Coverage / Reachability¶

Version: v1 · 2026-08-24 · History
Prime #: 759
Domain group: Applied Sciences & Engineering
Origin domain: Computer Science & Software Engineering
Subdomain: Formal Methods and Verification → Computer Science & Software Engineering

Core Idea¶

Coverage asserts that every required target in a target set is reachable from the system's inputs, pathways, or mechanisms — completeness in the surjective direction. The failure mode is a gap: a required target reached by no source. The claim is silent about uniqueness, efficiency, or whether multiple pathways exist.

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A Crayon for Every Colour

Imagine you have a box of crayons and a picture you want to finish. Coverage means: for every colour the picture needs, you have at least one crayon that can make it. If even one needed colour is missing from your box, that is a gap. You do not care how many crayons of that colour you have, just that none of the needed ones are missing.

Reach Everything, No Gaps

Coverage, or reachability, is the claim that every target you are supposed to reach can be reached from your system's inputs or pathways. It is a completeness claim: nothing required is left unreachable. The failure mode is a gap, a target that was supposed to be reachable but is not. You check it by listing all the required targets and making sure at least one path leads to each one. It deliberately says nothing about whether the path is the best one, or whether there are several paths, only that at least one exists for every target. A sneaky risk is that the list of required targets quietly grows, new cases get added, while everyone keeps trusting the old proof that coverage held.

Every Target Reachable

Coverage / reachability is the structural pattern of asserting that every required target in some target set is reachable from, or producible by, the system's inputs, pathways, or mechanisms. It is a completeness claim in the surjective direction: the system covers the target set and nothing required is left unreachable. The failure mode is a gap, a target the system was supposed to reach but cannot, and the diagnostic is to enumerate the required targets and check at least one pathway exists to each. It is deliberately silent about uniqueness, efficiency, or how many pathways exist. Formally it is the surjectivity of a relation from a source set onto a required-target set. Two properties follow: coverage is monotone, since adding inputs or pathways can only increase it, and it is relative to the target set, so it can be trivially gained by shrinking the targets or broken by expanding them, which makes the target set itself a load-bearing design choice rather than fixed background. It is the surjective sibling of injectivity, which forbids collisions; together they give the bijection properties, but each can hold or fail without the other.

Coverage / reachability is the structural pattern of asserting that every required target in some target set is reachable from, or producible by, the system's inputs, pathways, or mechanisms. The claim is one of completeness in the surjective direction: the system covers the target set, and nothing required is left unreachable. The failure mode is a gap, a target that the system was supposed to reach but cannot, and the diagnostic is enumeration of the required targets plus a check that at least one pathway exists to each. The pattern is the completeness claim about a relation (a mapping, routing, accessibility, or productive capacity) and is deliberately silent about the uniqueness or efficiency of the pathway and about whether multiple pathways exist. Formally it is the surjectivity of a relation R from a source set S to a required-target set T: for every t in T there exists some s in S with s R t, the negation being a t reached by no s. Two structural properties follow. Coverage is monotone: adding inputs or pathways can only increase it, removing them can only decrease it. And coverage is relative to T: it is trivially achieved by shrinking T and trivially broken by expanding it, which makes the target set itself a load-bearing design artefact rather than a fixed background. The standing risk is that a coverage claim rests on an implicit target set that drifts, new branches, beneficiary classes, or failure modes silently enlarging T while the old coverage proof is still cited. It is the surjective sibling of injectivity (no collisions, distinct inputs to distinct outputs); together they generate the standard bijection properties, but each carries an independent intervention catalogue and can hold or fail without the other.

Broad Use¶

Software testing: branch, path, and statement coverage — every code branch exercised by at least one test.
Network reachability: every required destination reachable from every required origin in the routing fabric.
Public policy: every entitled beneficiary class has at least one accessible pathway to the benefit; gaps are eligible-but-unreachable populations.
Access control: every required action permitted by at least one granted role; an unpermitted required action breaks the workflow.
Curriculum design: every required competency taught by at least one course; an uncovered competency is a graduation gap.
Failure-mode analysis: every documented failure mode has at least one mitigating control; an uncovered mode is a hole in the safety case.

Clarity¶

It distinguishes completeness in the surjective direction from neighbouring completeness claims, and foregrounds the target set as a design object: coverage is meaningful only relative to a stipulated set, so an honest claim must state its target set explicitly.

Manages Complexity¶

It reduces heterogeneous problems to one five-step procedure — enumerate targets, map pathways, identify gaps, close gaps, watch for target-set drift — that a test engineer, network architect, and safety analyst all run under different names.

Abstract Reasoning¶

Because coverage is monotone in pathways and relative to the target set, the non-obvious move is to treat target-set drift, not pathway failure, as the dominant silent failure: a quietly growing target set leaves an old, still-cited coverage proof that no longer covers.

Knowledge Transfer¶

Across domains: required-target set maps to branches / destinations / beneficiary classes / competencies; the relation maps to exercises / routes-to / reaches / teaches; the audit is invariant.
The portable lesson: the most dangerous coverage failures are usually unexamined growth of the target set, so the audit must re-anchor on it every time.

Example¶

A test report citing "92%" branch coverage silently goes stale the moment a developer adds a new if, because the target set enlarged while the old proof was measured against a smaller one — fixed honestly by writing a test, cosmetically by excluding the file.

