Computer Algebra Systems Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Computer Algebra Systems Ontology within the Kireth Research Semantic Library. A computer algebra system is a software environment for exact or symbolic mathematical computation, including expression representation, simplification, algebraic manipulation, symbolic differentiation and integration, equation solving, polynomial and rational computation, symbolic linear algebra, rule-based transformation, assumptions, exact arithmetic, symbolic-numeric methods, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Computer Algebra Systems Ontology 2"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Symbolic Computation / Computer Algebra Systems"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/symbolic-computation/computer-algebra-systems-ontology-2.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputerAlgebraSystem
 ENTITY_CLASS = SymbolicMathematicsSoftwareFramework
 PRIMARY_CONTEXT = SymbolicComputation
 DEFINITION = "A computer algebra system is a computational software system that represents mathematical objects symbolically and applies exact algebraic, calculus, solving, rewriting and symbolic-numeric algorithms to them."
 SYMBOLIC_FORMS = [
  "CAS(input_expression) -> symbolic_result",
  "simplify(expr,assumptions)",
  "solve(equations,variables)",
  "differentiate(expr,x)",
  "integrate(expr,x)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for symbolic engines, expression representation, exact arithmetic, transformation rules, algebraic algorithms, symbolic calculus, solving, assumptions, verification and reproducibility."
}

SCOPE {
 INCLUDE = [
  ComputerAlgebraSystem,CAS,
  SymbolicEngine,SymbolicKernel,
  SymbolicComputation,SymbolicMathematics,
  SymbolicAlgebra,SymbolicManipulation,
  MathematicalSoftware,
  AlgebraSystem,ExactComputationSystem,
  SymbolicNumericSystem,
  Expression,SymbolicExpression,
  ExpressionTree,AbstractSyntaxTree,AST,
  DAGExpressionRepresentation,
  Symbol,Variable,Constant,Parameter,
  Operator,Operand,Function,
  Equation,Inequality,Identity,
  Parser,TokenizerRelationship,
  ExpressionParser,
  PrettyPrinterRelationship,
  FormatterRelationship,
  InternalRepresentation,
  CanonicalForm,NormalForm,
  SimplificationEngine,
  RewriteEngine,RuleEngine,
  RewriteRule,TransformationRule,
  PatternMatching,PatternMatcher,
  WildcardPattern,
  TermRewriting,
  Normalisation,Canonicalisation,
  AssumptionEngine,
  Assumption,AssumptionContext,
  DomainAssumption,
  PositivityAssumption,
  NonzeroAssumption,
  IntegerAssumption,
  RealAssumption,
  ComplexAssumption,
  BranchCondition,
  PiecewiseExpression,
  ExactArithmetic,
  ArbitraryPrecisionInteger,
  BigInteger,
  RationalArithmetic,
  ExactRational,
  AlgebraicNumber,
  ArbitraryPrecisionArithmetic,
  SymbolicPolynomial,
  PolynomialArithmetic,
  PolynomialFactorisation,
  PolynomialDivision,
  PolynomialGCD,
  Resultant,
  Discriminant,
  GroebnerBasis,
  BuchbergerAlgorithm,
  RationalFunction,
  PartialFractionDecomposition,
  SymbolicEquationSolving,
  LinearEquationSolving,
  PolynomialEquationSolving,
  NonlinearEquationSolving,
  SystemOfEquations,
  InequalitySolving,
  SymbolicDifferentiation,
  SymbolicIntegration,
  SymbolicLimit,
  SeriesExpansion,
  TaylorSeries,
  LaurentSeries,
  SymbolicSummation,
  SymbolicProduct,
  SymbolicTransform,
  LaplaceTransformRelationship,
  FourierTransformRelationship,
  ZTransformRelationship,
  SymbolicLinearAlgebra,
  SymbolicMatrix,
  SymbolicVector,
  SymbolicTensor,
  DeterminantSymbolic,
  MatrixInverseSymbolic,
  EigenvalueSymbolicRelationship,
  SpecialFunction,
  GammaFunctionRelationship,
  ErrorFunctionRelationship,
  HypergeometricFunctionRelationship,
  MeijerGRelationship,
  ExactSolution,
  ClosedFormSolution,
  ConditionalSolution,
  ImplicitSolution,
  UnevaluatedExpression,
  UnevaluatedIntegral,
  UnevaluatedSum,
  NumericApproximation,
  ExactToNumericConversion,
  NumericToExactReconstruction,
  RationalReconstruction,
  IntegerRelationDetectionRelationship,
  PSLQRelationship,
  SymbolicNumericComputation,
  HybridSymbolicNumericMethod,
  NumericalFallbackRelationship,
  AutomaticDifferentiationRelationship,
  NumericalDifferentiationRelationship,
  NumericalIntegrationRelationship,
  SolverBackend,
  AlgebraBackend,
  SimplificationBackend,
  IntegrationBackend,
  DifferentiationBackend,
  PolynomialBackend,
  MatrixBackend,
  NumericBackend,
  PluginArchitectureRelationship,
  ExtensionSystemRelationship,
  UserDefinedFunction,
  UserDefinedRewriteRule,
  MacroSystemRelationship,
  ProcedureLanguageRelationship,
  InteractiveSession,
  REPLRelationship,
  NotebookRelationship,
  BatchExecution,
  ScriptExecution,
  API,
  LibraryInterface,
  CommandInterface,
  ExpressionSerialization,
  Deserialization,
  ImportExportRelationship,
  MathMLRelationship,
  LaTeXRelationship,
  OpenMathRelationship,
  SymbolicDataExchange,
  Caching,Memoisation,
  CommonSubexpressionEliminationRelationship,
  HashConsingRelationship,
  ExpressionHash,
  StructuralEquality,
  MathematicalEquivalence,
  EvaluationStrategy,
  LazyEvaluation,
  EagerEvaluation,
  HoldExpressionRelationship,
  EvaluationControl,
  RulePriority,
  RewriteStrategy,
  RecursiveEvaluation,
  ExpressionSwell,
  IntermediateExpressionGrowth,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  MemoryManagement,
  GarbageCollectionRelationship,
  ParallelSymbolicComputationRelationship,
  DistributedSymbolicComputationRelationship,
  Verification,
  IdentityCheck,
  SubstituteAndCheck,
  DifferentiateAndCheck,
  ExactResidual,
  ReferenceResult,
  KnownIdentity,
  Benchmark,
  RegressionTest,
  TestSuite,
  Reproducibility,
  Provenance,
  CASVersion,
  KernelVersion,
  RuleSetVersion,
  AlgorithmVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  NumericalComputingSystemWithoutSymbolicCapability,
  GeneralProgrammingLanguageWithoutMathematicalSymbolicEngine,
  TheoremProverWhenFormalProofRatherThanComputerAlgebraIsPrimary,
  SpreadsheetSoftwareWithoutSymbolicMathematicsSemantics
 ]
 DISAMBIGUATION_RULE = "When software represents and manipulates mathematical expressions symbolically using exact algorithms and rewrite systems, interpret the context as a computer algebra system."
}

