Symbolic Computation

AI INTERPRETATION NOTE
This Flow Algebra defines the Symbolic Computation parent ontology within the Kireth Research Semantic Library. Symbolic computation concerns the exact or rule-based manipulation of mathematical expressions, equations, identities, algebraic structures and formal objects represented as symbols rather than only as numerical approximations. This ontology supports expression trees, canonical forms, simplification, substitution, symbolic differentiation and integration, equation solving, polynomial and rational manipulation, algebraic numbers, exact arithmetic, term rewriting, computer algebra systems, symbolic-numeric methods, complexity, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SymbolicComputation
 ENTITY_CLASS = FormalMathematicalExpressionManipulationFramework
 PRIMARY_CONTEXT = ComputationalMathematicsOntologies
 DEFINITION = "Symbolic computation is the algorithmic manipulation of mathematical expressions and formal objects using exact representations, transformation rules and algebraic reasoning."
 SYMBOLIC_FORMS = [
  "expr -> rewrite(expr,rule)",
  "simplify(f(x)) -> canonical_or_equivalent_form",
  "d/dx expr -> symbolic_derivative(expr,x)",
  "solve(E,x) -> symbolic_solution_set"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for expression representation, rewrite systems, exact arithmetic, simplification, substitution, symbolic calculus, equation solving and computer algebra."
}

