Symbolic Algebra Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Symbolic Algebra Ontology within the Kireth Research Semantic Library. Symbolic algebra concerns the exact representation and transformation of algebraic expressions, equations, identities, polynomials, rational functions and algebraic structures using formal symbols and algebraic rules. This ontology supports expansion, factorisation, collection, substitution, simplification, polynomial arithmetic, rational-expression manipulation, Gröbner-basis relationships, exact equation solving, canonical forms, assumptions, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SymbolicAlgebra
 ENTITY_CLASS = ExactAlgebraicManipulationFramework
 PRIMARY_CONTEXT = SymbolicMathematics
 DEFINITION = "Symbolic algebra is the formal and exact manipulation of algebraic expressions and structures using symbols, algebraic identities, rewrite rules and exact arithmetic."
 SYMBOLIC_FORMS = [
  "expand((a+b)^2)=a^2+2ab+b^2",
  "factor(x^2-1)=(x-1)(x+1)",
  "collect(ax+bx,x)=(a+b)x",
  "substitute(f(x),x,c)=f(c)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for algebraic expressions, exact transformations, polynomial and rational manipulation, factorisation, expansion, substitution, canonicalisation and algebraic verification."
}

SCOPE {
 INCLUDE = [
  SymbolicAlgebra,AlgebraicExpression,SymbolicExpression,
  Variable,Constant,Parameter,Coefficient,
  Term,Monomial,Polynomial,
  MultivariatePolynomial,
  PolynomialRing,
  RationalExpression,RationalFunction,
  Fraction,ExactRational,
  AlgebraicNumber,
  AlgebraicIdentity,
  AlgebraicEquation,
  AlgebraicInequality,
  SystemOfEquations,
  LinearEquation,QuadraticEquation,PolynomialEquation,
  NonlinearEquation,
  SymbolicSystem,
  ExpressionTree,AbstractSyntaxTree,AST,
  Operator,Operand,
  Addition,Subtraction,Multiplication,Division,
  Exponentiation,Power,
  Negation,
  Associativity,Commutativity,Distributivity,
  AdditiveIdentity,MultiplicativeIdentity,
  AdditiveInverse,MultiplicativeInverse,
  Expansion,AlgebraicExpansion,
  BinomialExpansion,
  MultinomialExpansionRelationship,
  Factorisation,AlgebraicFactorisation,
  PolynomialFactorisation,
  DifferenceOfSquares,
  PerfectSquareFactorisation,
  GroupingFactorisationRelationship,
  CollectTerms,Collection,
  CombineLikeTerms,
  CoefficientCollection,
  Cancellation,
  CommonFactorExtraction,
  Simplification,AlgebraicSimplification,
  RationalSimplification,
  RadicalSimplificationRelationship,
  PowerSimplification,
  Substitution,SymbolicSubstitution,
  Replacement,PatternMatching,
  RewriteRule,RewriteSystem,TermRewriting,
  CanonicalForm,NormalForm,
  Canonicalisation,Normalisation,
  PolynomialNormalForm,
  RationalNormalForm,
  ExpandedForm,FactoredForm,
  HornerForm,
  LexicographicOrder,
  GradedLexicographicOrder,
  MonomialOrdering,
  LeadingTerm,LeadingCoefficient,
  Degree,TotalDegree,
  PolynomialAddition,
  PolynomialSubtraction,
  PolynomialMultiplication,
  PolynomialDivision,
  EuclideanDivision,
  SyntheticDivisionRelationship,
  PolynomialGCD,
  ExtendedPolynomialGCDRelationship,
  SquareFreeFactorisationRelationship,
  Content,PrimitivePartRelationship,
  Resultant,
  Discriminant,
  SubresultantRelationship,
  GroebnerBasis,GröbnerBasis,
  BuchbergerAlgorithm,
  SPolynomialRelationship,
  IdealRelationship,
  PolynomialIdeal,
  IdealMembership,
  Elimination,
  EliminationIdealRelationship,
  AlgebraicVarietyRelationship,
  RationalFunctionAddition,
  RationalFunctionMultiplication,
  RationalFunctionSimplification,
  PartialFractionDecomposition,
  CommonDenominator,
  Numerator,Denominator,
  LowestTerms,
  AlgebraicFraction,
  SymbolicEquationSolving,
  SolveByFactorisation,
  SolveBySubstitution,
  SolveByElimination,
  PolynomialRoot,
  ExactRoot,
  RootMultiplicity,
  SolutionSet,
  ConditionalSolution,
  ImplicitSolution,
  InequalitySolving,
  IntervalSolutionRelationship,
  Domain,AlgebraicDomain,
  IntegerDomain,RationalDomain,RealDomain,ComplexDomain,
  Field,IntegralDomainRelationship,
  Ring,CommutativeRingRelationship,
  Assumption,AssumptionContext,
  NonzeroAssumption,
  PositivityAssumption,
  RealAssumption,
  IntegerAssumption,
  BranchConditionRelationship,
  ExactArithmetic,
  ArbitraryPrecisionInteger,
  RationalArithmetic,
  AlgebraicNumberArithmetic,
  ModularArithmeticRelationship,
  SymbolicMatrixRelationship,
  MatrixExpressionRelationship,
  DeterminantSymbolicRelationship,
  SymbolicLinearAlgebraRelationship,
  FormalDerivativeRelationship,
  SymbolicDifferentiationRelationship,
  ComputerAlgebra,ComputerAlgebraSystem,CAS,
  SymbolicComputation,
  SymbolicMathematics,
  SymbolicManipulation,
  ExpressionComplexity,
  ExpressionSize,
  ExpressionSwell,
  IntermediateExpressionGrowth,
  Memoisation,
  CommonSubexpressionEliminationRelationship,
  StructuralEquality,
  SyntacticEquality,
  AlgebraicEquivalence,
  MathematicalEquivalence,
  IdentityCheck,
  SubstituteAndCheck,
  ExactResidual,
  RandomisedIdentityTestingRelationship,
  PolynomialIdentityTestingRelationship,
  Verification,
  ReferenceExpression,
  KnownIdentity,
  Benchmark,
  Reproducibility,
  Provenance,
  RuleSetVersion,
  CASVersion,
  AlgorithmVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  PureNumericalAlgebraWithoutSymbolicRepresentation,
  AbstractAlgebraWithoutComputationalSymbolManipulation,
  SymbolicCalculusWhenDifferentiationOrIntegrationIsPrimary,
  BooleanAlgebraWhenLogicalStructureRatherThanGeneralAlgebraicExpressionIsPrimary
 ]
 DISAMBIGUATION_RULE = "When algebraic expressions are represented and transformed exactly using symbolic identities, rewrite rules and algebraic algorithms, interpret the context as symbolic algebra."
}

