Symbolic Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Symbolic Mathematics Ontology within the Kireth Research Semantic Library. Symbolic mathematics concerns the formal representation and exact manipulation of mathematical objects, expressions, functions, equations and identities using symbols, algebraic rules and logical transformations rather than purely numerical approximation. This ontology supports symbolic expressions, expression trees, exact constants, variables, operators, canonical forms, simplification, substitution, algebraic transformation, symbolic calculus, equation solving, identities, assumptions, domains, exact arithmetic, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Symbolic Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Symbolic Computation / Symbolic Mathematics"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/symbolic-computation/symbolic-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SymbolicMathematics
 ENTITY_CLASS = FormalMathematicalSymbolRepresentationFramework
 PRIMARY_CONTEXT = SymbolicComputation
 DEFINITION = "Symbolic mathematics is the formal representation and exact transformation of mathematical objects using symbols, expressions, algebraic rules and logically valid equivalences."
 SYMBOLIC_FORMS = [
  "expr = op(arg1,...,argn)",
  "expr1 = expr2 under assumptions A",
  "simplify(expr) -> equivalent_form",
  "substitute(expr,x,a) -> expr[x:=a]"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for variables, constants, expressions, operators, assumptions, equivalence, exact arithmetic, symbolic transformations and validation."
}

SCOPE {
 INCLUDE = [
  SymbolicMathematics,SymbolicExpression,MathematicalExpression,
  FormalExpression,ExpressionTree,AbstractSyntaxTree,AST,
  Symbol,Variable,BoundVariable,FreeVariable,
  Constant,ExactConstant,Parameter,
  Coefficient,Literal,
  Operator,UnaryOperator,BinaryOperator,NaryOperator,
  Operand,Function,SymbolicFunction,
  FunctionApplication,
  Equation,Inequality,Identity,Relation,
  Formula,Constraint,
  AlgebraicExpression,RationalExpression,
  PolynomialExpression,TrigonometricExpression,
  ExponentialExpression,LogarithmicExpression,
  RadicalExpression,PiecewiseExpression,
  ConditionalExpression,
  SetExpressionRelationship,
  MatrixExpression,VectorExpression,TensorExpression,
  LogicExpression,BooleanExpression,
  Domain,MathematicalDomain,
  IntegerDomain,RationalDomain,RealDomain,ComplexDomain,
  PolynomialRingRelationship,FieldRelationship,
  Assumption,AssumptionContext,
  PositivityAssumption,NegativityAssumption,
  NonzeroAssumption,IntegerAssumption,
  RealAssumption,ComplexAssumption,
  OrderingAssumption,
  BranchCondition,PrincipalBranchRelationship,
  BranchCutRelationship,
  ExactArithmetic,ExactInteger,
  ArbitraryPrecisionInteger,BigInteger,
  RationalNumber,ExactRational,
  Fraction,AlgebraicNumber,
  IrrationalConstantRelationship,
  ImaginaryUnit,PiSymbol,EulerConstantRelationship,
  InfinitySymbolRelationship,
  StructuralEquality,SyntacticEquality,
  MathematicalEquality,MathematicalEquivalence,
  SemanticEquivalence,
  AlgebraicEquivalence,
  FunctionalEquivalence,
  IdentityEquivalence,
  CanonicalForm,NormalForm,
  Canonicalisation,Normalisation,
  SymbolicSimplification,
  AlgebraicSimplification,
  RationalSimplification,
  TrigonometricSimplification,
  LogarithmicSimplification,
  RadicalSimplification,
  ExpressionReduction,
  CombiningLikeTerms,
  CollectTerms,
  Expansion,AlgebraicExpansion,
  Factorisation,AlgebraicFactorisation,
  Cancellation,
  RationalisationRelationship,
  Substitution,SymbolicSubstitution,
  Replacement,PatternMatching,