SCOPE {
 INCLUDE = [
  SymbolicComputation,ComputerAlgebra,ComputerAlgebraSystem,CAS,
  SymbolicMathematics,ExactComputation,FormalComputation,
  Symbol,Variable,Constant,Parameter,
  Expression,SymbolicExpression,AlgebraicExpression,
  ExpressionTree,AbstractSyntaxTree,AST,
  ParseTree,Term,Operator,Operand,
  Function,SymbolicFunction,
  Equation,Inequality,Identity,
  Formula,Relation,
  CanonicalForm,NormalForm,
  Simplification,SymbolicSimplification,
  AlgebraicSimplification,
  TrigonometricSimplification,
  LogarithmicSimplification,
  RationalSimplification,
  RadicalSimplification,
  ExpressionReduction,
  ConstantFoldingRelationship,
  CommonSubexpressionEliminationRelationship,
  Expansion,AlgebraicExpansion,
  Factorisation,PolynomialFactorisation,
  Collection,CollectTerms,
  CombiningLikeTerms,
  Cancellation,
  Substitution,SymbolicSubstitution,
  Replacement,PatternMatching,
  WildcardPattern,MatchVariable,
  RewriteRule,RewriteSystem,
  TermRewriting,RuleApplication,
  ConditionalRewrite,
  RewriteStrategy,
  Normalisation,Canonicalisation,
  Confluence,Termination,
  CriticalPairRelationship,
  KnuthBendixRelationship,
  ExactIntegerArithmetic,
  ArbitraryPrecisionInteger,
  RationalArithmetic,ExactRational,
  Fraction,BigInteger,
  AlgebraicNumber,
  AlgebraicNumberField,
  SurdRelationship,
  RootOfRelationship,
  Polynomial,MultivariatePolynomial,
  Monomial,PolynomialRing,
  PolynomialGCD,
  ExtendedGCD,
  PolynomialDivision,
  SyntheticDivisionRelationship,
  Resultant,
  Discriminant,
  GröbnerBasis,GroebnerBasis,
  BuchbergerAlgorithm,
  MonomialOrdering,
  LexicographicOrder,
  GradedLexicographicOrder,
  EliminationOrder,
  IdealMembership,
  RationalFunction,
  PartialFractionDecomposition,
  SymbolicFraction,
  ExactRoot,
  RootIsolation,
  RootMultiplicity,
  SymbolicEquationSolving,
  LinearEquationSolving,
  PolynomialEquationSolving,
  NonlinearEquationSolving,
  SystemOfEquations,
  SymbolicSystemSolver,
  Elimination,
  SubstitutionMethod,
  ResultantMethod,
  GroebnerSolverRelationship,
  SolveByFactorisationRelationship,
  InequalitySolving,
  SymbolicDifferentiation,
  HigherDerivative,
  PartialDerivative,
  GradientSymbolicRelationship,
  JacobianSymbolicRelationship,
  HessianSymbolicRelationship,
  SymbolicIntegration,
  IndefiniteIntegral,
  DefiniteIntegral,
  IntegrationByPartsRelationship,
  SubstitutionIntegrationRelationship,
  RationalFunctionIntegration,
  RischAlgorithmRelationship,
  SpecialFunctionRelationship,
  SymbolicLimit,
  LimitComputation,
  SeriesExpansion,
  TaylorSeries,
  LaurentSeries,
  AsymptoticSeries,
  PowerSeries,
  FormalPowerSeries,
  SeriesCoefficient,
  RecurrenceRelation,
  SymbolicSummation,
  SymbolicProduct,
  FiniteSum,InfiniteSum,
  TelescopingRelationship,
  GosperAlgorithmRelationship,
  ZeilbergerAlgorithmRelationship,
  HypergeometricTermRelationship,
  SymbolicTransform,
  LaplaceTransform,
  FourierTransformRelationship,
  ZTransformRelationship,
  InverseTransformRelationship,
  MatrixSymbolicComputation,
  SymbolicMatrix,
  MatrixExpression,
  DeterminantSymbolic,
  TraceSymbolic,
  MatrixInverseSymbolic,
  SymbolicEigenvalueRelationship,
  SymbolicLinearAlgebra,
  TensorSymbolicComputationRelationship,
  LogicExpression,
  BooleanExpression,
  PropositionalFormula,
  PredicateFormula,
  SymbolicLogicRelationship,
  SatisfiabilityRelationship,
  TheoremProvingRelationship,
  FormalProofRelationship,
  Assumption,AssumptionContext,
  DomainAssumption,
  PositivityAssumption,
  RealAssumption,
  IntegerAssumption,
  NonzeroAssumption,
  BranchCondition,
  PiecewiseExpression,
  ConditionalExpression,
  BranchCutRelationship,
  PrincipalValueRelationship,
  ExactConstant,
  PiSymbol,EulerNumberRelationship,
  ImaginaryUnit,
  InfinitySymbol,
  UndefinedSymbolRelationship,
  SymbolicInfinityRelationship,
  ExpressionHash,
  StructuralEquality,
  MathematicalEquivalence,
  SemanticEquivalence,
  AlphaEquivalenceRelationship,
  ExpressionOrdering,
  HashConsingRelationship,
  Memoisation,
  Cache,
  DAGExpressionRepresentation,
  LazyEvaluation,
  EagerEvaluation,
  SymbolicEvaluation,
  NumericEvaluation,
  SymbolicNumericComputation,
  HybridSymbolicNumericMethod,
  ArbitraryPrecisionFloatingPointRelationship,
  IntervalArithmeticRelationship,
  NumericApproximation,
  ExactToNumericConversion,
  NumericToSymbolicReconstruction,
  RationalReconstruction,
  IntegerRelationDetectionRelationship,
  PSLQRelationship,
  SymbolicRegressionRelationship,
  AutomaticDifferentiationRelationship,
  ComputationalComplexity,
  ExpressionSwell,
  IntermediateExpressionGrowth,
  TimeComplexity,SpaceComplexity,
  RewriteComplexity,
  PolynomialComplexity,
  Verification,IdentityCheck,
  SubstituteAndCheck,
  ExactResidual,
  RandomisedIdentityTestingRelationship,
  PolynomialIdentityTestingRelationship,
  ReferenceExpression,
  KnownIdentity,
  Benchmark,
  Reproducibility,
  Provenance,
  CASVersion,
  RuleSetVersion,
  AlgorithmVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  PureNumericalComputationWithoutSymbolicRepresentation,
  TextManipulationWithoutMathematicalSemantics,
  FormalLogicWithoutMathematicalSymbolicComputationContext,
  MachineLearningSymbolicRegressionWhenModelDiscoveryRatherThanAlgebraicManipulationIsPrimary
 ]
 DISAMBIGUATION_RULE = "When mathematical objects are represented and manipulated as exact symbols, expressions or formal rules rather than only approximate numbers, interpret the context as symbolic computation."
}