CORE_CONCEPTS {
 SymbolicAlgebra = "Exact formal manipulation of algebraic expressions."
 AlgebraicExpression = "Expression composed of variables, constants and algebraic operators."
 Polynomial = "Finite sum of coefficient-weighted monomials."
 RationalExpression = "Quotient of algebraic expressions."
 Expansion = "Transformation from compact products or powers into distributed sums."
 Factorisation = "Transformation into a product of simpler algebraic factors."
 CanonicalForm = "Standard algebraic representation used for comparison or simplification."
 GroebnerBasis = "Generating set of a polynomial ideal adapted to a monomial ordering."
}

RELATIONS {
 SymbolicAlgebra --is_child_of--> SymbolicMathematics
 AlgebraicExpression --represented_by--> ExpressionTree
 Polynomial --is_type_of--> AlgebraicExpression
 RationalExpression --is_type_of--> AlgebraicExpression
 Expansion --preserves--> AlgebraicEquivalence
 Factorisation --preserves--> AlgebraicEquivalence
 Substitution --transforms--> AlgebraicExpression
 Simplification --uses--> AlgebraicIdentities
 PolynomialDivision --uses--> MonomialOrdering
 GroebnerBasis --represents--> PolynomialIdeal
 Resultant --relates--> PolynomialSystems
 SymbolicEquationSolving --uses--> AlgebraicTransformation
 Verification --checks--> AlgebraicResult
}

OPERATORS {
 EXPAND(expr) := distribute_products_and_powers(expr)
 FACTOR(expr) := factor_algebraic_expression(expr)
 COLLECT(expr,var) := collect_coefficients_by_power(expr,var)
 SIMPLIFY(expr,A) := algebraically_equivalent_reduced_form(expr,A)
 SUBSTITUTE(expr,var,value) := replace_symbol(expr,var,value)
 POLY_DIVIDE(f,g) := quotient_and_remainder(f,g)
 POLY_GCD(f,g) := greatest_common_polynomial_divisor(f,g)
 GROEBNER(F,order) := groebner_basis(F,order)
 SOLVE(F,vars) := exact_or_symbolic_solution_set(F,vars)
}

