Part A - Design what can be expressed and done

NOT.1 - Choose the Distinctions and Operations a Notation Must Support

Type: Method Status: Usable, evolving Normativity: Normative

NOT.1:1 - Problem frame

Use this when you are choosing or designing a notation and cannot yet say what its users must be able to express, recognize or do with an expression. A candidate may name all the right objects while hiding their order, dependence or duration. Another may express the distinction but make a frequent change require rebuilding the whole description.

Start with a piece of work: composing operations, comparing two models, tracing a dependency or performing a sequence are examples. Identify an operation that a person or computational agent needs to carry out using the notation. Then find two cases that this operation must treat differently. Trying to express and use those cases gives the first design requirement that can be acted on.

The reader needs enough knowledge of the practice to explain why the cases differ and what a useful result would be. That knowledge can come from a collaborator. Choosing the notation does not require the designer to perform every specialist inference unaided; it does require access to the distinctions that those inferences use.

The result is a choice of distinctions and operations to support, together with a small expression or interaction that shows how a proposed notation could support them. It may instead be a reason to keep an existing notation, preserve an unresolved choice, or obtain a missing account of the work before adding symbols.

If a suitable notation already exists and only one expression needs repair, construct that expression under its rules using A.6.3.RT.OE. Return here when the rules themselves lack a needed distinction or make the required operation impractical.

NOT.1:2 - Problem

How can a notation designer determine what the notation needs to make possible before committing to its symbols and rules?

A list of concepts leaves the work underdetermined. The same concepts can enter different compositions, questions and changes. A notation sufficient for recognizing an object may be insufficient for constructing it; a description sufficient for one question may discard what another question needs. Designing from familiar examples alone can conceal these differences until somebody tries a new case.

NOT.1:3 - Forces

NOT.1:4 - Solution

Choose a working operation → find a consequential difference → locate what carries that difference → try a notation choice → compare the work it enables and displaces → retain or revise the requirement.

NOT.1:4.1 - Choose an operation and the conditions for performing it

Describe what the user starts with, what they do and what they need to obtain. Prefer a question such as “after exchanging these two operations, what changes in the result?” to an aim such as “make the notation intuitive”. The former exposes what an expression must support.

Specify the preparation and access on which the operation relies. A mathematician may use established algebraic conventions; a performer may recognize a learned gesture; a program may receive only serialized text. An AI agent’s access to a rendered page, a text extraction and an editor’s internal structure are different conditions. Include the context that is actually available, rather than assuming that every reader receives the designer’s view.

For a small task, one operation is enough to begin. If the notation will be used for a family of operations, choose cases that exercise materially different demands. For example, interpreting a completed expression and changing a shared component can need different support. Do not turn this selection into an inventory of every conceivable use.

NOT.1:4.2 - Find cases whose difference changes the work

Construct or recover a pair of cases that require different answers, permitted changes or actions. Explain the difference in the practice before deciding how to write it. Two arrangements of the same components can have different outputs; two performances with the same starting times can have different sustained intervals.

Attempt to express both cases with the candidate notation and its allowed context. If they become indistinguishable while the required answers differ, something needed by that operation has been omitted. Recover it through an expression rule, an accessible annotation, a query to another representation or a narrower declared use. The choice depends on how the expression will be used.

Conversely, two visibly different cases may be interchangeable for the present operation. Keep their shared representation if that simplification preserves the needed result. The question determines which differences matter.

A deliberately unfinished choice can itself matter. If the work permits an unspecified order or duration, retain a way to leave it unspecified and a way to recognize when the next operation needs it settled. A convention that silently chooses a value would change the work being described.

NOT.1:4.3 - Locate the information and operations behind the marks

Follow the user through the proposed operation. Which part of the expression identifies a participant, establishes a relation or permits a transformation? Which part is supplied by a convention, the surrounding account or an interaction with a tool?

This examination can reveal three different needs:

• The difference is known but cannot be expressed or recovered under the available rules. Design a way to carry it.
• The expression carries the difference, but obtaining or changing it is too costly. Change the arrangement or the supported operation; NOT.7 develops that redesign.
• The practice has not supplied the needed distinction or inference. Obtain that contribution from the relevant practitioner or method before prescribing how it must be expressed.

