Symbolic Equation Solver Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Symbolic Equation Solver Ontology within the Kireth Research Semantic Library. A symbolic equation solver derives exact, conditional, implicit, parameterised or algebraically equivalent solution sets for equations and systems represented as formal mathematical expressions. This ontology supports linear, polynomial, rational, transcendental and nonlinear equations, systems of equations, elimination, substitution, factorisation, resultants, Gröbner-basis relationships, inequalities, assumptions, branch conditions, solution verification, completeness, complexity and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SymbolicEquationSolver
 ENTITY_CLASS = ExactEquationSolvingFramework
 PRIMARY_CONTEXT = SymbolicComputation
 DEFINITION = "A symbolic equation solver is an algorithmic system that derives formal solution sets for equations or systems using exact algebraic transformations, symbolic algorithms and declared assumptions."
 SYMBOLIC_FORMS = [
  "solve(f(x)=0,x) -> solution_set",
  "solve({f1=0,...,fn=0},{x1,...,xm})",
  "solution satisfies original_equations",
  "conditions -> branch_specific_solution"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for equation classes, variables, domains, exact solution sets, elimination, substitution, factorisation, conditional solutions, verification and solver completeness."
}

SCOPE {
 INCLUDE = [
  SymbolicEquationSolver,EquationSolver,SymbolicSolver,
  SymbolicEquationSolving,
  Equation,AlgebraicEquation,FunctionalEquationRelationship,
  LinearEquation,QuadraticEquation,CubicEquation,QuarticEquation,
  PolynomialEquation,RationalEquation,
  RadicalEquation,ExponentialEquation,
  LogarithmicEquation,TrigonometricEquation,
  HyperbolicEquationRelationship,
  TranscendentalEquation,
  NonlinearEquation,
  AbsoluteValueEquation,
  PiecewiseEquation,
  ImplicitEquation,
  ParametricEquationRelationship,
  DifferentialEquationRelationship,
  SystemOfEquations,
  LinearSystem,PolynomialSystem,
  NonlinearSystem,
  MixedEquationSystem,
  UnderdeterminedSystem,
  OverdeterminedSystem,
  DeterminedSystem,
  ConsistentSystem,InconsistentSystem,
  Variable,Unknown,DependentVariable,
  Parameter,Constant,Coefficient,
  Solution,SolutionSet,
  ExactSolution,ClosedFormSolution,
  ImplicitSolution,ConditionalSolution,
  ParameterisedSolution,
  PrincipalSolutionRelationship,
  GeneralSolutionRelationship,
  ExtraneousSolution,
  MissingSolutionRelationship,
  MultipleSolution,UniqueSolution,
  NoSolution,InfiniteSolutionSet,
  Root,Zero,
  RootMultiplicity,
  RepeatedRoot,
  RealRoot,ComplexRoot,
  AlgebraicRoot,TranscendentalRootRelationship,
  Domain,EquationDomain,
  IntegerDomain,RationalDomain,RealDomain,ComplexDomain,
  AlgebraicNumberDomainRelationship,
  Assumption,AssumptionContext,
  NonzeroAssumption,PositiveAssumption,
  RealAssumption,IntegerAssumption,
  ParameterCondition,
  BranchCondition,
  PrincipalBranch,
  ExcludedValue,
  DenominatorRestriction,
  DomainRestriction,
  SingularPoint,
  Factorisation,
  SolveByFactorisation,
  ZeroProductRule,
  Substitution,
  SolveBySubstitution,
  Elimination,
  SolveByElimination,
  GaussianEliminationRelationship,
  RowReductionRelationship,
  Resultant,
  ResultantMethod,
  Discriminant,
  PolynomialDivision,
  PolynomialGCD,
  GroebnerBasis,GröbnerBasis,
  BuchbergerAlgorithm,
  EliminationIdealRelationship,
  LexicographicOrder,
  MonomialOrdering,
  IdealMembershipRelationship,
