Random Event Ontology

FLOW ALGEBRA
RANDOM EVENT ONTOLOGY
Kireth Semantic Library | Entity Graph | Mathematical Entities | Probability Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/random-event-ontology.html
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent random events as measurable subsets of a sample space within a probability model, with explicit typing, mathematical constraints, semantic relations and machine-consumable inference.

2. ENTITY TYPES
RandomExperiment: procedure with uncertain outcome.
SampleSpace: set Omega of possible elementary outcomes.
Outcome: element omega in Omega.
SigmaAlgebra: collection F of measurable subsets of Omega.
RandomEvent: measurable subset A in F.
ProbabilityMeasure: P:F -> [0,1] satisfying probability axioms.
ProbabilitySpace: triple (Omega,F,P).
EventOccurrence: assertion that observed omega belongs to A.
CompositeEvent: union, intersection or complement of events.
ConditionalEvent: event evaluated under condition B with P(B)>0.

3. CORE SIGNATURE
flow "RandomEventOntology" v1.0
entity RandomEvent {
  id: URI
  sample_space: SampleSpace
  probability_space: ProbabilitySpace?
  outcome_set: Set[Outcome]
  measurable: Boolean
  event_kind: Enum[atomic,composite,certain,impossible,conditional]
  provenance: URI?
  confidence: Real[0,1]?
}
relation belongs_to(RandomEvent,SampleSpace)
relation measurable_in(RandomEvent,SigmaAlgebra)
relation occurs_when(RandomEvent,Outcome)
relation event_union(RandomEvent,RandomEvent,RandomEvent)
relation event_intersection(RandomEvent,RandomEvent,RandomEvent)
relation event_complement(RandomEvent,RandomEvent)
relation mutually_exclusive(RandomEvent,RandomEvent)
relation independent_under(RandomEvent,RandomEvent,ProbabilityMeasure)

4. AXIOMS AND INVARIANTS
A1: Omega is a nonempty set.
A2: F is a sigma-algebra over Omega.
A3: Every valid event A is in F, hence A subseteq Omega.
A4: P(Omega)=1 and P(empty_set)=0.
A5: For pairwise disjoint events A_i, P(union_i A_i)=sum_i P(A_i).
A6: 0 <= P(A) <= 1 for every A in F.
A7: Complement A^c=Omega\A is in F; P(A^c)=1-P(A).
A8: A union B and A intersection B are in F.
A9: An event occurs in a realised trial exactly when omega in A.
A10: P(A union B)=P(A)+P(B)-P(A intersection B).
A11: A and B are mutually exclusive iff A intersection B=empty_set.
A12: A and B are independent under P iff P(A intersection B)=P(A)P(B).
A13: P(A|B)=P(A intersection B)/P(B), only when P(B)>0.

5. TYPED RELATIONSHIPS
RandomExperiment -> has_sample_space -> SampleSpace
SampleSpace -> contains -> Outcome
ProbabilitySpace -> contains -> SampleSpace
ProbabilitySpace -> contains -> SigmaAlgebra
ProbabilitySpace -> contains -> ProbabilityMeasure
RandomEvent -> subset_of -> SampleSpace
RandomEvent -> member_of -> SigmaAlgebra
RandomEvent -> contains_outcome -> Outcome
RandomEvent -> has_probability_under -> ProbabilityMeasure
CompositeEvent -> derived_from -> RandomEvent
EventOccurrence -> observes -> Outcome
EventOccurrence -> satisfies -> RandomEvent

6. INFERENCE RULES
R1: omega in A AND A in F -> event_occurs(A,omega).
R2: A subseteq B AND event_occurs(A,omega) -> event_occurs(B,omega).
R3: A intersection B=empty_set -> mutually_exclusive(A,B).
R4: mutually_exclusive(A,B) -> P(A union B)=P(A)+P(B).
R5: independent_under(A,B,P) -> P(A intersection B)=P(A)P(B).
R6: A subseteq B -> P(A)<=P(B).
R7: A=Omega -> certain_event(A); A=empty_set -> impossible_event(A).
R8: P(A)=1 does NOT generally imply A=Omega; P(A)=0 does NOT generally imply A=empty_set.
R9: Independence and mutual exclusivity are distinct: nonzero-probability mutually exclusive events cannot be independent.

7. SEMANTIC DISTINCTIONS
RandomEvent != Outcome: an event is a measurable set; an outcome is an element.
RandomEvent != EventOccurrence: the latter is a realised membership fact.
RandomEvent != RandomVariable: a random variable is a measurable function on Omega.
Probability != Confidence: probability is defined relative to P; confidence metadata expresses epistemic reliability.
Independent != MutuallyExclusive: these encode different relations.
An arbitrary subset of an uncountable sample space need not be measurable.

8. EXAMPLE
Experiment: one fair six-sided die roll.
Omega={1,2,3,4,5,6}; F=power_set(Omega).
A={2,4,6} (even); B={4,5,6} (greater than three).
P(A)=1/2; P(B)=1/2; P(A intersection B)=2/6=1/3.
P(A union B)=1/2+1/2-1/3=2/3.
A and B are neither disjoint nor independent because 1/3 != 1/4.

9. VALIDATION
V1: Require event outcome_set subseteq Omega.
V2: Require event outcome_set in F before applying P.
V3: Reject probabilities outside [0,1].
V4: Reject conditional probabilities when conditioning event has probability zero.
V5: Check all event operations use a compatible probability space.
V6: Distinguish a mathematical probability value from an uncertain estimated value.
V7: Check countable additivity where applicable.
V8: Preserve source provenance and flag unresolved semantic mappings.

10. SEMANTIC_EQUIVALENCE
Equivalent terms: probabilistic event; measurable event (within a fixed probability space).
Contextual aliases: event A; outcome condition.
Non-equivalences: elementary outcome, random variable, observation, deterministic occurrence.
Equivalence must preserve sample space, sigma-algebra and interpretation.

11. FLOW_HASH AND UPDATE_MODEL
FLOW_HASH: SHA-256 of canonical UTF-8 serialisation of the ontology record, excluding the FLOW_HASH field itself; sort map keys and normalise line endings before hashing.
UPDATE_MODEL: versioned amendments with change log, source URL, timestamp, compatibility assessment and hash regeneration.
CONFIDENCE: [0,1] metadata for evidence quality, not a replacement for P(A).

12. AI_CONSUMPTION
Input: typed probability space, event definitions, optional observation and query.
Process: validate -> resolve entities -> check measurability -> apply sound inference -> attach provenance.
Output: event relations, derived probability statements, rule identifiers, confidence metadata, validation status and FLOW_HASH.
Guardrails: never infer independence from disjointness; never equate probability zero with impossibility; do not compute probabilities without a defined measure.

END FLOW ALGEBRA