  RewriteRule,RewriteSystem,
  TermRewriting,
  TransformationRule,
  AlgebraicTransformation,
  IdentityTransformation,
  DistributiveTransformation,
  AssociativeTransformation,
  CommutativeTransformation,
  PowerRuleTransformation,
  SignTransformation,
  SymbolicDifferentiation,
  Derivative,PartialDerivative,
  HigherDerivative,
  GradientSymbolicRelationship,
  JacobianSymbolicRelationship,
  HessianSymbolicRelationship,
  SymbolicIntegration,
  Integral,IndefiniteIntegral,DefiniteIntegral,
  SymbolicLimit,
  LimitExpression,
  SeriesExpansion,
  TaylorSeries,LaurentSeries,
  PowerSeries,
  SymbolicSummation,
  SymbolicProduct,
  SymbolicEquationSolving,
  LinearEquationSolving,
  PolynomialEquationSolving,
  NonlinearEquationSolving,
  SystemOfEquations,
  InequalitySolving,
  ExactSolution,
  ClosedFormSolution,
  ImplicitSolution,
  ConditionalSolution,
  SolutionSet,
  Root,ExactRoot,
  Multiplicity,
  Polynomial,Monomial,
  PolynomialCoefficient,
  PolynomialFactorisation,
  PolynomialDivision,
  PolynomialGCD,
  Resultant,
  Discriminant,
  GroebnerBasis,
  RationalFunction,
  PartialFractionDecomposition,
  SymbolicMatrix,
  SymbolicVector,
  SymbolicTensor,
  SymbolicLinearAlgebra,
  DeterminantSymbolic,
  TraceSymbolic,
  MatrixInverseSymbolic,
  SymbolicEigenvalueRelationship,
  SymbolicTransformRelationship,
  LaplaceTransformRelationship,
  FourierTransformRelationship,
  ZTransformRelationship,
  FormalProofRelationship,
  TheoremRelationship,
  LemmaRelationship,
  CorollaryRelationship,
  LogicalImplication,
  MathematicalStatement,
  PredicateRelationship,
  QuantifierRelationship,
  UniversalQuantifierRelationship,
  ExistentialQuantifierRelationship,
  ComputerAlgebra,ComputerAlgebraSystem,CAS,
  SymbolicComputation,
  SymbolicNumericComputation,
  NumericApproximation,
  ExactToNumericConversion,
  NumericToExactReconstruction,
  RationalReconstruction,
  IntegerRelationDetectionRelationship,
  PSLQRelationship,
  ExpressionComplexity,
  ExpressionSize,TreeDepth,
  ExpressionSwell,
  CommonSubexpression,
  CommonSubexpressionEliminationRelationship,
  Memoisation,
  HashConsingRelationship,
  DAGExpressionRepresentation,
  LazyEvaluation,
  EagerEvaluation,
  SymbolicEvaluation,
  Verification,IdentityCheck,
  SubstitutionCheck,
  SimplificationCheck,
  ExactResidual,
  ReferenceExpression,
  KnownIdentity,
  Reproducibility,
  Provenance,
  RuleSetVersion,
  CASVersion,
  AlgorithmVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  PureNumericalApproximationWithoutSymbolicRepresentation,
  NaturalLanguageMathematicsWithoutFormalSymbolicStructure,
  ProgrammingSyntaxWithoutMathematicalSemantics,
  FormalLogicWithoutMathematicalExpressionContext
 ]
 DISAMBIGUATION_RULE = "When mathematical meaning is represented through symbols and exact formal expressions that are transformed by mathematical rules, interpret the context as symbolic mathematics."
}

CORE_CONCEPTS {
 SymbolicMathematics = "Formal representation and transformation of mathematical meaning using symbols."
 SymbolicExpression = "Structured mathematical expression composed of symbols and operators."
 Symbol = "Atomic formal identifier representing a variable, constant or mathematical object."
 Operator = "Formal operation applied to one or more symbolic operands."
 Assumption = "Contextual condition restricting the domain or meaning of symbols."
 MathematicalEquivalence = "Semantic equality between expressions under declared assumptions."
 CanonicalForm = "Chosen standard representation of an equivalence class of expressions."
 ExactArithmetic = "Arithmetic preserving exact symbolic values."
}

