Symbolic AI Mathematics Ontology

FLOW ALGEBRA
TITLE: Symbolic AI Mathematics Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Discrete AI Mathematics / Symbolic AI
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-ai-mathematics/symbolic-ai-mathematics-ontology.html
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Mathematical synthesis; source page content not independently verified.

1. PURPOSE
Model symbolic representations, deductive inference, unification, constraints, and planning with explicit types, semantics, proof obligations and provenance.

2. TYPES
Symbol := Identifier
Variable := Symbol
Constant := Symbol
FunctionSymbol := (name: Symbol, arity: Nat)
PredicateSymbol := (name: Symbol, arity: Nat)
Term := Variable | Constant | FunctionApplication(FunctionSymbol, List[Term])
Atom := PredicateApplication(PredicateSymbol, List[Term])
Formula := Atom | Not(Formula) | And(Formula,Formula) | Or(Formula,Formula) | Implies(Formula,Formula) | ForAll(Variable,Formula) | Exists(Variable,Formula)
Substitution := finite mapping Variable -> Term
Interpretation := (domain: nonempty Set, functions: Mapping, predicates: Mapping)
KnowledgeBase := finite Set[Formula]
Proof := finite sequence of justified inference steps
Query := Formula
Model := Interpretation satisfying a KnowledgeBase

3. WELL-FORMEDNESS
WF(FunctionApplication(f,args)) iff length(args)=f.arity and every arg is a well-formed Term.
WF(PredicateApplication(p,args)) iff length(args)=p.arity and every arg is a well-formed Term.
WF(ForAll(x,F)) iff x is a Variable and WF(F).
WF(Exists(x,F)) iff x is a Variable and WF(F).
WF(KB) iff every member is a well-formed Formula.
Substitution application must avoid variable capture; alpha-rename bound variables where necessary.

4. SEMANTICS
I,g |= P(t1,...,tn) iff (eval(I,g,t1),...,eval(I,g,tn)) belongs to I(P).
I,g |= Not(F) iff not (I,g |= F).
I,g |= And(F,G) iff (I,g |= F) and (I,g |= G).
I,g |= Implies(F,G) iff not (I,g |= F) or (I,g |= G).
I,g |= ForAll(x,F) iff every d in I.domain satisfies F under g[x:=d].
I,g |= Exists(x,F) iff some d in I.domain satisfies F under g[x:=d].
KB |= Q iff every model of KB satisfies Q.
KB |- Q denotes derivability under a specified proof calculus; distinguish syntax from semantic entailment.

5. TYPED RELATIONSHIPS
represents: Symbol -> Concept
has_argument: (FunctionApplication | PredicateApplication) -> Term
asserts: KnowledgeBase -> Formula
satisfies: Interpretation -> Formula
entails: KnowledgeBase -> Formula
unifies_with: Term x Term -> Bool
substitutes: Substitution x Variable -> Term
proves: Proof -> Formula
depends_on: InferenceStep -> Set[Formula]
refines: Theory -> Theory
contradicts: Formula x Formula -> Bool (relative to background theory)

6. INFERENCE OPERATORS
MODUS_PONENS: {P, P -> Q} |- Q.
CONJUNCTION_ELIMINATION: {P and Q} |- P; {P and Q} |- Q.
UNIVERSAL_INSTANTIATION: {forall x P(x)} |- P(t), for a term t free for x.
RESOLUTION: {A or C, not A or D} |- C or D, with first-order unification where needed.
HORN_RULE: P1 and ... and Pn -> Q; if all Pi established, derive Q.
UNIFICATION: unify(s,t) returns a most general unifier when one exists; occurs check prevents cyclic first-order terms.
FORWARD_CHAINING: repeatedly fire sound rules to a fixed point under finite-domain, function-free Datalog assumptions.
BACKWARD_CHAINING: reduce a goal to rule premises, with loop handling and explicit search strategy.