Test the whole expression and its usable context. A symbol may be ambiguous in isolation yet clear in a composition. A diagram may be clear to a prepared human reader while its plain-text export loses the grouping on which that reading depends. Preserve the effective context or revise the notation for the receiving conditions.

NOT.1:4.4 - Try a design choice through use

Propose a small change capable of supporting the selected operation. Examples include marking operand groups, adding a reference to a shared component, exposing event duration, or making an unresolved alternative visible. Construct the contrasting cases with that change and carry out the operation using the proposed reading rule.

Where modification matters, change an input, component or relation and follow the resulting edits. A notation that displays one finished example well may make the required family of changes error-prone or laborious. Include a contrasting case that challenges the same rule, rather than polishing the first example until it looks convincing.

Separate the notation’s contribution from supplied subject knowledge. In a diagram for composing functions, arrows can expose the order of application; the functions and the ability to apply them supply the numerical result. If the result cannot be obtained because an algorithm or physical law is missing, return that question to the corresponding practice.

An expert walkthrough can establish an explicit loss or a useful candidate change. When it remains uncertain whether the intended reader can recover or perform the operation, use C.2.8 to choose a bounded reading probe with the intended preparation and access. Obtain further evidence only when its outcome could change the design or the permitted use.

NOT.1:4.5 - Compare the work enabled and the work displaced

Compare candidate notations on the selected operations under comparable conditions. Look for a practical trade-off: a shared definition may make one update cheap while requiring the reader to follow references; expanded expressions may be easy to read locally while repeating changes in several places.

Keep both alternatives when they serve different necessary operations and the correspondence between them can be maintained. NOT.6 develops that arrangement. For a choice among alternatives, compare only characteristics that affect the work and state the selection criterion. A simple contrast can settle the local choice: one notation loses an answer that the other preserves.

When the comparison must be reused across alternatives or design iterations, A.19.CPM governs comparison under a declared comparator; A.19.SelectorMechanism governs selecting a set from those results under stated criteria. A Pareto criterion can retain alternatives that are not dominated on the chosen characteristics. It does not by itself resolve a trade-off between reading and editing effort. Keep that trade-off visible until the work supplies a reason to choose, or retain both notations for their different uses. A vocabulary of design trade-offs can help notice a difficulty, but a vocabulary or weighted checklist cannot establish that a user can perform the operation.

Trying a candidate can reveal that the original question was incomplete. Revise the selected operations or cases when the newly visible distinction changes a useful next move. Then test the affected requirement again. This is a development of the problem and its proposed solution, for which C.40.CD supplies the common method.

NOT.1:4.6 - Retain the decision at the scale needed for the next construction

State the operation to support, the distinction it needs, how a candidate carries it and the burden or unresolved choice that remains. A short explanation beside the candidate can suffice. Keep the contrasting cases when they will help construct, teach or revise the rule; no separate requirements document is necessary for an ordinary small use.

Stop when the next notation construction or choice is clear enough to undertake. Detailed expression formation, interpretation and transformation can then be developed where needed. Do not reopen every design choice merely because a new example is available; reopen the choice whose supported operation or conditions changed.

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.

NOT.1:6 - Bias-Annotation

Fluency can hide a notation’s demands. A designer who knows the answer may supply order, grouping or duration from memory while believing the expression carries it. Test the selected operation with only the expression, declared conventions and available context. C.2.8 distinguishes this expert judgement from an actual reader’s recovery.

A successful notation for one audience can require unavailable preparation in another. Preserve the audience and operation when drawing conclusions; human reading results do not establish an AI agent’s interpretation, or vice versa. Check the receiving conditions that can change the choice.

NOT.1:7 - Conformance Checklist

Use these questions to examine the design decision when its adequacy matters. They do not require a separate written answer for each ordinary application.

• Can the intended user identify the working operation, its starting information and the result they need?
• Does a consequential difference between cases explain why the selected distinction is required? If a difference is omitted, is the resulting loss acceptable for that use?
• Is the necessary information available in the expression, its declared conventions or an accessible interaction?
• Has the proposed rule been used on the contrasting cases, including a change when modification is part of the work?
• Are the subject knowledge and reader preparation needed for that use recoverable?
• Does the next construction follow from the decision, with remaining uncertainty and displaced work visible where they matter?