RELATIONS {
 SymbolicMathematics --is_child_of--> SymbolicComputation
 SymbolicExpression --contains--> SymbolsAndOperators
 ExpressionTree --represents--> SymbolicExpression
 Symbol --can_represent--> VariableOrConstant
 Assumption --constrains--> SymbolicExpression
 RewriteRule --transforms--> SymbolicExpression
 Simplification --preserves--> MathematicalEquivalence
 Canonicalisation --produces--> CanonicalForm
 Substitution --replaces--> Symbol
 SymbolicDifferentiation --maps--> ExpressionToDerivativeExpression
 SymbolicIntegration --maps--> ExpressionToIntegralExpression
 SymbolicEquationSolving --maps--> EquationToSolutionSet
 ExactArithmetic --supports--> SymbolicMathematics
 Verification --checks--> SymbolicTransformation
}

OPERATORS {
 PARSE(text) := construct_symbolic_expression(text)
 SUBSTITUTE(expr,symbol,value) := replace_symbol(expr,symbol,value)
 SIMPLIFY(expr,assumptions) := equivalent_reduced_form(expr,assumptions)
 EXPAND(expr) := algebraic_expansion(expr)
 FACTOR(expr) := algebraic_factorisation(expr)
 DIFFERENTIATE(expr,x) := symbolic_derivative(expr,x)
 INTEGRATE(expr,x) := symbolic_integral(expr,x)
 SOLVE(expr_or_system,vars) := symbolic_solution_set(expr_or_system,vars)
 EQUIVALENT(a,b,A) := true_if_mathematically_equivalent_under_assumptions(a,b,A)
}

AXIOMS {
 A1 = "Symbolic mathematics represents mathematical meaning through formal symbols and expressions."
 A2 = "Syntactic equality and mathematical equivalence are distinct."
 A3 = "Symbolic transformations must preserve semantics under declared assumptions."
 A4 = "Canonical form depends on domain, representation and ordering rules."
 A5 = "Exact arithmetic preserves symbolic value independently of floating-point approximation."
 A6 = "Branch-sensitive expressions require explicit domain and branch semantics."
 A7 = "Equivalent expressions can have different structural complexity."
 A8 = "Symbolic-mathematics results require explicit domain, assumptions, representation, transformation and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(expression_type,symbols,domain,assumptions,target_operation,exactness_goal)
 R2_PARSE = construct_expression_tree_or_DAG)
 R3_DOMAIN = resolve(integer,rational,real,complex,polynomial,functional_or_other_domain)
 R4_ASSUMPTIONS = attach_domain_sign_nonzero_branch_and_ordering_conditions)
 R5_NORMALISE = canonicalise_structure_under_declared_domain)
 R6_TRANSFORM = choose(simplify,substitute,expand,factor,differentiate,integrate,solve_or_other_operation)
 R7_VERIFY = confirm_semantic_equivalence_or_solution_validity)
 R8_COMPLEXITY = monitor_expression_size_and_expression_swell)
 R9_OUTPUT = preserve_exact_form_or_convert_to_numeric_only_if_requested)
 R10_REPRODUCE = document_assumptions_rules_CAS_and_versions)
}

CONSTRAINTS {
 C1 = "Simplification can be invalid if required assumptions are omitted."
 C2 = "Complex branch cuts can invalidate naive algebraic transformations."
 C3 = "Canonical forms are domain dependent."
 C4 = "Not every symbolic integral has an elementary closed form."
 C5 = "Exact solutions can be implicit rather than explicit."
 C6 = "Expression swell can make exact transformations expensive."
 C7 = "Factorisation depends on the coefficient domain."
 C8 = "Piecewise expressions require branch-aware transformations."
 C9 = "Numeric approximation can conceal exact symbolic distinctions."
 C10 = "Syntactic simplicity does not guarantee semantic generality."
}

