Discrete Mathematics Ontologies

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Discrete Mathematics Ontologies — Main Page
Version: 1.0
Source: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies.html
Authors: Mark Higgins — Sky (Recursive AI)
Generated: 2026-10-11
Status: Domain-level formalisation based on verified main-page section headings; child-page specifications not independently verified.

flow "DiscreteMathematicsOntologies" v1.0

DOMAIN:
  id: discrete_mathematics_ontologies
  type: MathematicalDomainCollection
  parent: ai_core
  purpose: Formal representation, validation and reasoning over finite and countable mathematical structures.

SECTIONS:
  - Foundations of Discrete Mathematics
  - Set Theory
  - Relations
  - Functions
  - Logic
  - Graph Theory
  - Combinatorics
  - Number Theory
  - Algebraic Structures
  - Discrete Probability
  - Coding Theory
  - Cryptography
  - Automata Theory
  - Formal Languages
  - Computational Complexity
  - Algorithmic Mathematics
  - Discrete AI Mathematics

TYPES:
  Set: {elements: Collection[Any]}
  Relation: {domain: Set, codomain: Set, pairs: Set[Tuple]}
  Function: {domain: Set, codomain: Set, mapping: Map}
  Proposition: {expression: LogicalExpression, truth_value: Boolean?}
  Graph: {vertices: Set, edges: Set, directed: Boolean}
  CombinatorialStructure: {objects: Set, constraints: Set}
  Integer: {value: Z}
  AlgebraicStructure: {carrier: Set, operations: Set, axioms: Set}
  DiscreteDistribution: {support: Set, probability_mass: Map}
  Code: {alphabet: Set, codewords: Set}
  CryptographicPrimitive: {algorithm: Identifier, assumptions: Set}
  Automaton: {states: Set, alphabet: Set, transition: Relation, initial: Any, accepting: Set}
  FormalLanguage: {alphabet: Set, strings: Set}
  ComplexityClass: {name: Identifier, resource_bound: Expression}
  Algorithm: {input_type: Type, output_type: Type, steps: Sequence, termination: Predicate}
  DiscreteAIModel: {representation: Any, reasoning_rules: Set}
  Proof: {premises: Set, conclusion: Proposition, derivation: Sequence}
  Evidence: {source: URI, timestamp: DateTime?, method: String}

RELATIONSHIPS:
  contains(Set, Any)
  subset_of(Set, Set)
  relates(Relation, Any, Any)
  maps_to(Function, Any, Any)
  implies(Proposition, Proposition)
  equivalent_to(Proposition, Proposition)
  connected_by(Graph, Any, Any)
  satisfies(Any, Predicate)
  derived_from(Any, Evidence)
  models(DiscreteAIModel, Any)
  recognised_by(FormalLanguage, Automaton)
  bounded_by(Algorithm, ComplexityClass)
  encoded_by(Any, Code)

INVARIANTS:
  - Every relation pair belongs to its declared domain × codomain.
  - Every function assigns exactly one codomain value to each domain element.
  - Every graph edge references existing vertices.
  - Every probability mass is non-negative and the masses sum to one (within declared tolerance).
  - Every automaton's initial and accepting states belong to its state set.
  - Inferences retain premises, rule identifiers and provenance.
  - Undefined, unproven and false are distinct states.

INFERENCE_RULES:
  R01_SUBSET_TRANSITIVITY:
    IF subset_of(A,B) AND subset_of(B,C) THEN subset_of(A,C).
  R02_RELATION_COMPOSITION:
    IF relates(R,a,b) AND relates(S,b,c) THEN relates(compose(R,S),a,c).
  R03_FUNCTION_COMPOSITION:
    IF maps_to(f,a,b) AND maps_to(g,b,c) THEN maps_to(compose(g,f),a,c).
  R04_MODUS_PONENS:
    IF true(P) AND implies(P,Q) THEN true(Q).
  R05_GRAPH_REACHABILITY:
    IF connected_by(G,a,b) AND reachable(G,b,c) THEN reachable(G,a,c).
  R06_PROBABILITY_COMPLEMENT:
    IF valid_probability(P,A) THEN probability(not(A)) = 1 - probability(A).
  R07_AUTOMATON_ACCEPTANCE:
    IF transition_star(M,initial,input,state) AND state IN accepting(M) THEN accepts(M,input).
  R08_COMPLEXITY_BOUND:
    IF proved_upper_bound(algorithm,resource) THEN bounded_by(algorithm,resource).
  R09_NO_UNSUPPORTED_EQUIVALENCE:
    IF semantic_similarity(A,B) THEN NOT automatically semantic_equivalence(A,B).
  R10_PROVENANCE_PROPAGATION:
    IF derived(conclusion,premises,rule) THEN attach_provenance(conclusion,premises,rule).

CONFIDENCE:
  range: [0.0, 1.0]
  rule: Confidence measures evidence or extraction reliability, not mathematical truth.
  verified_proof: "valid under explicit axioms and assumptions"
  uncertain_assertion: "retain confidence, source and uncertainty"
  no_fabrication: true

SEMANTIC_EQUIVALENCE:
  requires: [compatible_types, preserved_meaning, explicit_context, supporting_proof_or_mapping]
  distinguish: [identity, isomorphism, logical_equivalence, semantic_similarity]
  preserve_domain_assumptions: true

PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies.html
  source_scope: verified_main_page_headings
  record: [source, author, extraction_date, rule_id, assumptions, version]

FLOW_HASH:
  algorithm: SHA-256
  input: canonical_serialisation(domain,types,relationships,invariants,inference_rules,version)
  note: Calculate at ingestion; no digest is asserted here.

UPDATE_MODEL:
  strategy: versioned_incremental_updates
  actions: [validate_types, compare_versions, recompute_affected_inferences, retain_provenance, recompute_flow_hash]
  rollback: previous_valid_version

AI_CONSUMPTION:
  format: typed_semantic_flow
  modes: [ontology_navigation, symbolic_reasoning, proof_assistance, knowledge_graph_integration, retrieval]
  requirements: [type_checking, rule_validation, provenance_tracking, uncertainty_reporting]
  restrictions: [no_unproven_claims_as_theorems, no_implicit_axiom_changes, no_unjustified_cross_domain_equivalence]

END FLOW ALGEBRA