NOT.1:8 - Common Anti-Patterns and How to Avoid Them

NOT.1:9 - Consequences

The next notation decision becomes a choice about work that can be attempted. A designer can explain why a distinction, convention or editing operation is needed and can recognize when a simpler notation suffices.

The method costs representative construction and use. It can reveal a gap in the subject account that notation design alone cannot settle. Its result is local to the selected operations and conditions; extending use can expose another requirement. Keeping the contrast behind a decision makes that extension easier to reason about.

NOT.1:10 - Architectural Rationale

A notation provides ways to form and use expressions. What to support depends on what the user needs to obtain or change. Starting with that operation connects the subject matter, reader preparation and representation without making any one of them a substitute for the others.

Contrasting cases make an omitted distinction consequential. If two cases with the same available representation require different answers, a reader using only that representation cannot be guaranteed the right answer in both. The design must recover the difference, obtain additional information or accept the resulting limitation. This argument identifies a requirement; trying the supported operation is still needed to learn whether the candidate is usable at the available effort.

The pattern chooses requirements through small constructive trials. NOT.2-.4 develop formation, interpretation and transformation rules. A.6.3.RT.OE addresses a different starting condition: the scheme is available and an operative expression must be built under it. C.2.8 characterizes what a prepared observer can extract; the present method uses that question to construct or choose a scheme. These contributions can be combined without making notation design a compulsory step in every reading task.

NOT.1:11 - SoTA-Echoing

Choosing the design move. When the needed distinction is still unknown, construct a contrasting pair first. A gallery-based comparison already needs expressions that implement chosen tasks; preparing that gallery cannot replace discovering the missing task or distinction. Once several viable notations support a stable family of work, a shared set of examples becomes useful: compare the same readings and edits, use metrics to find costly cases, then inspect those cases. This costs more than the first pair but can expose repeated-change burdens hidden by one successful example. Reopen the small comparison when that broader burden could change the choice; NOT.7 develops the corresponding redesign. Cognitive Dimensions and PUX can also help discover or reformulate the operations to compare.

The contrast-based requirement construction and the two worked cases are the present synthesis. They connect notation design to available FPF representation, extraction and problem-development methods without prescribing one notation across practices.

NOT.1:12 - Relations

NOT.1:End

NOT.2 - Define Expression Rules for Recoverable Binding and Composition

Type: Method Status: Usable, evolving Normativity: Normative

NOT.2:1 - Problem frame

Use this when a notation needs rules for combining expressions or referring to their parts, and the proposed marks leave those rules unclear. Readers may disagree about an operation’s operands, the declaration an occurrence of a name uses, or which connections join two components.

The working question is “How can another user construct and recover the intended arrangement under these rules?” Start with one expression that admits two consequentially different readings. Give its components explicit places, make their grouping and references recoverable, and use the resulting construction to perform the intended operation.

The result is a small set of formation, binding and connection rules demonstrated by an expression and a contrasting case. These rules can support mathematical formulas, diagrammatic descriptions or structured sequences. A complete grammar or software parser is needed only when the work requires it.

The reader must know what the relevant components and operations mean, or obtain that account from a practitioner. The examples below require elementary arithmetic and the stated conventions. Theory of formal languages and category theory are optional tools for extending or proving properties of a scheme.

Use the rules already provided by a suitable notation when they settle the question. A.6.3.RT.OE helps construct an operative expression under such a scheme. Use NOT.1 first if the work has not yet determined which distinctions or operations the notation must support.

NOT.2:2 - Problem

How can the rules for forming an expression expose its components, references and permitted composition well enough to support the required reading or change?

A printed formula can conceal two groupings. The same letter can name an external parameter and a locally introduced variable. Diagram ports with the same value type can have different roles. Correctly recognizing the individual signs therefore leaves open which whole has been expressed.

NOT.2:3 - Forces

NOT.2:4 - Solution

Identify the constructors and their places → make grouping recoverable → define how references are resolved → define permitted connections → construct and read contrasting cases → abbreviate only what the reader can recover.

NOT.2:4.1 - Identify the expression constructors and their operands

Take the needed operation and distinctions from the working question or NOT.1. Describe the elementary expressions and the ways to construct a larger expression from them. For each construction, identify the required parts and what may occupy each place.

For example, a summation expression needs an index declaration, bounds and a body. A two-input transformation needs both inputs and a way to distinguish their roles. A repeated sequence needs a repeat count and the sequence being repeated. These are different constructors with different rules.

