Event Ontology

FLOW ALGEBRA
TITLE: Event Ontology
DOMAIN: Discrete Mathematics / Discrete Probability
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-probability/event-ontology-2.html
VERSION: 1.0
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Mathematical synthesis; live source page not independently verified.

1. ONTOLOGY
ontology EventOntology {
  imports: SampleSpaceOntology, DiscreteProbabilityOntology, SetTheoryOntology
  concept Event: subset of a sample space representing outcomes satisfying a condition
  scope: finite or countable discrete probability spaces
}

2. TYPED ENTITIES
  SampleSpace Ω: Set[Outcome]
  Outcome ω: Element[Ω]
  Event A, B, C: Subset[Ω]
  ImpossibleEvent: EmptySet
  CertainEvent: Ω
  ProbabilityMeasure P: Event -> Real[0,1]
  EventIndicator 1_A: Ω -> {0,1}
  EventFamily F: Set[Event]
  EventPartition Π: Family[Event]

3. FORMAL DEFINITIONS
  is_event(A, Ω) := A ⊆ Ω
  occurs(A, ω) := ω ∈ A
  impossible_event(Ω) := ∅
  certain_event(Ω) := Ω
  complement(A) := Ω \ A
  union(A,B) := {ω ∈ Ω : ω ∈ A or ω ∈ B}
  intersection(A,B) := {ω ∈ Ω : ω ∈ A and ω ∈ B}
  difference(A,B) := A \ B
  mutually_exclusive(A,B) := A ∩ B = ∅
  exhaustive(F,Ω) := ⋃F = Ω
  partition(F,Ω) := exhaustive(F,Ω) and pairwise_disjoint(F) and all_nonempty(F)
  indicator(A,ω) := 1 if ω ∈ A else 0

4. PROBABILITY AXIOMS AND OPERATIONS
  P(Ω) = 1
  P(∅) = 0
  0 ≤ P(A) ≤ 1
  A ⊆ B => P(A) ≤ P(B)
  P(Aᶜ) = 1 - P(A)
  P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  A ∩ B = ∅ => P(A ∪ B) = P(A) + P(B)
  P(A \ B) = P(A) - P(A ∩ B)
  P(A | B) = P(A ∩ B)/P(B), provided P(B)>0
  independent(A,B) := P(A ∩ B) = P(A)P(B)
  for discrete Ω: P(A) = Σ_{ω∈A} P({ω})
  for pairwise disjoint countable A_i: P(⋃_i A_i) = Σ_i P(A_i)

5. ALGEBRAIC LAWS
  A ∪ B = B ∪ A
  A ∩ B = B ∩ A
  (A ∪ B) ∪ C = A ∪ (B ∪ C)
  (A ∩ B) ∩ C = A ∩ (B ∩ C)
  A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
  (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
  A ∪ Aᶜ = Ω
  A ∩ Aᶜ = ∅

6. SEMANTIC RELATIONSHIPS
  Event --is_subset_of--> SampleSpace
  Event --contains--> Outcome
  Event --has_probability--> ProbabilityValue
  Event --complement_of--> Event
  Event --mutually_exclusive_with--> Event [conditional]
  Event --independent_of--> Event [conditional]
  Event --combined_by--> Union / Intersection / Difference
  EventPartition --partitions--> SampleSpace
  EventIndicator --represents_membership_in--> Event

7. INFERENCE RULES
  R1: ω ∈ A => occurs(A,ω)
  R2: ω ∈ A and A ⊆ B => ω ∈ B
  R3: A ∩ B = ∅ => mutually_exclusive(A,B)
  R4: mutually_exclusive(A,B) => P(A∪B)=P(A)+P(B)
  R5: independent(A,B) and P(B)>0 => P(A|B)=P(A)
  R6: A ⊆ B => P(A) ≤ P(B)
  R7: P(A)=0 does NOT generally imply A=∅
  R8: P(A)=1 does NOT generally imply A=Ω
  R9: mutual exclusivity does NOT generally imply independence

8. EXECUTION FLOW
flow EventEvaluation(Ω, A, ω, P):
  require Ω is a valid discrete sample space
  require A ⊆ Ω
  require ω ∈ Ω
  require P is nonnegative and normalised over Ω
  membership := (ω ∈ A)
  probability := sum(P({x}) for x in A)
  complement_probability := 1 - probability
  return {membership, probability, complement_probability}

9. VALIDATION
  V1: Reject event with outcomes outside Ω.
  V2: Reject negative atom probabilities or nonnormalised distribution.
  V3: Verify union, intersection and complement remain subsets of Ω.
  V4: Verify P(A) in [0,1] with numerical tolerance when floating point.
  V5: Distinguish mutually exclusive events from independent events.
  V6: Permit conditioning only where conditioning event has positive probability.
  V7: Check countable sums for convergence and total probability 1.

10. SEMANTIC_EQUIVALENCE
  event = measurable subset of Ω (under the selected event sigma-algebra)
  impossible event ≡ ∅
  certain event ≡ Ω
  event complement ≡ not-event within Ω
  event union ≡ at least one event occurs
  event intersection ≡ both events occur
  Note: equivalent expressions need not denote identical event objects across distinct sample spaces.

11. CONFIDENCE
  mathematical_definitions: 1.00
  algebraic_laws: 1.00
  source_page_alignment: unverified
  confidence_policy: Do not infer empirical probabilities without evidence.

12. FLOW_HASH
  method: SHA-256 over canonical UTF-8 representation of the final published flow
  status: compute on publication; no hash asserted before canonicalisation

13. UPDATE_MODEL
  versioning: semantic
  triggers: corrected definitions, improved proof constraints, ontology linkage changes
  procedure: validate -> compare -> version -> canonicalise -> hash -> publish
  preserve: provenance, previous versions, mathematical invariants

14. AI_CONSUMPTION
  machine_readable: true
  reasoning_tasks: event classification, probability calculation, event algebra, consistency checks
  requirements: explicit sample space, event membership, distribution and assumptions
  limitations: probabilities must not be invented; independence must be proved or assumed explicitly
  output: typed events, computed probabilities, derivation, validation status, provenance

END FLOW ALGEBRA