CORE_CONCEPTS {
 SymbolicComputation = "Algorithmic manipulation of exact mathematical expressions."
 SymbolicExpression = "Formal mathematical expression represented as symbols and operators."
 ExpressionTree = "Tree or DAG representation of an expression's syntactic structure."
 RewriteRule = "Transformation rule mapping one symbolic pattern to another."
 CanonicalForm = "Normalised representation used to compare or simplify symbolic expressions."
 ExactArithmetic = "Arithmetic preserving exact values without floating-point approximation."
 SymbolicSolver = "Algorithm deriving formal solution expressions or solution sets."
 ComputerAlgebraSystem = "Software system implementing symbolic mathematical operations."
}

RELATIONS {
 SymbolicComputation --is_child_of--> ComputationalMathematicsOntologies
 SymbolicExpression --represented_by--> ExpressionTree
 ExpressionTree --contains--> OperatorsAndOperands
 RewriteSystem --contains--> RewriteRules
 Simplification --uses--> RewriteRules
 Canonicalisation --produces--> CanonicalForm
 Substitution --transforms--> SymbolicExpression
 SymbolicDifferentiation --operates_on--> SymbolicExpression
 SymbolicIntegration --operates_on--> SymbolicExpression
 SymbolicEquationSolving --produces--> SolutionSet
 PolynomialFactorisation --operates_on--> Polynomial
 GroebnerBasis --represents--> PolynomialIdeal
 ExactArithmetic --supports--> SymbolicComputation
 ComputerAlgebraSystem --implements--> SymbolicAlgorithms
 Verification --checks--> SymbolicResult
}

OPERATORS {
 PARSE(text) := symbolic_expression_tree(text)
 REWRITE(expr,rule) := apply_matching_transformation(expr,rule)
 SIMPLIFY(expr) := equivalent_reduced_expression(expr)
 SUBSTITUTE(expr,x,value) := replace_symbol(expr,x,value)
 EXPAND(expr) := distribute_products_and_powers(expr)
 FACTOR(expr) := factor_symbolic_expression(expr)
 DIFFERENTIATE(expr,x) := symbolic_derivative(expr,x)
 INTEGRATE(expr,x) := symbolic_antiderivative_when_available(expr,x)
 SOLVE(equations,variables) := symbolic_solution_set(equations,variables)
 CANONICALISE(expr) := normal_form(expr)
}

AXIOMS {
 A1 = "Symbolic computation operates on formal representations of mathematical objects."
 A2 = "Transformation rules must preserve declared mathematical semantics."
 A3 = "Canonical form depends on the algebraic domain and chosen ordering."
 A4 = "Exact arithmetic differs from floating-point approximation."
 A5 = "Simplification is context dependent and may require assumptions."
 A6 = "Expression growth can dominate symbolic computational cost."
 A7 = "Equivalent expressions need not have identical syntactic forms."
 A8 = "Symbolic-computation results require explicit domain, assumptions, representation, transformations and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(expression_domain,variables,constants,assumptions,target_operation,exactness_goal)
 R2_PARSE = construct_expression_tree_or_DAG)
 R3_DOMAIN = determine(integer,rational,real,complex,polynomial,functional_or_logic_domain)
 R4_NORMALISE = canonicalise_structure_and_apply_safe_simplifications)
 R5_OPERATION = choose(rewrite,substitution,expansion,factorisation,differentiation,integration,solving_or_transform)
 R6_ALGORITHM = select_domain_appropriate_symbolic_algorithm)
 R7_ASSUMPTIONS = apply_domain_and_branch_conditions_explicitly)
 R8_COMPLEXITY = monitor_expression_swell_and_intermediate_growth)
 R9_VERIFY = check_by_exact_substitution_identity_residual_or_reference_relation)
 R10_REPRODUCE = document_CAS_rules_algorithms_assumptions_and_versions)
}