  TriangularDecompositionRelationship,
  CylindricalAlgebraicDecompositionRelationship,
  CADRelationship,
  RationalRootTheoremRelationship,
  QuadraticFormula,
  CubicFormulaRelationship,
  QuarticFormulaRelationship,
  RootIsolation,
  RootSeparationRelationship,
  SturmSequenceRelationship,
  DescartesRuleOfSignsRelationship,
  ExactRootRepresentation,
  RootOfRelationship,
  AlgebraicNumber,
  LambertWRelationship,
  SpecialFunctionSolution,
  InverseFunction,
  InverseTrigonometricFunction,
  InverseHyperbolicFunctionRelationship,
  LogarithmicTransformation,
  ExponentialTransformation,
  TrigonometricIdentity,
  PeriodicSolution,
  IntegerParameter,
  GeneralPeriodicSolution,
  Inequality,
  SymbolicInequalitySolving,
  IntervalSolution,
  UnionOfIntervals,
  SignChartRelationship,
  PolynomialInequality,
  RationalInequality,
  AbsoluteValueInequality,
  EquationTransformation,
  EquivalentEquation,
  ImplicationPreservingTransformation,
  EquivalencePreservingTransformation,
  SquaringTransformation,
  RaisingToPower,
  TakingLogarithm,
  TakingExponential,
  ClearingDenominators,
  CrossMultiplication,
  Simplification,
  ExpressionSimplification,
  Expansion,
  Collection,
  CanonicalForm,
  NormalForm,
  SymbolicAlgebra,
  SymbolicMathematics,
  SymbolicComputation,
  ComputerAlgebraSystem,CAS,
  ExactArithmetic,
  RationalArithmetic,
  AlgebraicNumberArithmetic,
  SymbolicNumericSolvingRelationship,
  NumericalRootFindingRelationship,
  NewtonRaphsonRelationship,
  BisectionMethodRelationship,
  IntervalVerificationRelationship,
  SolverStrategy,
  DecisionTreeRelationship,
  EquationClassification,
  CaseSplit,
  BranchAndBoundRelationship,
  BacktrackingRelationship,
  RewriteRule,
  PatternMatching,
  HeuristicSolverRelationship,
  CompleteSolverRelationship,
  IncompleteSolverRelationship,
  SolverCompleteness,
  Soundness,
  Correctness,
  Termination,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  ExpressionSwell,
  IntermediateExpressionGrowth,
  Verification,
  SubstituteAndCheck,
  Residual,
  ExactResidual,
  IdentityCheck,
  SolutionFiltering,
  ExtraneousRootRemoval,
  DomainCheck,
  BranchCheck,
  ReferenceSolution,
  KnownAnswerTest,
  Benchmark,
  RegressionTest,
  Reproducibility,
  Provenance,
  RuleSetVersion,
  CASVersion,
  SolverVersion,
  AlgorithmVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  NumericalEquationSolvingWhenApproximateRootsArePrimary,
  DifferentialEquationSolvingWhenODEOrPDEStructureIsPrimary,
  OptimisationWhenObjectiveExtremaRatherThanEquationSolutionsArePrimary,
  TheoremProvingWhenLogicalProofRatherThanSolutionDerivationIsPrimary
 ]
 DISAMBIGUATION_RULE = "When formal equations are solved through exact symbolic transformations to produce mathematical solution sets, interpret the context as a symbolic equation solver."
}

CORE_CONCEPTS {
 SymbolicEquationSolver = "Exact symbolic framework for deriving equation solution sets."
 Equation = "Mathematical equality whose unknown values are to be determined."
 SolutionSet = "Set of values satisfying all declared equation and domain conditions."
 Elimination = "Removal of variables to reduce a system."
 Substitution = "Replacement of variables or expressions to reduce or solve equations."
 Factorisation = "Transformation exposing roots through multiplicative factors."
 ConditionalSolution = "Solution valid only under specified parameter or domain conditions."
 SolverCompleteness = "Extent to which a solver finds all valid solutions in a declared problem class."
}