7. EXECUTABLE FLOW
flow SymbolicReasoning(KB: KnowledgeBase, query: Query, calculus: ProofSystem):
  require WF(KB) and WF(query)
  normalised := alpha_normalise(KB union {query})
  index := build_predicate_index(normalised)
  agenda := initialise_goals_or_facts(index, query, calculus)
  while agenda not empty and resource_budget_available:
    step := select(agenda)
    result := apply_sound_rule(step, index, calculus)
    record(result, premises, rule_id, substitution)
    if verified_proof_of(query): return PROVED(proof, provenance)
    if verified_refutation_of(query): return REFUTED(proof, provenance)
  return UNKNOWN(reason="incomplete search, resource bound, or undecided")
NOTE: Failure to prove is not proof of falsity under open-world semantics.
NOTE: A SAT/SMT UNSAT result can establish unsatisfiability only with a trusted solver or checked certificate.

8. KNOWLEDGE REPRESENTATION
FACT: Human(socrates).
RULE: forall x (Human(x) -> Mortal(x)).
QUERY: Mortal(socrates).
DERIVATION: instantiate rule at socrates; apply modus ponens.
RESULT: KB |- Mortal(socrates).
PROVENANCE: {Human(socrates), forall x (Human(x) -> Mortal(x))}.

9. CONSTRAINTS AND SEARCH
CSP := (Variables, Domains, Constraints).
Solution(theta) iff each variable receives a domain value and all constraints hold.
SAT(phi) iff there exists a Boolean assignment satisfying phi.
SMT(phi,T) iff phi is satisfiable in background theory T.
Search may use backtracking, propagation, resolution, DPLL or CDCL, subject to each algorithm's preconditions.

10. SYMBOLIC PLANNING
PlanningState := Set[GroundAtom]
Action := (preconditions: Set[Literal], add: Set[GroundAtom], delete: Set[GroundAtom])
Applicable(a,s) iff all positive preconditions hold in s and all negative preconditions are absent under the chosen closed-world convention.
Transition(s,a) := (s minus a.delete) union a.add, if Applicable(a,s).
Plan := List[Action]
ValidPlan(plan,s0,goal) iff every transition is defined and final state satisfies goal.

11. COMPUTATIONAL BOUNDARIES
Propositional SAT is NP-complete; propositional entailment is coNP-complete.
First-order validity is semi-decidable in general and not decidable.
Finite function-free Datalog evaluation terminates over a finite extensional database.
Complexity statements must identify representation, assumptions, and decision problem.

12. SEMANTIC EQUIVALENCE
Formula equivalence: F == G iff for every interpretation I and assignment g, I,g |= F iff I,g |= G.
Alpha-equivalence: bound-variable renaming preserves meaning.
Theory-relative equivalence: T |= (F <-> G).
Do not conflate syntactic identity, logical equivalence, and approximate semantic similarity.

13. CONFIDENCE
CONFIDENCE.proven := 1.0 only for a verified derivation under declared sound rules and assumptions.
CONFIDENCE.heuristic := calibrated score in [0,1], not a logical truth value.
CONFIDENCE.unknown := null when evidence or calibration is absent.
Proof validity and empirical reliability are separate attributes.

14. VALIDATION
CHECK_01: all symbols and arities well-typed.
CHECK_02: free and bound variables correctly scoped.
CHECK_03: substitutions capture-avoiding.
CHECK_04: inference steps cite sound rules and premises.
CHECK_05: proof checker independently verifies derived conclusion.
CHECK_06: contradictory KB handled without silently treating explosion as practical query answering.
CHECK_07: open-world versus closed-world assumptions explicit.
CHECK_08: solver timeouts reported as UNKNOWN, not UNSAT.
CHECK_09: symbolic plan preconditions and transitions validated.
CHECK_10: provenance and version recorded.

15. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical UTF-8 serialization of ontology content excluding FLOW_HASH.value.
FLOW_HASH.value: COMPUTE_ON_PUBLICATION (not fabricated).

16. UPDATE_MODEL
UPDATE_MODEL.version: 1.0
UPDATE_MODEL.change_policy: semantic versioning
UPDATE_MODEL.revalidate_on: rule change | schema change | semantic change | new evidence
UPDATE_MODEL.provenance: source URL, editor, timestamp, revision and verification status.

17. AI_CONSUMPTION
AI_CONSUMPTION.format: plain-text typed Flow Algebra
AI_CONSUMPTION.use_cases: knowledge graphs | expert systems | formal reasoning | planning | proof checking | constraint solving
AI_CONSUMPTION.requirements: preserve type declarations, assumptions, uncertainty and proof provenance.
AI_CONSUMPTION.safety: do not promote heuristic conclusions to verified theorems.

END FLOW ALGEBRA