Discrete AI Mathematics

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

FLOW ALGEBRA
DISCRETE AI MATHEMATICS — MAIN PAGE ONTOLOGY
Version: 1.0
Date: 2026-10-11
Source: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-ai-mathematics.html
Authors: Mark Higgins — Sky (Recursive AI)
Publisher: Kireth Research | https://kirethresearch.org
Status: Research ontology specification; conceptual formalism, not a verified executable implementation.

1. PURPOSE AND SCOPE
Discrete AI Mathematics formalises finite and countable mathematical structures supporting AI knowledge representation, search, inference, optimisation, learning and verification. This main-page ontology integrates discrete objects, operators, typed relations and auditable reasoning flows. It does not assert that all AI methods are discrete or that all problems are decidable.

2. TYPE SYSTEM
TYPE Entity := identifiable object
TYPE Set[T] := collection of elements of type T
TYPE FiniteSet[T] <: Set[T]
TYPE Relation[A,B] := subset of A × B
TYPE Function[A,B] := total single-valued mapping A → B
TYPE PartialFunction[A,B] := mapping defined on a subset of A
TYPE Proposition := formula with truth value under an interpretation
TYPE Graph[V,E] := vertices V with edges E and incidence relation
TYPE State := discrete configuration of an AI system
TYPE Action := state transition operator
TYPE Algorithm := finite description of a computational procedure
TYPE ComplexityBound := asymptotic resource bound
TYPE Proof := derivation under a stated deductive system
TYPE ProbabilityDistribution[Ω] := nonnegative mass function summing to 1 on countable Ω
TYPE KnowledgeBase := finite or recursively representable set of assertions
TYPE Model := interpretation satisfying selected assertions
TYPE Constraint := predicate restricting admissible assignments

3. ONTOLOGY ENTITIES
ENTITY DiscreteAIMathematics: Domain
ENTITY SetTheory: MathematicalDiscipline
ENTITY MathematicalLogic: MathematicalDiscipline
ENTITY Combinatorics: MathematicalDiscipline
ENTITY GraphTheory: MathematicalDiscipline
ENTITY RelationTheory: MathematicalDiscipline
ENTITY DiscreteProbability: MathematicalDiscipline
ENTITY AutomataTheory: MathematicalDiscipline
ENTITY FormalLanguageTheory: MathematicalDiscipline
ENTITY AlgorithmicMathematics: MathematicalDiscipline
ENTITY ComplexityTheory: MathematicalDiscipline
ENTITY KnowledgeRepresentation: AIDiscipline
ENTITY ConstraintSatisfaction: AIProblemClass
ENTITY DiscreteOptimisation: AIProblemClass
ENTITY SearchAndPlanning: AIDiscipline
ENTITY SymbolicReasoning: AIDiscipline
ENTITY AIModelVerification: AIDiscipline
NOTE: These are conceptual domain entities, not a claim about the exact child-page navigation on the source website.

4. TYPED RELATIONSHIPS
RELATION isSubdomainOf(Domain, Domain) -> Boolean
RELATION represents(Entity, MathematicalObject) -> Boolean
RELATION hasElement(Set[T], T) -> Boolean
RELATION relates(Relation[A,B], A, B) -> Boolean
RELATION satisfies(Model, Proposition) -> Boolean
RELATION entails(KnowledgeBase, Proposition) -> Boolean
RELATION transitions(Action, State, State) -> Boolean
RELATION constrains(Constraint, State) -> Boolean
RELATION hasComplexity(Algorithm, ComplexityBound) -> Boolean
RELATION verifiedBy(Assertion, Proof) -> Boolean
RELATION encodedBy(AIProblem, MathematicalStructure) -> Boolean
RELATION dependsOn(Concept, Concept) -> Boolean
RELATION semanticallyEquivalent(Concept, Concept, Context) -> Boolean

5. MATHEMATICAL FOUNDATIONS
SET: A ∪ B = {x | x ∈ A OR x ∈ B}
SET: A ∩ B = {x | x ∈ A AND x ∈ B}
SET: A \ B = {x | x ∈ A AND x ∉ B}
RELATION: R ⊆ A × B
FUNCTION: f: A → B; ∀a∈A, ∃!b∈B such that f(a)=b
GRAPH: G=(V,E), with E⊆V×V for directed simple graphs
LOGIC: Γ ⊨ φ iff every model satisfying Γ also satisfies φ
PROBABILITY: Σ_{ω∈Ω} P(ω)=1 for countable sample space Ω
SEARCH: successor(s,a)=s' where a is applicable to s
OPTIMISATION: minimise c(x) subject to x∈F

6. FLOW OPERATORS
OPERATOR DEFINE: symbol × type × constraints -> declared entity
OPERATOR LINK: entity × typed relation × entity -> graph assertion
OPERATOR NORMALISE: -expression × equivalence rules -> canonical candidate
OPERATOR INFER: knowledge base × rule -> derived assertion
OPERATOR CHECK: assertion × model or proof -> validation result
OPERATOR ENUMERATE: finite domain × predicate -> satisfying assignments
OPERATOR SEARCH: initial state × successors × goal -> path or failure
OPERATOR OPTIMISE: feasible assignments × objective -> optimum or bound
OPERATOR ANALYSE: algorithm × cost model -> complexity claim
OPERATOR EXPLAIN: conclusion × provenance -> explanation trace