RELATIONS {
 SymbolicEquationSolver --is_child_of--> SymbolicComputation
 SymbolicEquationSolver --solves--> Equation
 SymbolicEquationSolver --produces--> SolutionSet
 Solution --satisfies--> OriginalEquation
 Factorisation --supports--> PolynomialEquationSolving
 Substitution --reduces--> EquationSystem
 Elimination --reduces--> NumberOfUnknowns
 Resultant --eliminates--> Variable
 GroebnerBasis --supports--> PolynomialSystemSolving
 QuadraticFormula --solves--> QuadraticEquation
 RootIsolation --locates--> RealOrComplexRoots
 ConditionalSolution --depends_on--> ParameterCondition
 Verification --checks--> CandidateSolution
 SolutionFiltering --removes--> InvalidOrExtraneousSolutions
}

OPERATORS {
 SOLVE(eq,var,A) := symbolic_solution_set(eq,var,A)
 SOLVE_SYSTEM(F,vars,A) := symbolic_system_solution_set(F,vars,A)
 SUBSTITUTE_CHECK(eq,var,value) := simplify(substitute(eq,var,value))
 ELIMINATE(F,var) := eliminate_variable(F,var)
 RESULTANT_ELIMINATE(f,g,x) := resultant(f,g,x)
 FACTOR_SOLVE(f,x) := roots_of_factors(f,x)
 FILTER_SOLUTIONS(S,conditions) := {s in S | satisfies(s,conditions)}
 VERIFY_SOLUTION(eq,s) := exact_residual(eq,s) == 0
}

AXIOMS {
 A1 = "A symbolic equation solver seeks mathematically valid solution sets rather than merely approximate roots."
 A2 = "Equivalent transformations preserve the solution set only under their stated conditions."
 A3 = "Squaring, exponentiation or clearing denominators can introduce extraneous candidates."
 A4 = "Division by expressions can remove valid solutions when zero cases are not handled separately."
 A5 = "Solution validity depends on the declared mathematical domain."
 A6 = "Transcendental equations may require special functions, implicit forms or infinite families."
 A7 = "Solver completeness is equation-class dependent."
 A8 = "Symbolic-equation-solving results require explicit domain, assumptions, transformations, completeness and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(equations,unknowns,parameters,domain,assumptions,target_solution_form)
 R2_CLASSIFY = classify(linear,polynomial,rational,radical,exponential,logarithmic,trigonometric,transcendental_or_mixed)
 R3_NORMALISE = simplify_and_move_to_canonical_equation_form)
 R4_RESTRICTIONS = derive_denominator_domain_branch_and_parameter_conditions)
 R5_METHOD = choose(factorisation,formula,substitution,elimination,resultant,Groebner,CAD,inverse_function_or_special_function_method)
 R6_CASES = split_parameter_or_branch_cases_when_required)
 R7_SOLVE = derive_candidate_solution_set)
 R8_FILTER = remove_extraneous_or_domain_invalid_candidates)
 R9_VERIFY = substitute_candidates_into_original_equations_and_conditions)
 R10_REPRODUCE = document_domain_assumptions_solver_strategy_CAS_and_versions)
}

CONSTRAINTS {
 C1 = "Clearing denominators can introduce candidates at forbidden denominator zeros."
 C2 = "Squaring both sides can introduce extraneous solutions."
 C3 = "Taking logarithms requires positivity or complex branch semantics."
 C4 = "Inverse trigonometric functions capture principal values, not automatically complete periodic solution families."
 C5 = "Polynomial systems can have high algebraic complexity."
 C6 = "Gröbner-basis computation can suffer severe intermediate expression growth."
 C7 = "Transcendental equations may not have elementary closed-form solutions."
 C8 = "Parameter values can change the number or nature of solutions."
 C9 = "A solver can be sound for returned solutions while incomplete for the full solution set."
 C10 = "Numeric root checks alone cannot establish symbolic completeness."
}

SEMANTIC_EQUIVALENCE {
 "SymbolicEquationSolver" <=> "SymbolicSolver"
 "ComputerAlgebraSystem" <=> "CAS"
 "GroebnerBasis" <=> "GröbnerBasis"
 "CylindricalAlgebraicDecomposition" <=> "CAD"
 "SolutionSet" <=> "SetOfSolutions"
}