SEMANTIC_EQUIVALENCE {
 "SymbolicMathematics" <=> "FormalSymbolicMathematics"
 "ComputerAlgebraSystem" <=> "CAS"
 "AbstractSyntaxTree" <=> "AST"
 "MathematicalEquivalence" <=> "SemanticMathematicalEquivalence"
 "Canonicalisation" <=> "Canonicalization"
}

VALIDATION {
 V1 = "Expression domain is documented."
 V2 = "All relevant assumptions are explicit."
 V3 = "Parsing preserves intended operator precedence and structure."
 V4 = "Transformations preserve equivalence under assumptions."
 V5 = "Solutions are substituted into original equations where feasible."
 V6 = "Differentiation results are checked against standard identities or inverse operations where appropriate."
 V7 = "Integration results are checked by differentiation where feasible."
 V8 = "Exact output remains exact unless approximation is explicitly requested."
 V9 = "Branch-sensitive results document conditions or piecewise cases."
 V10 = "Results are reproducible under documented rule set, CAS and algorithm versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_MATHEMATICS_DEFINITION = 0.99
 EXPRESSION_REASONING = 0.99
 EQUIVALENCE_REASONING = 0.99
 SIMPLIFICATION_REASONING = 0.98
 SYMBOLIC_CALCULUS_REASONING = 0.98
 EQUATION_SOLVING_REASONING = 0.98
 ASSUMPTION_REASONING = 0.99
 POLICY = "Reduce confidence when domain, assumptions, branch conditions, expression parsing or transformation rules are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SymbolicMathematicsReasoning,
  SymbolReasoning,ExpressionReasoning,ExpressionTreeReasoning,
  AssumptionReasoning,EquivalenceReasoning,CanonicalFormReasoning,
  SimplificationReasoning,SubstitutionReasoning,ExpansionReasoning,
  FactorisationReasoning,SymbolicCalculusReasoning,
  EquationSolvingReasoning,ExactArithmeticReasoning,
  PolynomialReasoning,SymbolicLinearAlgebraReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "symbolic mathematics" -> SymbolicMathematics
  "symbolic math" -> SymbolicMathematics
  "CAS" -> ComputerAlgebraSystem
  "AST" -> AbstractSyntaxTree
  "canonicalization" -> Canonicalisation
 }
}

EXECUTION_RULES {
 E1 = "If mathematical meaning is represented and transformed as formal symbols, activate SymbolicMathematicsOntology."
 E2 = "Determine mathematical domain before simplification or equivalence testing."
 E3 = "Attach relevant assumptions before applying conditional transformations."
 E4 = "Preserve exact arithmetic by default."
 E5 = "Distinguish syntactic equality from mathematical equivalence."
 E6 = "Use canonical forms appropriate to the declared domain."
 E7 = "Verify symbolic solutions and transformations by substitution or inverse operations where feasible."
 E8 = "Monitor expression swell for computationally expensive exact transformations."
 E9 = "Document assumptions, rule sets and CAS versions for reproducibility."
}

