Configuration Space Ontology

FLOW ALGEBRA
CONFIGURATION SPACE ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Coordinate and Spatial Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities/configuration-space-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13

1. PURPOSE
Represent the set of all admissible configurations of a system, its degrees of freedom, constraints, coordinate charts, topology, and links to motion planning and mechanics. Distinguish configurations (positions/arrangements) from full dynamical states (which may include velocities and momenta).

2. TYPED ENTITIES
ConfigurationSpace: Space
Configuration: Point
System: PhysicalOrAbstractSystem
DegreeOfFreedom: IndependentParameter
GeneralisedCoordinate: Coordinate
Constraint: Predicate
HolonomicConstraint: Equation
NonholonomicConstraint: DifferentialConstraint
CoordinateChart: LocalParameterisation
ConfigurationManifold: SmoothManifold
FreeConfigurationSpace: Subset
ObstacleConfigurationSpace: Subset
Path: ContinuousMap
Trajectory: TimeParameterisedPath
Metric: DistanceFunction
Topology: OpenSetStructure

3. CORE DEFINITIONS
For system S, Q(S) = {q | q is a physically or logically admissible configuration of S}.
For n independent local degrees of freedom, q = (q1,...,qn) in a local coordinate chart.
A configuration space need not be globally Euclidean; examples include circles S^1, tori (S^1)^n, and manifolds with singularities or boundaries.
A configuration is not automatically a phase-space point: in classical mechanics the phase space is often T*Q, the cotangent bundle of Q.

4. AXIOMS AND INVARIANTS
A1: Every Configuration q belongs to exactly the declared ambient Q for a given model context.
A2: Q contains only configurations satisfying all active admissibility constraints.
A3: Coordinate representations may differ without changing the underlying configuration.
A4: A continuous feasible path gamma:[0,1] -> Q_free must remain in Q_free for all parameter values.
A5: Dimension equals the number of independent local coordinates at regular points; constraints may reduce it.
A6: A metric, differentiable structure, or global coordinates are optional enrichments, not prerequisites for an abstract configuration space.

5. RELATIONSHIPS
HAS_CONFIGURATION(System, Configuration)
CONFIGURATION_BELONGS_TO(Configuration, ConfigurationSpace)
HAS_DEGREE_OF_FREEDOM(System, DegreeOfFreedom)
REPRESENTED_BY(Configuration, GeneralisedCoordinate)
SATISFIES(Configuration, Constraint)
EXCLUDES(Constraint, Configuration)
HAS_TOPOLOGY(ConfigurationSpace, Topology)
HAS_CHART(ConfigurationManifold, CoordinateChart)
HAS_FREE_SUBSPACE(ConfigurationSpace, FreeConfigurationSpace)
HAS_OBSTACLE_SUBSPACE(ConfigurationSpace, ObstacleConfigurationSpace)
PATH_CONNECTS(Path, Configuration, Configuration)
IS_BASE_OF(ConfigurationSpace, PhaseSpace)

6. FORMAL OPERATORS
Q = {q in Q_ambient | forall c in C, c(q)=true}
Q_obs = {q in Q | geometry(q) intersects obstacles}
Q_free = Q \ Q_obs
gamma:[0,1] -> Q_free; gamma(0)=q_start; gamma(1)=q_goal
For independent regular holonomic constraints f_i(q)=0, local dimension = n - rank(Df).
For planar rigid-body pose: Q = SE(2) = R^2 x S^1 as a manifold.
For a revolute joint: Q = S^1; for n independent revolute joints: Q = (S^1)^n.

7. INFERENCE RULES
R1: IF q in Q_ambient AND all active constraints hold THEN q in Q.
R2: IF q in Q AND q not in Q_obs THEN q in Q_free.
R3: IF a path is continuous and all its points lie in Q_free THEN it is a collision-free configuration path under the declared obstacle model.
R4: IF a system has n independent revolute joints and no coupling constraints THEN its configuration manifold is an n-torus.
R5: IF a coordinate transformation is a valid chart transition THEN distinct coordinate tuples may denote the same underlying configuration.
R6: IF a dynamical model includes momentum covectors THEN its classical Hamiltonian phase space can be represented by T*Q.

8. SEMANTIC DISTINCTIONS
ConfigurationSpace != Workspace: workspace describes physical positions or task outcomes, not every internal configuration.
ConfigurationSpace != StateSpace: state may include velocity, mode, temperature, memory or other variables.
ConfigurationSpace != PhaseSpace: phase space commonly includes conjugate momenta.
ConfigurationPath != DynamicalTrajectory: a geometric path need not satisfy timing, actuation or dynamics constraints.
DegreesOfFreedom != CoordinateCountGlobally: local dimension need not imply a single global coordinate system.

9. VALIDATION
V1: Check every configuration against declared constraints.
V2: Check dimensionality and independence assumptions at regular points.
V3: Check coordinate charts and transitions where manifold structure is asserted.
V4: Check path endpoints, continuity and membership in Q_free.
V5: Flag ambiguous system boundaries, obstacle models and collision geometry.
V6: Do not infer a metric, differentiability, connectedness or compactness without evidence.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: 1.0 for deductive consequences under explicitly satisfied assumptions; otherwise attach a calibrated estimate and evidence.
PROVENANCE: source URL, ontology version, mathematical assumptions, modelling context, derivation and validation timestamp.
SEMANTIC_EQUIVALENCE: compare underlying mathematical structure and constraints, not only labels or coordinate tuples.
FLOW_HASH: SHA-256 of canonical UTF-8 normalisation of ontology content excluding the FLOW_HASH value itself; compute on deployment.
UPDATE_MODEL: version changes when entity types, constraints, inference rules or semantics change; preserve migration mapping.

11. AI_CONSUMPTION
Input: system specification, coordinate model, constraints, optional geometry, start and goal configurations.
Process: type-check -> build Q -> validate constraints -> derive Q_free if geometry exists -> reason over paths and topology -> attach assumptions and confidence.
Output: typed configuration-space graph, feasible configurations, relationships, derivations, validation status and provenance.
Fail closed on inconsistent constraints; report uncertainty rather than inventing topology or reachability.

END FLOW ALGEBRA