VALIDATION {
 V1 = "Unknown variables and parameters are explicit."
 V2 = "Mathematical domain is documented."
 V3 = "Assumptions, denominator restrictions and branch conditions are documented."
 V4 = "Each candidate solution is checked against the original equations."
 V5 = "Extraneous solutions introduced by transformations are removed."
 V6 = "Periodic solution families include integer-parameter semantics where required."
 V7 = "Parameter-dependent cases are documented."
 V8 = "System solutions satisfy all equations simultaneously."
 V9 = "Solver completeness or known limitations are documented where relevant."
 V10 = "Results are reproducible under documented solver, CAS, rule-set and algorithm versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_EQUATION_SOLVER_DEFINITION = 0.99
 LINEAR_SOLVING_REASONING = 0.99
 POLYNOMIAL_SOLVING_REASONING = 0.99
 RATIONAL_SOLVING_REASONING = 0.99
 GROEBNER_REASONING = 0.98
 TRANSCENDENTAL_SOLVING_REASONING = 0.97
 COMPLETENESS_REASONING = 0.97
 POLICY = "Reduce confidence when domain, parameters, branch conditions, excluded values, solver class or completeness guarantees are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SymbolicEquationSolverReasoning,
  EquationClassificationReasoning,LinearEquationReasoning,
  PolynomialEquationReasoning,RationalEquationReasoning,
  TranscendentalEquationReasoning,SystemSolvingReasoning,
  FactorisationReasoning,SubstitutionReasoning,
  EliminationReasoning,ResultantReasoning,
  GroebnerBasisReasoning,CADReasoning,
  ConditionalSolutionReasoning,BranchReasoning,
  ExtraneousSolutionReasoning,CompletenessReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "symbolic equation solver" -> SymbolicEquationSolver
  "symbolic solver" -> SymbolicEquationSolver
  "CAS" -> ComputerAlgebraSystem
  "Groebner basis" -> GroebnerBasis
  "CAD" -> CylindricalAlgebraicDecomposition
 }
}

EXECUTION_RULES {
 E1 = "If equations are solved through exact symbolic transformations, activate SymbolicEquationSolverOntology."
 E2 = "Identify unknowns, parameters, domain and assumptions before selecting a method."
 E3 = "Normalise equations while preserving domain restrictions."
 E4 = "Track transformations that can introduce or remove candidate solutions."
 E5 = "Use factorisation, substitution, elimination, resultants, Gröbner bases, CAD or inverse-function methods according to equation class."
 E6 = "Represent non-elementary solutions implicitly or with supported special functions rather than inventing closed forms."
 E7 = "Filter candidate solutions against original equations and branch conditions."
 E8 = "Document solver completeness or limitations when material."
 E9 = "Document solver strategy, CAS and algorithm versions for reproducibility."
}