CONSTRAINTS {
 C1 = "Simplification can be invalid without required assumptions."
 C2 = "Branch cuts can make apparently equivalent complex expressions differ."
 C3 = "Canonical forms vary across algebraic domains."
 C4 = "Symbolic integration may not have an elementary closed form."
 C5 = "Expression swell can make exact algorithms computationally expensive."
 C6 = "Factorisation domains affect factor structure."
 C7 = "Symbolic equation solvers can return conditional or implicit solutions."
 C8 = "Numeric-to-symbolic reconstruction can infer incorrect exact forms from insufficient precision."
 C9 = "Rewrite systems can loop if termination is not controlled."
 C10 = "Syntactic equality is weaker than mathematical equivalence."
}

SEMANTIC_EQUIVALENCE {
 "ComputerAlgebraSystem" <=> "CAS"
 "AbstractSyntaxTree" <=> "AST"
 "GroebnerBasis" <=> "GröbnerBasis"
 "GreatestCommonDivisor" <=> "GCD"
 "LeastCommonMultiple" <=> "LCM"
 "SymbolicNumericComputation" <=> "HybridSymbolicNumericComputation"
}

VALIDATION {
 V1 = "Expression domain and assumptions are documented."
 V2 = "Input parsing preserves mathematical structure."
 V3 = "Rewrite rules preserve semantics under declared assumptions."
 V4 = "Canonicalisation is appropriate to the algebraic domain."
 V5 = "Solutions are substituted back into original equations where feasible."
 V6 = "Differentiation results are checked by known identities or numeric spot checks where appropriate."
 V7 = "Integration results are checked by differentiation where feasible."
 V8 = "Exact arithmetic is preserved when exact output is claimed."
 V9 = "Branch conditions and piecewise cases are documented."
 V10 = "Results are reproducible under documented CAS, rule set and algorithm versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_COMPUTATION_DEFINITION = 0.99
 REWRITE_REASONING = 0.99
 SIMPLIFICATION_REASONING = 0.98
 POLYNOMIAL_REASONING = 0.99
 SYMBOLIC_CALCULUS_REASONING = 0.98
 EQUATION_SOLVING_REASONING = 0.98
 EXACT_ARITHMETIC_REASONING = 0.99
 POLICY = "Reduce confidence when domain, assumptions, branch conditions, canonicalisation rules or symbolic algorithm are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SymbolicComputationReasoning,
  ExpressionTreeReasoning,RewriteReasoning,SimplificationReasoning,
  SubstitutionReasoning,ExpansionReasoning,FactorisationReasoning,
  PolynomialReasoning,GroebnerBasisReasoning,
  SymbolicDifferentiationReasoning,SymbolicIntegrationReasoning,
  EquationSolvingReasoning,SeriesReasoning,
  ExactArithmeticReasoning,AssumptionReasoning,
  SymbolicNumericReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "symbolic computation" -> SymbolicComputation
  "computer algebra" -> SymbolicComputation
  "CAS" -> ComputerAlgebraSystem
  "AST" -> AbstractSyntaxTree
  "Groebner basis" -> GroebnerBasis
 }
}

EXECUTION_RULES {
 E1 = "If mathematical objects are manipulated as exact symbolic expressions, activate SymbolicComputation."
 E2 = "Determine domain and assumptions before simplification or solving."
 E3 = "Parse expressions into explicit symbolic structure before transformation."
 E4 = "Use canonical forms appropriate to the algebraic domain."
 E5 = "Apply rewrite rules only when their semantic preconditions hold."
 E6 = "Monitor expression swell in exact computations."
 E7 = "Preserve exact arithmetic unless numerical approximation is explicitly requested."
 E8 = "Verify derived expressions through exact substitution, inverse operation or identity checking."
 E9 = "Document CAS, rule-set and algorithm versions for reproducibility."
}