7. CORE INFERENCE RULES
RULE ModusPonens:
  IF P AND (P -> Q) THEN Q
RULE RelationComposition:
  IF R(a,b) AND S(b,c) THEN (S∘R)(a,c)
RULE GraphReachability:
  IF Edge(u,v) THEN Reachable(u,v)
  IF Reachable(u,v) AND Edge(v,w) THEN Reachable(u,w)
RULE ConstraintFiltering:
  IF candidate x violates constraint C THEN reject x
RULE SearchGoal:
  IF state s is reachable from initial state AND Goal(s) THEN solutionCandidate(s)
RULE SoundInference:
  IF Γ ⊨ φ AND Γ is satisfied in model M THEN M ⊨ φ
RULE ComplexityComposition:
  IF sequential phases cost T1(n), T2(n), THEN T(n)=T1(n)+T2(n)
  PRECONDITION: phases share a specified input-size measure and cost model

8. DISCRETE AI REASONING PIPELINE
FLOW DiscreteAIReasoning(input, schema, rules, goal):
  1. parse input into typed entities and assertions
  2. validate types and domain constraints
  3. construct relation or knowledge graph
  4. normalise representations only under proven equivalence rules
  5. select logic, graph, search, constraint or optimisation procedure
  6. execute bounded reasoning with recorded assumptions
  7. test candidate conclusions or solutions
  8. record complexity, uncertainty and provenance
  9. emit result, explanation and validation status
END FLOW

9. WORKED EXAMPLE — GRAPH SEARCH
INPUT:
  V={A,B,C,D}
  E={(A,B),(B,C),(A,D)}
  start=A; goal=C
  graph directed and unweighted
METHOD: breadth-first search
TRACE:
  frontier=[A]
  expand A -> discover B,D
  expand B -> discover C
  goal C found; reconstruct A→B→C
OUTPUT: shortest path by edge count = [A,B,C], length=2
COMPLEXITY: O(|V|+|E|) time and O(|V|) auxiliary space under adjacency-list representation

10. WORKED EXAMPLE — PROPOSITIONAL CONSTRAINTS
VARIABLES: p,q ∈ {false,true}
CONSTRAINTS: (p OR q) AND (NOT p OR q)
DERIVATION: -expression is equivalent to q
SATISFYING ASSIGNMENTS: (false,true), (true,true)
VALIDATION: truth-table evaluation over all four assignments

11. CORRECTNESS AND SAFETY CONDITIONS
INVARIANT TypeSafety: every relation argument matches its declared type
INVARIANT Soundness: derived claims must follow from stated rules and assumptions
INVARIANT Traceability: each derived assertion links to premises and rule identifier
INVARIANT Scope: finite-domain enumeration claims require finite input domain
INVARIANT Complexity: cost bounds specify machine model and input representation
INVARIANT Probability: probabilities are normalised and nonnegative
INVARIANT Termination: bounded algorithms declare termination condition or timeout
INVARIANT Uncertainty: heuristics are not silently promoted to proofs

12. SEMANTIC EQUIVALENCE
EQUIVALENCE BooleanConjunction: P AND Q ≡ Q AND P
EQUIVALENCE Implication: P -> Q ≡ NOT P OR Q
EQUIVALENCE SetEquality: A=B iff ∀x(x∈A ↔ x∈B)
EQUIVALENCE GraphEncoding: two graph encodings are equivalent only if they preserve the graph structure relevant to the task
NON_EQUIVALENCE: reachability ≠ adjacency
NON_EQUIVALENCE: satisfiable ≠ valid
NON_EQUIVALENCE: algorithmic upper bound ≠ exact runtime

13. CONFIDENCE
CONFIDENCE formal_theorem: 1.0 only relative to a valid proof and explicit assumptions
CONFIDENCE computed_result: contingent on implementation and input validation
CONFIDENCE heuristic_result: calibrated or empirically estimated; never presumed certain
CONFIDENCE unsupported_claim: unknown

14. VALIDATION TESTS
TEST T01: union({1,2},{2,3}) = {1,2,3}
TEST T02: relation composition produces correctly typed pairs
TEST T03: graph BFS returns A→B→C for example graph
TEST T04: SAT example has exactly two satisfying assignments
TEST T05: ill-typed graph edge is rejected
TEST T06: an unreachable goal is reported without inventing a path
TEST T07: all output claims include premises or source provenance
TEST T08: no unproven complexity claim is labelled as a theorem

15. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical UTF-8 serialisation of the final flow text, excluding the FLOW_HASH.value field
FLOW_HASH.value: UNCOMPUTED
FLOW_HASH.note: compute only after final editorial approval; do not fabricate digest

16. UPDATE_MODEL
UPDATE_MODEL.version: 1.0
UPDATE_MODEL.strategy: semantic-versioned, provenance-preserving updates
UPDATE_MODEL.change_types: [definition, relation, rule, example, validation, metadata]
UPDATE_MODEL.on_change: revalidate dependencies; rerun tests; recompute hash
UPDATE_MODEL.compatibility: flag breaking changes to types, semantics or inference rules

17. AI_CONSUMPTION
AI_CONSUMPTION.format: plain-text typed Flow Algebra
AI_CONSUMPTION.language: en-GB
AI_CONSUMPTION.domain: discrete mathematics for artificial intelligence
AI_CONSUMPTION.tasks: [indexing, retrieval, graph construction, symbolic inference, algorithm selection, verification, teaching]
AI_CONSUMPTION.required_behaviour: preserve types, assumptions, provenance and uncertainty
AI_CONSUMPTION.prohibited_behaviour: fabricate proofs, source contents, complexity guarantees or hash values
AI_CONSUMPTION.crosslinks: [set theory, logic, graph theory, formal languages, algorithms, complexity, AI knowledge representation]

18. LIMITATIONS AND PROVENANCE
This is an original main-page synthesis based on the URL topic. The live page could not be fetched in this session; exact website headings and child-page links have not been verified. The flow is a conceptual research specification and should be reviewed before use as executable syntax or as a definitive inventory of the website.

END FLOW ALGEBRA