Reachability Matrix or Table¶
Coverage record — instantiates Generated Span Closure Design
Aggregates many membership verdicts into a table of which targets the generators can and cannot reach, making the coverage holes visible.
One membership answer settles one target; a decision usually needs the shape of the whole coverage. Reachability Matrix or Table is the aggregate record: rows are the targets you care about, columns capture how they can be reached (or from which starting points, or within how many steps), and each cell is a reachable / not-reachable / not-yet-known verdict. Its job is to turn a scatter of individual membership results into a map, so the holes — the targets nothing reaches, the regions the generators can't cover — are visible at a glance. It is a summary artifact, not a constructor: a cell tells you that a target is reachable, not the recipe for reaching it. That distinction is what separates it from the per-element certificate, which supplies the recipe but never the map.
Example¶
A regional airline wants to know which city pairs a traveler can complete using only its own flights, given a fixed hub-and-spoke network and a cap of two connections. Reachability Matrix or Table lays it out: rows are origin cities, columns are destination cities, each cell marks "reachable within ≤2 connections," "reachable but only with 3+," or "not reachable on our metal." Filling the table runs a membership check per cell — can you get from A to B under the connection cap? The finished grid shows the coverage picture instantly: a dense block around the hub, a diagonal of easy same-region pairs, and a conspicuous cluster of empty cells for cross-network city pairs that need a partner airline.
The table doesn't hand a traveler an itinerary — it doesn't say which two connections — but it shows the airline exactly where its own network covers demand and where the holes are, which is the decision the planners actually face.
How it works¶
- Choose the axes. Decide what the rows and columns enumerate — targets against sources, targets against step-budgets, or capabilities against feature sets.
- Bound the enumeration. Because the target space is often huge, set the truncation policy: how many steps, how large a target list, what region of interest — and label the table as covering that scope.
- Fill each cell with a verdict. Run the membership criterion for each cell and record reachable / unreachable / unknown, distinguishing "checked and false" from "not yet checked."
- Read the coverage. Surface the holes and the dense regions; the table's payload is the pattern, not any single cell.
The defining move is aggregation under an explicit scope: it collects many verdicts into one picture and is honest that the picture covers a bounded slice, not the infinite target space.
Tuning parameters¶
- Cell resolution — a coarse reachable/unreachable flag versus a graded cell (steps required, cost, probability). Finer cells carry more but cost more verdicts to fill.
- Target enumeration scope — how many targets and which region the table covers. Wider scope reveals more holes but multiplies the membership checks.
- Unknown handling — whether unchecked cells are blank, marked "unknown," or optimistically assumed. Marking them honestly prevents an untested cell from reading as a reachability claim.
- Refresh trigger — when the table is recomputed (on generator change, on new targets). Stale tables silently misreport coverage after the generators move.
When it helps, and when it misleads¶
Its strength is the bird's-eye view: it converts many one-off reachability questions into a coverage map that makes gaps obvious, which is exactly what "is the generator set expressive enough?" really asks. It is the natural place to spot that a search system can't reach a whole region, or that a feature kit leaves a class of screens unbuildable.
Its two failure modes are both about reading too much into it. First, a blank or unknown cell is not a proof of unreachability — it may simply be untested or beyond the truncation depth — yet empty cells read as "impossible" to a hurried eye.[n1] Second, the target space is usually far larger than any table, so a tidy grid invites the illusion of completeness; the covered slice gets mistaken for the whole. The guarding discipline is to label the truncation scope on the table's face, to distinguish "checked false" from "not checked," and to treat holes as questions — candidates for a membership certificate or a wider generator set — rather than verdicts.
How it implements the components¶
Reachability Matrix or Table realizes the coverage-summary machinery:
coverage_or_reachability_record— it is that record, laid out as a table whose pattern of filled and empty cells is the coverage picture.span_membership_criterion— each cell is one application of the membership test, reused across the whole grid.truncation_depth_policy— the table's scope (step-budget, target region) is an explicit, labeled truncation, so it never poses as the full space.
It does NOT construct the explicit generator combination that reaches a target (combination_expression_template) — that is Span Membership Certificate, its nearest twin: the certificate proves one element with a recipe, this table summarizes many as verdicts. Nor does it declare the generators (generator_set_specification) — that is Generator Inventory.
Related¶
- Instantiates: Generated Span Closure Design — the table supplies the coverage picture the appraisal uses to judge expressiveness and spot gaps.
- Consumes: Span Membership Certificate supplies the per-cell membership verdicts the table aggregates.
- Sibling mechanisms: Generator Inventory · Closure Generation Workflow · Span Membership Certificate · Dependency Elimination Test · Normal Form Reduction Procedure · Bounded Depth Generation Template · Basis Sensitivity Review
Editorial Notes¶
Form Classification¶
Form family: Representation, Specification & Plan
Rationale: Reachability Matrix Or Table operates by externalizes bounded reachability relations on explicit row and column axes in a labeled matrix. That concrete deployed or enacted form is Representation, Specification & Plan under the frozen taxonomy.
Nearest alternative: Analysis, Modeling & Optimization — Although Analysis, Modeling & Optimization can support this mechanism, the frozen evidence makes its operative form the act that externalizes bounded reachability relations on explicit row and column axes in a labeled matrix; the alternative is therefore secondary rather than defining.
Review outcome: Adjudicated after independent review; medium confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Convergent development
Present-day reach: Multi-domain
Rationale: Tabulating whether generators span targets is a linear-algebraic and abstract-algebraic coverage construction.
Related originating lineages:
- Computer Science & Software Engineering — Algorithmic graph-reachability tables provide a parallel computational formulation.
Review resolution: Both blind reviewers agree that mathematics is the primary origin. Explicit reconciliation of alternate origin disagreement, origin mode disagreement, encyclopedia synthesis disagreement adopts reviewer_b's classification because tabulating whether generators span targets is a linear-algebraic and abstract-algebraic coverage construction. The resulting lineage records alternates=computer_science, origin_mode=convergent, and domain_reach=multi_domain; these describe formative provenance separately from later applicability.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The reachability matrix of a graph — computable by transitive-closure methods such as Warshall's algorithm — records for every pair whether a path exists. Its cells are strictly "path exists / no path within the modeled edges"; a false cell means unreachable in the model, not unreachable in principle, which is why an untested or truncated cell must never be read as an impossibility proof. ↩