AXIOMS {
 A1 = "Symbolic algebra manipulates algebraic expressions exactly."
 A2 = "Algebraic transformations must preserve equivalence under declared domain assumptions."
 A3 = "Polynomial canonical forms depend on monomial ordering."
 A4 = "Factorisation depends on the coefficient domain."
 A5 = "Rational simplification requires valid nonzero-denominator conditions."
 A6 = "Equivalent expressions need not have identical syntax."
 A7 = "Intermediate expression growth can dominate symbolic-algebra cost."
 A8 = "Symbolic-algebra results require explicit domain, assumptions, transformation and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(expression_type,variables,coefficient_domain,assumptions,target_operation,exactness_goal)
 R2_PARSE = construct_expression_tree)
 R3_DOMAIN = determine(integer,rational,real,complex,polynomial_ring_or_other_algebraic_domain)
 R4_NORMALISE = canonicalise_terms_coefficients_and_operator_structure)
 R5_OPERATION = choose(expand,factor,collect,substitute,simplify,divide,GCD,Groebner_or_solve)
 R6_ALGORITHM = select_domain_and_structure_appropriate_algorithm)
 R7_CONDITIONS = preserve_nonzero_domain_and_branch_conditions)
 R8_COMPLEXITY = monitor_expression_swell_degree_growth_and_intermediate_terms)
 R9_VERIFY = check_by_expansion_substitution_identity_or_exact_residual)
 R10_REPRODUCE = document_domain_ordering_rules_CAS_and_versions)
}

CONSTRAINTS {
 C1 = "Factorisation can differ across integer, rational, real and complex domains."
 C2 = "Cancelling factors can be invalid at points where the cancelled denominator is zero."
 C3 = "Polynomial Gröbner-basis complexity can grow rapidly with variable count and degree."
 C4 = "Expanded form can be much larger than factored form."
 C5 = "Canonical form depends on monomial ordering and domain."
 C6 = "Symbolic equation systems can have conditional, implicit or parameterised solutions."
 C7 = "Radical simplification can require sign or branch assumptions."
 C8 = "Exact polynomial algorithms can generate large intermediate coefficients."
 C9 = "A syntactically shorter expression is not necessarily computationally cheaper."
 C10 = "Numeric approximation is not a substitute for exact algebraic equivalence."
}

SEMANTIC_EQUIVALENCE {
 "SymbolicAlgebra" <=> "ComputerAlgebraicManipulation"
 "ComputerAlgebraSystem" <=> "CAS"
 "AbstractSyntaxTree" <=> "AST"
 "GroebnerBasis" <=> "GröbnerBasis"
 "PolynomialGreatestCommonDivisor" <=> "PolynomialGCD"
}

VALIDATION {
 V1 = "Coefficient domain is documented."
 V2 = "Variables and assumptions are documented."
 V3 = "Expanded and factored forms are checked for equivalence."
 V4 = "Cancelled denominators preserve excluded-value conditions."
 V5 = "Polynomial division satisfies dividend = divisor*quotient + remainder."
 V6 = "GCD results divide all relevant polynomials."
 V7 = "Gröbner-basis results are checked against the selected ordering where feasible."
 V8 = "Equation solutions are substituted into original equations."
 V9 = "Exact output remains exact unless approximation is explicitly requested."
 V10 = "Results are reproducible under documented domain, ordering, CAS and algorithm versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_ALGEBRA_DEFINITION = 0.99
 EXPANSION_REASONING = 0.99
 FACTORISATION_REASONING = 0.99
 POLYNOMIAL_REASONING = 0.99
 RATIONAL_REASONING = 0.99
 GROEBNER_REASONING = 0.98
 EQUATION_SOLVING_REASONING = 0.98
 POLICY = "Reduce confidence when coefficient domain, assumptions, monomial ordering, denominator conditions or solver semantics are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SymbolicAlgebraReasoning,
  AlgebraicExpressionReasoning,ExpansionReasoning,
  FactorisationReasoning,CollectionReasoning,
  SubstitutionReasoning,SimplificationReasoning,
  PolynomialArithmeticReasoning,PolynomialGCDReasoning,
  RationalExpressionReasoning,GroebnerBasisReasoning,
  EliminationReasoning,EquationSolvingReasoning,
  CanonicalFormReasoning,AssumptionReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "symbolic algebra" -> SymbolicAlgebra
  "CAS" -> ComputerAlgebraSystem
  "AST" -> AbstractSyntaxTree
  "Groebner basis" -> GroebnerBasis
 }
}

EXECUTION_RULES {
 E1 = "If algebraic expressions are transformed exactly through symbolic algebraic rules, activate SymbolicAlgebraOntology."
 E2 = "Determine coefficient domain and assumptions before factorisation or simplification."
 E3 = "Canonicalise term ordering before comparison where appropriate."
 E4 = "Preserve denominator exclusions during rational simplification."
 E5 = "Choose polynomial algorithms according to degree, variable count and monomial ordering."
 E6 = "Monitor expression swell during expansion, elimination and Gröbner-basis computation."
 E7 = "Preserve exact arithmetic unless numerical approximation is requested."
 E8 = "Verify transformations through expansion, substitution or exact identity testing."
 E9 = "Document domain, monomial ordering, CAS and algorithm versions for reproducibility."
}