State the restrictions that the intended use needs. The restrictions may concern the kind of an operand, the number and order of connections, or a relation between component interfaces. Keep a missing part visible when partial construction is allowed. Identify the operation that requires the part to be supplied.

NOT.2:4.2 - Make the composition structure recoverable

Choose how a user can determine which parts belong to each construction. Text may use delimiters, indentation or precedence; a diagram may use enclosures, named ports or connections. A notation unfolding in time may use a learned boundary gesture or a segmentation convention. Explain the selected rule where its effect is not already established for the intended reader.

Try a nested construction. Recover the outer operation and its immediate parts, then repeat that reading within each part. This is a useful structural reading for tree-like expressions. When a component is shared or a connection returns to an earlier component, expose its reference or connection instead of pretending that every use is an independent nested copy.

Choose the structural distinctions needed by the operation. Some schemes identify several bracketings or drawings as the same expression. Establish the relevant equivalence before omitting their differences. If alternatives are intentionally unresolved, show that unresolved choice and postpone only the operation that needs it decided.

NOT.2:4.3 - Define which declaration or external value a reference uses

Identify the constructions that introduce local names and the parts in which those names apply. State how a name occurrence reaches its declaration or an input supplied from outside. The visible extent of a box or line becomes a scope boundary only through an established rule of the notation.

For a notation with lexical scope, a common rule resolves a name at its nearest enclosing declaration with that name; if there is none, the value must come from the declared external context. Adopt that rule only when it fits the intended interpretation. Other schemes can make references available through earlier statements, explicit identifiers or connections. Define their resolution accordingly.

Trace the references in one expression with repeated names and one expression with a free input. Here a free input is one whose value must be supplied from outside the expression. If insertion, copying or renaming changes which declaration an occurrence reaches, expose that change before treating the result as equivalent. NOT.4 develops transformations that preserve the intended use.

For a reusable component, distinguish its locally introduced names from the inputs and results made available at its boundary. Compose components by those boundary references. A displayed label can be repeated without identifying its occurrences as the same declaration; the reference rule must settle that question.

NOT.2:4.4 - Define the permitted connections between components

Specify which component places can be connected and what the connection means for forming the expression. If a connection passes a result to an operand, include the operand’s position or role as well as its accepted kind. Two inputs of the same kind may still be ordered.

Construct the compound expression and determine its remaining external inputs and outputs. Check each newly connected boundary; an unconnected required input remains an input to be supplied. If the scheme permits shared values, show how repeated references obtain that value. If it describes resources whose copying requires an operation, include that operation rather than deriving copying from a forked line alone.

A connection that forms a cycle needs a rule admitting and interpreting that cycle. Depending on the practice, it might describe an equation, feedback through time or a repeated computation. Choose the intended construction before using one picture for these different operations. NOT.3 develops its interpretation, and CMP.12 supplies an effective evaluator when one is required.

Formation rules establish a permitted expression. Whether its described mathematical construction exists, its algorithm terminates or its physical realization works remains a question for the corresponding methods. The rules should expose the information those questions need.

NOT.2:4.5 - Construct and read the cases that challenge the rule

Build the intended expression from the declared parts. Recover its grouping, resolve the references and identify the component boundaries. Then perform the required operation with the supplied interpretation.

Use a contrasting case that could reveal a mistaken rule. Depending on the chosen construction, change the grouping, reuse a name inside another scope, exchange two ports or introduce a shared component. Select the challenge because its interpretation can change the result.

If two readers still recover different arrangements that matter to the work, locate the point of divergence. Add or repair the corresponding delimiter, reference rule or connection convention. If both arrangements express the intended equivalence, make that equivalence part of the scheme instead of forcing one arbitrary drawing.

NOT.2:4.6 - Retain usable conventions and controlled abbreviations

Keep the rule at the scale required by its next use. A legend and a worked construction can suffice for a local diagram; repeated automated processing can justify a grammar, parser or structural editor. A user must be able to recover any omitted structure that changes the operation.

For an abbreviation, give the expansion or another way to establish its meaning. If several expansions are allowed, show why the difference does not affect the intended use. NOT.4 handles the corresponding preservation argument. Retain a visible distinction when no such argument is available and the choice matters.

