State Space Ontology

FLOW ALGEBRA
STATE 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/state-space-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13

1. PURPOSE
Represent a state space as a typed mathematical collection of possible system states, together with optional dynamics, observations, constraints, and uncertainty. Distinguish a state space from a coordinate representation and from a physical location.

2. TYPED ENTITIES
ENTITY StateSpace {id: IRI, carrier: Set<State>, system: System?, structure: Structure?, coordinates: CoordinateChart?, time_domain: TimeDomain?, provenance: Provenance}
ENTITY State {id: IRI, space: StateSpace, values: Tuple?, valid_at: Time?}
ENTITY System {id: IRI, name: String, model: Model?}
ENTITY StateVariable {id: IRI, domain: Set, unit: Unit?, role: String?}
ENTITY Transition {source: State, target: State, input: Input?, interval: Duration?}
ENTITY TransitionFunction {domain: StateSpace × InputSpace, codomain: StateSpace, map: Function}
ENTITY Trajectory {domain: TimeDomain, codomain: StateSpace, map: Function}
ENTITY Observation {state: State?, value: ObservationValue, time: Time?}
ENTITY ObservationFunction {domain: StateSpace, codomain: ObservationSpace, map: Function}
ENTITY Constraint {predicate: State -> Boolean}
ENTITY ReachableSet {initial: Set<State>, horizon: TimeDomain?, members: Set<State>}
ENTITY Equilibrium {state: State, dynamics: TransitionFunction, input: Input?}
ENTITY ProbabilityDistribution {support: StateSpace, measure: ProbabilityMeasure}
ENTITY CoordinateChart {domain: Set<State>, codomain: CoordinateSpace, map: InjectiveFunction}

3. FOUNDATIONAL AXIOMS
AXIOM S1: StateSpace(S) -> S.carrier is a set of admissible states.
AXIOM S2: State(x) and x.space = S -> x in S.carrier.
AXIOM S3: Transition(x,y) -> x.space = y.space unless an explicit inter-space mapping is supplied.
AXIOM S4: Deterministic discrete dynamics F: S × U -> S must map every declared admissible input pair to exactly one next state.
AXIOM S5: A trajectory x: T -> S satisfies x(t) in S for every t in its domain.
AXIOM S6: A coordinate chart represents states; it does not by itself define the identity of a state.
AXIOM S7: Topology, metric, vector operations and probability are optional structures, not universal requirements of a state space.
AXIOM S8: A valid state must satisfy every mandatory constraint declared for its model.

4. FLOW OPERATIONS
FLOW DefineStateSpace(system, variables, constraints):
  construct candidate product of variable domains
  filter by declared constraints
  return StateSpace(carrier=admissible_states, system=system)

FLOW ValidateState(state, space):
  require state.space = space
  require state in space.carrier
  require all mandatory constraints hold
  return VALID or structured violations

FLOW Step(state, input, transition_function):
  require state in transition_function.domain.state_space
  require input admissible
  next := transition_function.map(state,input)
  require next in transition_function.codomain.carrier
  return next

FLOW Evolve(initial_state, inputs, times, dynamics):
  validate initial_state
  for each consecutive step: apply dynamics and validate resulting state
  return Trajectory(times, states)

FLOW Observe(state, observation_function):
  require state in observation_function.domain.carrier
  return observation_function.map(state)

FLOW Reachability(initial_states, dynamics, horizon):
  begin with validated initial states
  expand successor states under admissible inputs within horizon
  return visited states with horizon and model provenance

5. SEMANTIC RELATIONSHIPS
RELATION has_state_space(System, StateSpace)
RELATION contains_state(StateSpace, State)
RELATION described_by(State, StateVariable)
RELATION constrained_by(StateSpace, Constraint)
RELATION transitions_to(State, State)
RELATION generated_by(Trajectory, TransitionFunction)
RELATION observed_through(StateSpace, ObservationFunction)
RELATION represented_in(StateSpace, CoordinateSpace)
RELATION equipped_with(StateSpace, Structure)
RELATION has_reachable_subset(StateSpace, ReachableSet)
RELATION semantically_related_to(StateSpace, PhaseSpace)

6. INFERENCE RULES
RULE R1: valid_state(x,S) AND valid_transition(x,y,F) -> candidate_reachable(y,from=x).
RULE R2: trajectory(x,T,S) AND t in T -> x(t) in S.
RULE R3: equilibrium(x,F,u) AND discrete_dynamics(F) -> F(x,u)=x.
RULE R4: equivalent_models(M1,M2,phi) AND state(x,M1) -> corresponding_state(phi(x),M2), subject to declared equivalence conditions.
RULE R5: metric_on(S,d) -> topological_structure(S,topology_induced_by(d)).
RULE R6: probability_measure_on(S,P) -> probabilistic_state_model(S,P).

7. IMPORTANT DISTINCTIONS
State space: all admissible states of a specified model.
Phase space: a state space commonly used in dynamical systems, often with position and momentum coordinates.
Configuration space: admissible configurations, which may omit velocity or momentum.
Vector space: a set with addition and scalar multiplication satisfying vector axioms; a general state space need not be one.
Metric space: a set equipped with a distance function satisfying metric axioms; a general state space need not have a metric.
Topological space: a set equipped with a topology; a general state space need not have one.
Observation space: possible measured outputs, potentially distinct from hidden internal states.

8. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE:
  exact: same carrier, state interpretation, dynamics, constraints and declared structures, modulo identifier renaming
  structural: bijective state mapping preserving declared transition and observation relationships
  contextual: alternative coordinate encodings with explicit invertible chart mappings on their applicable domains
  non_equivalent: same dimensionality alone does not establish equivalence

9. CONFIDENCE AND PROVENANCE
CONFIDENCE:
  value: Float in [0,1]
  basis: axiomatic | formally_verified | measured | inferred | heuristic
  scope: entity | relationship | rule | model
  note: confidence scores describe evidence quality, not mathematical truth of axioms
PROVENANCE:
  source_url: canonical page URL
  source_version: optional
  created_at: ISO8601
  validation_method: optional

10. VALIDATION
CHECK nonempty_or_explicitly_empty_carrier
CHECK state_membership_and_constraint_consistency
CHECK transition_domain_codomain_types
CHECK deterministic_function_totality_when_claimed
CHECK trajectory_time_domain_consistency
CHECK observation_function_codomain
CHECK chart_injectivity_where_required
CHECK metric_or_topology_axioms_only_when_declared
CHECK provenance_present

11. FLOW HASH / UPDATE MODEL
FLOW_HASH:
  algorithm: SHA-256
  input: canonical UTF-8 serialisation of ontology identifier, version, entity schemas, axioms, relations and rules
  output: computed by implementation; no unverified digest asserted
UPDATE_MODEL:
  strategy: versioned_semantic_diff
  detect: entity | axiom | relationship | constraint | dynamics | provenance changes
  invalidate: affected inferences, reachable sets and cached trajectories
  preserve: historical versions and source attribution

12. AI CONSUMPTION
AI_CONSUMPTION:
  representation: typed graph plus executable validation rules
  retrieval_keys: state_space, state, system, dynamics, trajectory, observation, reachability
  reasoning: type checking, constraint satisfaction, transition inference, model comparison
  guardrails: do not infer vector, metric, topological, or probabilistic structure without explicit declaration
  output: grounded entities, relationships, provenance, confidence and validation outcomes

END FLOW ALGEBRA