Relationships to Other Abstractions¶

Current abstraction Coverage / Reachability Prime

Parents (2) — outgoing typed relations

Coverage / Reachability is a kind of Completeness Prime

Coverage/reachability is explicitly the surjective-direction specialization of the general no-gaps completeness claim.
Coverage / Reachability is a kind of Surjectivity Prime

Coverage / Reachability is a specialization of Surjectivity, retaining the parent's defining structure while adding the child's specific commitments.

Children (23) — abstractions linked here by typed relations

Content validity Domain-specific is a kind of Coverage / Reachability

It remains its own entry because its identity is fixed by the target construct and use, domain blueprint and facets, item pool, expert qualifications and sampling, rating criteria, agreement or index calculation, revision decisions and residual coverage gaps.
Cop-win graph Domain-specific is a kind of Coverage / Reachability

It remains its own entry because its identity is fixed by the graph class and loop convention, player counts and initial-choice order, legal move and information rules, capture condition, finite assumption, winning strategy and dismantling or constructibility certificate.
Cover (topology) Domain-specific is a kind of Coverage / Reachability

What makes it its own entry: A family of subsets whose union contains a specified set or space, with open covers restricting the members to open subsets.
Covering design Domain-specific is a kind of Coverage / Reachability

Covering Design instantiates Coverage / Reachability because every required t-subset must be reached by at least one selected block through the containment relation, with minimization applied after completeness is secured.
Covering number Domain-specific is a kind of Coverage / Reachability

It remains its own entry because its identity is fixed by the metric or pseudometric space, target subset, radius and positivity, open or closed balls, internal or external center restriction, candidate cover and proof, minimized cardinality and relation to packing number or metric entropy.

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Covering Set Domain-specific is a kind of Coverage / Reachability
Covering Set instantiates **Coverage / Reachability** in a strict number-theoretic form: the required targets are all allowed indices, and each must be reached by at least one prime-divisibility rule.
Danzer set Domain-specific is a kind of Coverage / Reachability
What makes it its own entry: A set of points in Euclidean space that intersects every convex body of unit volume, studied through the unresolved question of whether bounded-density examples exist.
Discoverability Domain-specific is a kind of Coverage / Reachability
HCI discoverability is coverage and reachability specialized to the goal-weighted portion of a capability set reachable through bounded perception and exploration.
Dominant functor Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the source and target categories, functor, quantified target object, source witness, section and retraction arrows and identity composite.
Erdős–Woods number Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences.
Eternal dominating set Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by graph, initial guard set, attack visibility and legality, movement model, occupancy, response rule, domination after every turn, adversary power, and minimum objective.
Mode collapse Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the target data distribution and relevant modes, generative model and latent source, training or fine-tuning objective, diversity and coverage metrics, omitted-mode evidence, distinction from memorization and sampling settings.
News desert Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the community and geographic boundary, population and language, local-information needs, qualifying outlet and original-reporting criteria, publication frequency, staffing and beats, access and affordability, ownership, time period, data source, and threshold.
Opaque set Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the planar body and boundary, allowed barrier sets, line-crossing definition, intersection convention, connectedness, total-length measure, coverage proof, lower and upper bounds and optimality status.
Pancyclic graph Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the finite graph and order n, directed or undirected cycle convention, simplicity, every integer length from three to n, witness cycles, Hamiltonian implication and vertex edge or bipancyclic qualifications.
Piecewise syndetic set Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the ambient semigroup or natural numbers, subset, translate convention, finite witness or gap bound, arbitrary finite-pattern or long-interval quantifiers and equivalence with thick-intersection and partition-regular formulations.
Reachability problem Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the state space and representation, initial and target states or sets, transition rules, path semantics, finite or infinite horizon, exact or approximate decision and complexity or decidability claim.
Γ-space Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the topological space, cover family and openness, omega-cover finite-subset condition and exclusion, selected subfamily, gamma-cover eventual-membership condition, countability assumptions and equivalent combinatorial characterization.
Tarski's Plank Problem Domain-specific is a kind of Coverage / Reachability
**Coverage / Reachability** is the strict parent by specialization.
Web navigation Domain-specific is a kind of Coverage / Reachability
It remains its own entry because its identity is fixed by the information space and user goal, nodes and hyperlinks, global and local hierarchy, contextual and supplemental controls, labels and orientation cues, browser state, path and usability evidence.
NAT Traversal Domain-specific presupposes Coverage / Reachability
NAT Traversal presupposes Coverage / Reachability because its techniques exist to establish and maintain endpoint reachability across address-translating gateways.
Revocation Closure Prime includes Coverage / Reachability
Complete reachability over the grant's actual live effect set is a strict constituent of Revocation Closure.
Schema-Bounded Blind Spot Prime presupposes Coverage / Reachability
A schema-bounded blind spot exists only relative to an intended coverage claim connecting elicitation prompts or categories to the cases that should be surfaced.

Hierarchy paths (2) — routes to 2 parentless roots

Coverage / Reachability → is a kind of Completeness

Show alternative path (1)

Not to Be Confused With¶

Coverage / Reachability is not Traceability because traceability is a per-element provenance claim (each artefact links back to its source), whereas coverage is surjective completeness (every target reached by some source).
Coverage / Reachability is not Redundancy because coverage demands at least one pathway and is silent about a second, whereas redundancy demands more than one and adds resilience coverage never offers.
Coverage / Reachability is not general Completeness because coverage is the specific surjective form — completeness of a Relation relative to a stipulated target set, monotone in pathways.