The method finishes with rules that permit the required construction and make its use recoverable, or with a specific unresolved formation or interpretation question. Teaching those rules and comparing their reading cost use the relevant learning and explanation methods; a more explicit syntax alone does not establish that a reader has learned it.

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.

NOT.2:6 - Bias-Annotation

A designer can mistake their own intended grouping for a rule already available to the reader. Recover the expression from its declared conventions before relying on the intended result. The sum example separates those two readings with different outputs.

Familiar programming conventions can also be imported into another notation without justification. State the resolution policy that fits the represented work. An arrow, enclosure or repeated name acquires its binding and composition role through that policy.

NOT.2:7 - Conformance Checklist

Use the applicable questions when the construction rules need checking.

• Can the reader recover the components and the places they occupy from the expression and available conventions?
• Are the groupings that change the operation distinguishable, or explicitly retained as unresolved alternatives?
• Does each needed reference reach its declaration, external input or unresolved place according to the stated policy?
• Do composition rules account for the roles of connected participants and the resulting external boundary?
• Does a contrasting construction expose a consequential change of grouping, reference or connection?
• Can an abbreviation be expanded or otherwise interpreted without losing a distinction needed by the operation?

NOT.2:8 - Common Anti-Patterns and How to Avoid Them

NOT.2:9 - Consequences

Users can construct expressions whose parts and references can be recovered, and can locate where a change alters the described operation. The same rules provide an input to interpretation, translation and automated processing.

Explicit structure can make an expression longer or impose conventions to learn. Controlled abbreviations and alternative views can reduce that burden once their interpretation is established. A scheme can also require a richer account of scope or interfaces than the first examples revealed.

NOT.2:10 - Architectural Rationale

Formation, binding and composition answer connected questions. Formation identifies how an expression is built. Binding determines how its references obtain their values or participants. Composition determines how component boundaries are joined. A grammar that answers only the first question can leave the other two undecided.

This method develops the missing rules. NOT.1 selects what the work needs them to support; NOT.3 gives the interpretation and reading operations. A.6.3.RT.OE supplies construction under an available scheme. MATH.17/.18 can describe the operations and their interpretations mathematically, while CMP.12 constructs an effective interpreter or translation when required. The general notation-design result remains usable without first constructing all of those formal accounts.

Different graphical presentations or local names can express the same construction under the chosen rules. Treating them as equivalent requires the corresponding argument; it cannot be inferred from visual similarity. Conversely, giving two ports the same type cannot identify their roles. Keeping those questions explicit allows the notation to help reasoning about the construction itself.

NOT.2:11 - SoTA-Echoing

Choosing the rule design. Retain familiar conventions when they already determine the needed grouping, reference and connection. A constructor list or an implicit layout is cheaper to state, but the sum and ratio cases show the price when it leaves two consequential readings. For that local difficulty, add the missing scope, delimiter or operand role and try the changed expression. This costs more signs or conventions to learn, while preserving a first use through a legend and worked construction.

When static references cross imports or other boundaries that simple nested environments do not handle conveniently, a scope-graph model can make the resolution policy explicit and reusable. It requires constructing those scopes, paths and priorities, and assessing the available implementation. Use that richer account when the reference problem calls for it, rather than adding it to the elementary sum. Likewise, choose a formal binding theory or diagram calculus when its laws are needed for repeated transformation or reasoning.

Revisit the retained choice when a needed expression cannot recover its references or composition, the interpreted operation changes, or a less costly available scheme preserves the same needed distinction. The synthesis is a way to construct and revise those rules; it does not select one notation or one binding policy for every practice.

NOT.2:12 - Relations

NOT.2:End

NOT.3 - Give Expressions an Operative Interpretation

Type: Method Status: Usable, evolving Normativity: Normative

NOT.3:1 - Problem frame

Use this when a notation can form expressions, but its users lack a clear way to obtain the consequence those expressions are supposed to support. A legend may explain the signs while leaving the reader to invent how to combine, inspect or manipulate them.

Start with one question an expression should help answer. Give its relevant parts an interpretation, supply the operations needed to read them together, and follow that reading to a result. A useful first result can also be a located gap: a value, interpretation rule or reader capability that must be supplied before the question can be answered.