RECURSION {
 ENTRY = SymbolicMathematics
 EXPAND(SymbolicMathematics) -> [
  SymbolicExpression,ExpressionTree,AST,Symbol,Variable,Constant,Parameter,
  Operator,Function,Equation,Inequality,Identity,Formula,
  AlgebraicExpression,RationalExpression,PolynomialExpression,
  TrigonometricExpression,PiecewiseExpression,Domain,Assumption,
  ExactArithmetic,ExactInteger,RationalNumber,AlgebraicNumber,
  MathematicalEquality,MathematicalEquivalence,CanonicalForm,
  Canonicalisation,Simplification,Expansion,Factorisation,
  Substitution,PatternMatching,RewriteRule,TermRewriting,
  SymbolicDifferentiation,SymbolicIntegration,SymbolicLimit,
  TaylorSeries,SymbolicSummation,SymbolicEquationSolving,
  ExactSolution,ConditionalSolution,Polynomial,GroebnerBasis,
  RationalFunction,SymbolicMatrix,SymbolicVector,SymbolicTensor,
  ComputerAlgebra,CAS,SymbolicNumericComputation,
  ExpressionSwell,Verification,IdentityCheck,Reproducibility,Provenance
 ]
 EXPAND(SymbolicMathematicsFamilies) -> [
  SymbolAndExpressionRepresentation,
  DomainsAssumptionsAndEquivalence,
  SimplificationAndRewriting,
  ExactArithmetic,
  SymbolicAlgebra,
  SymbolicCalculus,
  SymbolicEquationSolving,
  VerificationAndSymbolicNumericRelationships
 ]
 STOP_CONDITION = "Stop when representation, domain, assumptions, operation, equivalence, exactness and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Symbolic Computation"
 RELATED_ONTOLOGIES = [
  "Symbolic Computation",
  "Algebra Ontologies",
  "Computational Algebra Ontology",
  "Symbolic Manipulation Ontology",
  "Computer Algebra Systems Ontology",
  "Polynomial Ontology",
  "Equation Ontology",
  "Function Ontology",
  "Calculus and Analysis Ontologies",
  "Computational Matrix Ontology",
  "Tensor Computation Ontology",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Symbolic Mathematics -> Symbolic Computation for parent computational semantics",
  "Symbolic Mathematics -> Algebra Ontologies for symbolic algebraic foundations",
  "Symbolic Mathematics -> Computational Algebra for exact algebraic algorithms",
  "Symbolic Mathematics -> Symbolic Manipulation for rewrite semantics",
  "Symbolic Mathematics -> Computer Algebra Systems for software implementation semantics",
  "Symbolic Mathematics -> Polynomial, Equation and Function ontologies for object-specific reasoning",
  "Symbolic Mathematics -> Calculus and Analysis for symbolic calculus",
  "Symbolic Mathematics -> Computational Matrix and Tensor ontologies for structured symbolic objects",
  "Symbolic Mathematics -> Computational Complexity for expression-size and algorithm-cost semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible symbolic object classes, transformations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve symbol-versus-expression distinction.",
  "Preserve syntax-versus-mathematical-semantics distinction.",
  "Preserve exact-versus-approximate representation distinction.",
  "Preserve domain-free-versus-assumption-dependent transformation distinction.",
  "Preserve structural-equality-versus-mathematical-equivalence distinction.",
  "Preserve expression-versus-canonical-form distinction.",
  "Preserve machine-readable symbolic-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Symbolic mathematics represents and transforms mathematical objects exactly using formal symbols, expressions and semantic rules."
 REPRESENTATION_SUMMARY = "Symbols, operators and expression trees encode mathematical structure."
 ASSUMPTION_SUMMARY = "Domains and assumptions control the validity of simplification, equivalence and branch-sensitive transformations."
 ALGEBRA_SUMMARY = "Expansion, factorisation, substitution, polynomial manipulation and rewriting provide core symbolic algebra."
 CALCULUS_SUMMARY = "Differentiation, integration, limits, series and summation extend symbolic reasoning to analysis."
 VALIDATION_SUMMARY = "Symbolic mathematics should be checked through exact equivalence, substitution, inverse operations, branch review and reproducible rule settings."
}

FLOW {
 INPUT
 -> detect_symbolic_mathematics_context
 -> identify_expression_symbols_domain_assumptions_target_operation_and_exactness_goal
 -> parse_expression_into_tree_or_DAG
 -> resolve_domain_and_branch_conditions
 -> attach_assumptions
 -> canonicalise_structure_under_declared_domain
 -> apply_simplification_substitution_expansion_factorisation_calculus_solving_or_other_symbolic_operation
 -> verify_equivalence_solution_or_inverse_operation
 -> monitor_expression_size_and_swell
 -> preserve_exact_output_unless_approximation_is_requested
 -> document_assumptions_rule_set_CAS_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-SYMBOLIC-COMPUTATION-SYMBOLIC-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW