined. If the intended instruction instead bound A and B permanently to the original cups, the same two transfers would give (10, 20, 0). The designer must choose the rule matching the work. Changing the tags is an action on the represented situation; renaming a letter in the notation while preserving its reference is a different change.