An operative interpretation here connects what an expression stands for with what a prepared reader can do to obtain a needed consequence from it. Reading can involve calculation, inspection of a diagram, rule-based manipulation or enactment. The reader may be a person or a computational agent; the applicable operations and preparation depend on that reader.

The designer needs the intended subject account and the operations on which the reading depends. The first example uses elementary Boolean reasoning, explained in the case; the temporal example uses addition and interval comparison. A machine implementation is a further construction when required.

Use an existing interpretation directly when it already supplies the needed reading. A.6.3.RT.OE addresses building an operative expression under an available scheme. Use the present method to develop or repair the scheme’s interpretation and its reading operations. NOT.2 addresses missing grouping, binding and connection rules.

NOT.3:2 - Problem

How can a notation’s interpretation give its intended reader a usable way to obtain a consequence, while keeping the conditions of that consequence recoverable?

Assigning names to marks can leave their combined use unexplained. Assigning a mathematical meaning can specify an answer without providing a way to obtain it. A familiar reader may bridge either gap from experience while a new reader cannot tell which operation is missing.

NOT.3:3 - Forces

NOT.3:4 - Solution

Choose the consequence → interpret the expression’s parts and composition → supply the reading operations → perform a case → test a consequential change → retain the usable reading and its conditions.

NOT.3:4.1 - Choose what the reading must produce

State the question, the expression and the information available outside it. Identify what would count as a useful result. Examples include the value of a formula, a permitted next action, whether two intervals overlap, or a consequence of a diagram’s relations.

Distinguish obtaining the expressed answer from establishing its use in the subject matter. Reading an interval from a schedule can answer what is planned; observing what happened needs the corresponding observation. The interpretation should identify which question it answers.

Specify the reader’s available operations. It may be reasonable to presuppose arithmetic, following a visible connection or a trained movement. Expose a less familiar operation if the intended reader could otherwise mistake its absence for a defect in the expression. A short prerequisite explanation can suffice; a capability that requires practice belongs in the appropriate learning method.

NOT.3:4.2 - Interpret components, composition and context

For the parts used by the question, specify what they denote or instruct and how their combination is interpreted. Include grouping and reference rules from NOT.2. An arrow may supply an input to an operation, express a precedence condition or prescribe a movement; choose its role in this scheme before deriving a consequence.

Give the interpretation of a compound expression in terms that permit its use. For a compositional mathematical scheme, interpret the parts and then apply the corresponding operation to their interpretations. MATH.5/.18 develops that construction and its preservation questions. When context contributes to interpretation, include the relevant context among the inputs rather than assigning a context-free meaning that the notation does not support.

If the expression admits alternatives or has an unresolved part, specify what can still be concluded. A reader may be able to obtain a common consequence across the alternatives. Another question may have to wait for the missing information. Preserve that distinction instead of silently completing the expression.

NOT.3:4.3 - Construct the reading operations

Show how the intended reader moves from the expression and available inputs to the result. For each indispensable step, identify what is inspected or changed and how the next step uses what was obtained. The procedure may reuse an established method rather than explain that method again.

For a formula, this can mean resolving a parameter, evaluating selected subexpressions and combining their values. For a diagram, it can mean identifying a boundary, following the admitted connections and applying the corresponding inference. For a performed notation, it can mean recognizing a cue and executing a learned action. Choose operations that the reader can actually perform under the stated conditions.

Keep the reasoning behind a manipulation available at the level required by the work. A rule may be applied fluently after training, while changing the rule requires understanding what it preserves. NOT.4 develops expression transformations; B.5.RC/RA helps recover an unfamiliar construction or argument when that is the missing contribution.

If a mathematical interpretation leaves the required obtaining procedure unavailable, name that remaining problem. CMP.12 constructs effective evaluation when the expression’s operations admit it; other CMP methods can help construct the needed algorithm. Denotation alone does not establish computability or affordability. A human reading or trained perceptual operation also needs its actual capability and access conditions.

NOT.3:4.4 - Perform the reading and locate where it depends on additional knowledge

Take an expression whose relevant parts and input values are supplied. Carry the reading through to its stated result. At a step that cannot be performed, distinguish a missing input, an undefined interpretation and an unavailable operation. These lead to different repairs.

Compare the result with the intended consequence under the subject account. Repair an interpretation rule when it yields a different answer. Repair the reader preparation or access when the rule is usable in principle but cannot be carried out under the current conditions. If the subject account itself leaves the consequence unresolved, return that question to the corresponding mathematical, physical, computational or other practice.