RECURSION {
 ENTRY = SymbolicEquationSolver
 EXPAND(SymbolicEquationSolver) -> [
  Equation,LinearEquation,QuadraticEquation,PolynomialEquation,RationalEquation,
  RadicalEquation,ExponentialEquation,LogarithmicEquation,TrigonometricEquation,
  TranscendentalEquation,NonlinearEquation,SystemOfEquations,
  Unknown,Parameter,Solution,SolutionSet,ExactSolution,ImplicitSolution,
  ConditionalSolution,ParameterisedSolution,Root,RootMultiplicity,
  RealRoot,ComplexRoot,Domain,Assumption,BranchCondition,ExcludedValue,
  Factorisation,SolveByFactorisation,Substitution,SolveBySubstitution,
  Elimination,SolveByElimination,Resultant,Discriminant,
  GroebnerBasis,BuchbergerAlgorithm,RootIsolation,
  QuadraticFormula,CylindricalAlgebraicDecomposition,CAD,
  InverseFunction,SpecialFunctionSolution,
  SymbolicInequalitySolving,IntervalSolution,
  Simplification,ExpressionSimplification,SymbolicAlgebra,
  ComputerAlgebraSystem,CAS,SolverCompleteness,
  SolutionFiltering,ExtraneousSolution,Verification,
  Reproducibility,Provenance
 ]
 EXPAND(SymbolicEquationSolverFamilies) -> [
  LinearAndPolynomialEquationSolving,
  RationalAndRadicalEquationSolving,
  ExponentialLogarithmicAndTrigonometricSolving,
  PolynomialSystemAndGroebnerMethods,
  EliminationResultantsAndCAD,
  ConditionalImplicitAndSpecialFunctionSolutions,
  InequalityAndDomainAwareSolving,
  VerificationCompletenessAndReproducibility
 ]
 STOP_CONDITION = "Stop when equation class, domain, assumptions, solver method, solution conditions, filtering, completeness and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Symbolic Computation"
 RELATED_ONTOLOGIES = [
  "Symbolic Computation",
  "Symbolic Mathematics Ontology",
  "Symbolic Algebra Ontology",
  "Expression Simplification Ontology",
  "Computer Algebra Systems Ontology 2",
  "Equation Ontology",
  "Polynomial Equation Ontology",
  "System of Equations Ontology",
  "Factorisation Ontology",
  "Substitution Ontology",
  "Computational Algebra Ontology",
  "Numerical Root Finding Ontologies",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Symbolic Equation Solver -> Symbolic Computation for parent execution semantics",
  "Symbolic Equation Solver -> Symbolic Mathematics for formal expression semantics",
  "Symbolic Equation Solver -> Symbolic Algebra for algebraic transformations",
  "Symbolic Equation Solver -> Expression Simplification for canonicalisation and reduction",
  "Symbolic Equation Solver -> Computer Algebra Systems for solver implementation",
  "Symbolic Equation Solver -> Equation, Polynomial Equation and System of Equations ontologies for problem-specific semantics",
  "Symbolic Equation Solver -> Factorisation and Substitution ontologies for method-specific reasoning",
  "Symbolic Equation Solver -> Computational Algebra for resultants, Gröbner bases and elimination",
  "Symbolic Equation Solver -> Numerical Root Finding for approximation and verification relationships",
  "Symbolic Equation Solver -> Computational Complexity for solver-cost semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible equation classes, solver methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve candidate-solution-versus-verified-solution distinction.",
  "Preserve exact-versus-numerical solution distinction.",
  "Preserve principal-value-versus-general-solution distinction.",
  "Preserve equation-transformation-versus-solution-set-equivalence distinction.",
  "Preserve soundness-versus-completeness distinction.",
  "Preserve domain-independent-versus-parameter-dependent solution distinction.",
  "Preserve machine-readable symbolic-equation-solver semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A symbolic equation solver derives exact, conditional, implicit or parameterised solution sets from formal equations."
 METHOD_SUMMARY = "Factorisation, formulas, substitution, elimination, resultants, Gröbner bases and CAD provide major exact solving pathways."
 DOMAIN_SUMMARY = "Domains, excluded values, assumptions and branch conditions determine solution validity."
 TRANSCENDENTAL_SUMMARY = "Transcendental equations may require inverse functions, special functions, implicit forms or periodic solution families."
 COMPLETENESS_SUMMARY = "Returned solutions must be distinguished from claims that the complete solution set has been found."
 VALIDATION_SUMMARY = "Symbolic equation solutions should be checked against original equations, domain restrictions, branch conditions and reproducible solver settings."
}

FLOW {
 INPUT
 -> detect_symbolic_equation_solver_context
 -> identify_equations_unknowns_parameters_domain_assumptions_and_target_solution_form
 -> classify_equation_or_system_type
 -> simplify_and_normalise_equations
 -> derive_denominator_domain_branch_and_parameter_restrictions
 -> choose_factorisation_formula_substitution_elimination_resultant_Groebner_CAD_inverse_function_or_special_function_method
 -> split_parameter_and_branch_cases_when_required
 -> derive_candidate_solution_set
 -> filter_extraneous_domain_invalid_or_branch_invalid_candidates
 -> verify_candidates_against_original_equations_and_conditions
 -> assess_solver_completeness_or_known_limitations
 -> document_domain_assumptions_solver_strategy_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-EQUATION-SOLVER-ONTOLOGY-V1"
}

}

END_FLOW