RECURSION {
 ENTRY = SymbolicAlgebra
 EXPAND(SymbolicAlgebra) -> [
  AlgebraicExpression,Polynomial,Monomial,RationalExpression,
  AlgebraicIdentity,Equation,Inequality,SystemOfEquations,
  ExpressionTree,Expansion,Factorisation,CollectTerms,
  CombineLikeTerms,Cancellation,Simplification,Substitution,
  PatternMatching,RewriteRule,CanonicalForm,NormalForm,
  PolynomialAddition,PolynomialMultiplication,PolynomialDivision,
  PolynomialGCD,Resultant,Discriminant,GroebnerBasis,
  BuchbergerAlgorithm,MonomialOrdering,PolynomialIdeal,
  RationalFunction,PartialFractionDecomposition,
  SymbolicEquationSolving,ExactRoot,SolutionSet,
  IntegerDomain,RationalDomain,RealDomain,ComplexDomain,
  Assumption,ExactArithmetic,ExpressionSwell,
  ComputerAlgebra,CAS,Verification,IdentityCheck,
  Reproducibility,Provenance
 ]
 EXPAND(SymbolicAlgebraFamilies) -> [
  ExpressionSimplificationAndCanonicalisation,
  ExpansionAndFactorisation,
  PolynomialArithmetic,
  RationalExpressionComputation,
  GroebnerBasesAndElimination,
  ExactEquationSolving,
  AlgebraicDomainsAndAssumptions,
  VerificationAndComputerAlgebra
 ]
 STOP_CONDITION = "Stop when domain, expression representation, algebraic operation, assumptions, canonical form and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Symbolic Mathematics Ontology"
 RELATED_ONTOLOGIES = [
  "Symbolic Mathematics Ontology",
  "Symbolic Computation",
  "Algebra Ontologies",
  "Computational Algebra Ontology",
  "Symbolic Manipulation Ontology",
  "Computer Algebra Systems Ontology",
  "Polynomial Ontology",
  "Rational Expression Ontology",
  "Equation Ontology",
  "Factorisation Ontology",
  "Expansion Ontology",
  "Simplification Ontology",
  "Substitution Ontology",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Symbolic Algebra -> Symbolic Mathematics for parent semantics",
  "Symbolic Algebra -> Symbolic Computation for general symbolic execution semantics",
  "Symbolic Algebra -> Algebra Ontologies for algebraic foundations",
  "Symbolic Algebra -> Computational Algebra for exact algebraic algorithms",
  "Symbolic Algebra -> Symbolic Manipulation for rewrite operations",
  "Symbolic Algebra -> Computer Algebra Systems for implementation semantics",
  "Symbolic Algebra -> Polynomial, Rational Expression and Equation ontologies for object-specific semantics",
  "Symbolic Algebra -> Factorisation, Expansion, Simplification and Substitution ontologies for transformation-specific semantics",
  "Symbolic Algebra -> Computational Complexity for expression-growth and algorithm-cost semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible algebraic algorithms, domains or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve expression-versus-canonical-form distinction.",
  "Preserve expanded-versus-factored-form distinction.",
  "Preserve polynomial-versus-rational-expression distinction.",
  "Preserve syntactic-equality-versus-algebraic-equivalence distinction.",
  "Preserve domain-independent-versus-domain-dependent factorisation distinction.",
  "Preserve exact-versus-approximate algebra distinction.",
  "Preserve machine-readable symbolic-algebra semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Symbolic algebra performs exact algebraic manipulation of expressions, equations, polynomials and rational functions."
 TRANSFORMATION_SUMMARY = "Expansion, factorisation, collection, cancellation, substitution and simplification provide core symbolic transformations."
 POLYNOMIAL_SUMMARY = "Polynomial arithmetic, GCDs, resultants and Gröbner bases support exact computational algebra."
 RATIONAL_SUMMARY = "Rational expressions require numerator-denominator normalisation and explicit excluded-value semantics."
 SOLVER_SUMMARY = "Symbolic algebraic solvers derive exact, conditional or implicit solutions through factorisation, substitution, elimination or polynomial methods."
 VALIDATION_SUMMARY = "Symbolic-algebra results should be checked through exact equivalence, substitution, polynomial identities, domain conditions and reproducible CAS settings."
}

FLOW {
 INPUT
 -> detect_symbolic_algebra_context
 -> identify_expression_variables_coefficient_domain_assumptions_target_operation_and_exactness_goal
 -> parse_expression_into_symbolic_tree
 -> canonicalise_terms_coefficients_and_operator_structure
 -> choose_expand_factor_collect_substitute_simplify_polynomial_rational_Groebner_elimination_or_solve_operation
 -> select_domain_appropriate_exact_algorithm
 -> preserve_nonzero_domain_and_branch_conditions
 -> monitor_expression_swell_degree_growth_and_intermediate_coefficients
 -> verify_by_expansion_substitution_identity_or_exact_residual
 -> document_domain_monomial_ordering_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-ALGEBRA-ONTOLOGY-V1"
}

}

END_FLOW