When more than one reading route is offered, perform the routes on a case that could expose a difference. Establish the agreement needed by the use, or explain why the routes answer different questions. A route that only returns a final value may lose the dependency structure another operation needs.

NOT.3:4.5 - Change one condition that matters to the interpretation

Change a value, grouping, reference, context or intended question that could alter the result. Use the same rules to recover the new answer, or locate the rule that must change. The challenge should test the chosen interpretation rather than introduce an unrelated subject problem.

For a formal family, a general argument can establish that the reading operations implement the interpretation across that family. MATH.18 supplies interpretation-preservation questions; CMP.12 supplies executable translation and evaluation. For human or embodied use, C.2.8 can characterize what the prepared reader recovers and distinguish an expert walkthrough from observed use. Choose further checking when it can change the interpretation or permitted reliance.

NOT.3:4.6 - Keep the operative reading available to its users

Provide the interpretation, the indispensable reading operations and their prerequisites where the intended user can find them. A legend, one worked reading and a reference to a known method may be enough. A new operator needs enough explanation to use it, rather than only a new name.

Stop when the required consequence can be obtained at the declared scope, or when the missing contribution and the next way to obtain it are clear. Retain a slower explanatory route alongside a fluent route when users need to learn, justify or change the operation. A further notation redesign belongs to NOT.7 if the reading works but imposes avoidable difficulty.

NOT.3:5 - Archetypal Grounding

NOT.3:5.1 - Turn a condition diagram into a reading procedure

A team designs a notation for combinations of conditions. Labels P, Q and R refer to conditions with supplied truth values. An ALL enclosure holds only when every enclosed condition holds. An ANY enclosure holds when at least one enclosed condition holds. The conditions are stable while the expression is read, and reading them does not change their values.

Consider:

ALL {
  P
  ANY { Q R }
}

The legend defines the operators, but a new reader still needs a way to apply them to a nested expression. Supply this reading procedure: resolve each condition label from the given inputs; evaluate the innermost enclosure; replace it by its Boolean result; continue outward.

With P = true, Q = false and R = true, the inner enclosure is true because R is true. The outer enclosure combines true with true, so the whole condition holds. This result concerns the supplied conditions. Establishing their truth in an actual project is separate work.

A second route examines only values that can still change the answer. For ALL, one false child settles the result as false; for ANY, one true child settles it as true. With the same inputs, the reader checks P, then Q and R, and obtains the same result. If P changes to false, the outer ALL is false without inspecting Q or R. The stable, side-effect-free condition premise makes this omitted reading legitimate.

Now change the inner operator from ANY to ALL while keeping the original inputs. The inner result becomes false and therefore the outer result becomes false. The reader has followed the changed expression through the same interpretation rules. Replacing a label without resolving its value would leave a different gap: the input is missing, not the nesting rule.

The two reading routes expose a choice about obtaining the answer. They agree on these Boolean results; one shows every intermediate value, while the other can avoid work. A user who also needs all intermediate values should retain the first route or extend the second to produce them.

With P false and the values of Q and R not supplied, the whole expression is still false. A request for every intermediate value still needs those missing inputs.

NOT.3:5.2 - Recover what an interval notation means before using it

A signal plan contains the pair (2, 4). The notation designer must say whether the second component is an ending time or a duration. Under the first interpretation, the planned active interval is [2, 4); under the second it is [2, 6). Half-open intervals include their start and exclude their end.

Choose (start, duration). Give the reader two operations: add duration to start to obtain the end, then test whether a queried time is at least the start and less than the end. At time 5 the planned signal is active, because 2 <= 5 < 6. If duration changes from 4 to 2, the new end is 4, and the signal is inactive at time 5.

The pair’s two numeric entries did not establish this interpretation by themselves. The chosen convention and the comparison procedure make the answer obtainable. A timeline can support another reading by locating the queried point against a drawn interval, provided its scale and endpoints carry the same values.

This answers a question about the plan. To determine whether a device actually emitted the signal at time 5, obtain the relevant observation and the conditions relating it to emission. The notation can retain that observation or a model of the device, but the planned interval alone supplies neither.

NOT.3:6 - Bias-Annotation