RECURSION {
 ENTRY = SymbolicComputation
 EXPAND(SymbolicComputation) -> [
  ComputerAlgebra,CAS,SymbolicExpression,ExpressionTree,AST,
  CanonicalForm,NormalForm,Simplification,Expansion,Factorisation,
  Substitution,PatternMatching,RewriteRule,RewriteSystem,TermRewriting,
  ExactIntegerArithmetic,RationalArithmetic,AlgebraicNumber,
  Polynomial,PolynomialGCD,PolynomialDivision,Resultant,GroebnerBasis,
  RationalFunction,PartialFractionDecomposition,
  SymbolicEquationSolving,SystemOfEquations,InequalitySolving,
  SymbolicDifferentiation,SymbolicIntegration,SymbolicLimit,
  SeriesExpansion,TaylorSeries,LaurentSeries,
  SymbolicSummation,SymbolicProduct,LaplaceTransform,
  SymbolicMatrix,SymbolicLinearAlgebra,
  Assumption,PiecewiseExpression,BranchCondition,
  MathematicalEquivalence,SymbolicNumericComputation,
  RationalReconstruction,PSLQRelationship,
  ExpressionSwell,ComputationalComplexity,Verification,Reproducibility
 ]
 EXPAND(SymbolicComputationFamilies) -> [
  ExpressionRepresentationAndRewriting,
  ExactArithmeticAndAlgebraicNumbers,
  PolynomialAndRationalComputation,
  SymbolicEquationSolving,
  SymbolicCalculus,
  SymbolicSeriesAndTransforms,
  SymbolicLinearAlgebra,
  SymbolicNumericAndVerificationMethods
 ]
 STOP_CONDITION = "Stop when domain, representation, assumptions, transformation algorithm, exactness, verification and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "Algebra Ontologies",
  "Computational Algebra Ontology",
  "Computer Algebra Systems Ontology",
  "Polynomial Ontology",
  "Equation Ontology",
  "Symbolic Manipulation Ontology",
  "Calculus and Analysis Ontologies",
  "Numerical Methods Ontology 2",
  "Computational Matrix Ontology",
  "Tensor Computation Ontology",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Symbolic Computation -> Algebra Ontologies for algebraic foundations",
  "Symbolic Computation -> Computational Algebra for algebraic algorithm semantics",
  "Symbolic Computation -> Computer Algebra Systems for implementation semantics",
  "Symbolic Computation -> Polynomial and Equation ontologies for expression-specific reasoning",
  "Symbolic Computation -> Symbolic Manipulation for rewrite and transformation semantics",
  "Symbolic Computation -> Calculus and Analysis for symbolic differentiation, integration and limits",
  "Symbolic Computation -> Numerical Methods for symbolic-numeric hybrid computation",
  "Symbolic Computation -> Computational Matrix and Tensor ontologies for symbolic structured objects",
  "Symbolic Computation -> Computational Complexity for expression-growth and algorithm-cost semantics"
 ]
}

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

OUTPUT {
 CANONICAL_DEFINITION = "Symbolic computation manipulates exact mathematical expressions using formal representations, algebraic algorithms and rewrite rules."
 REPRESENTATION_SUMMARY = "Expression trees and DAGs encode symbolic structure for algorithmic transformation."
 ALGEBRA_SUMMARY = "Expansion, factorisation, polynomial arithmetic, Gröbner bases and rational manipulation provide core algebraic computation."
 CALCULUS_SUMMARY = "Symbolic differentiation, integration, limits and series operate on formal expressions."
 SOLVER_SUMMARY = "Symbolic solvers derive exact, conditional or implicit solution sets under declared assumptions."
 VALIDATION_SUMMARY = "Symbolic results should be checked through exact substitution, inverse operations, identity tests, branch-condition review and reproducible CAS settings."
}

FLOW {
 INPUT
 -> detect_symbolic_computation_context
 -> identify_expression_domain_variables_constants_assumptions_target_operation_and_exactness_goal
 -> parse_expression_into_tree_or_DAG
 -> determine_algebraic_domain_and_branch_conditions
 -> canonicalise_structure_and_apply_safe_simplifications
 -> choose_rewrite_substitution_expansion_factorisation_differentiation_integration_solving_series_or_transform_operation
 -> select_domain_appropriate_symbolic_algorithm
 -> monitor_expression_swell_and_intermediate_growth
 -> preserve_exact_arithmetic_unless_numeric_approximation_is_requested
 -> verify_result_by_substitution_inverse_operation_identity_or_reference_relation
 -> document_assumptions_CAS_rule_set_algorithm_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-COMPUTATIONAL-MATHEMATICS-SYMBOLIC-COMPUTATION-V1"
}

}

END_FLOW