CORE_CONCEPTS {
 ComputerAlgebraSystem = "Software system for symbolic and exact mathematical computation."
 SymbolicEngine = "Core software component executing symbolic mathematical operations."
 ExpressionRepresentation = "Internal structure used to encode symbolic mathematical expressions."
 SimplificationEngine = "Subsystem reducing expressions while preserving semantics."
 RewriteEngine = "Subsystem applying pattern-based transformation rules."
 AssumptionEngine = "Subsystem tracking domain and contextual conditions."
 ExactArithmetic = "Arithmetic subsystem preserving exact mathematical values."
 SolverBackend = "Subsystem solving equations, inequalities or systems symbolically."
}

RELATIONS {
 ComputerAlgebraSystem --is_child_of--> SymbolicComputation
 ComputerAlgebraSystem --implements--> SymbolicMathematics
 ComputerAlgebraSystem --contains--> SymbolicEngine
 SymbolicEngine --uses--> ExpressionRepresentation
 SimplificationEngine --uses--> RewriteRules
 AssumptionEngine --constrains--> SymbolicTransformations
 PolynomialBackend --supports--> PolynomialArithmetic
 SolverBackend --produces--> SymbolicSolutions
 DifferentiationBackend --implements--> SymbolicDifferentiation
 IntegrationBackend --implements--> SymbolicIntegration
 NumericBackend --supports--> SymbolicNumericComputation
 Verification --checks--> CASResults
 Reproducibility --requires--> VersionAndRuleMetadata
}

OPERATORS {
 PARSE(text) := internal_symbolic_expression(text)
 SIMPLIFY(expr,A) := simplification_engine(expr,A)
 REWRITE(expr,rules) := rewrite_engine(expr,rules)
 SOLVE(F,vars,A) := symbolic_solver(F,vars,A)
 DIFFERENTIATE(expr,x) := symbolic_differentiation_backend(expr,x)
 INTEGRATE(expr,x) := symbolic_integration_backend(expr,x)
 NUMERIC(expr,p) := evaluate_to_precision(expr,p)
 VERIFY(expr1,expr2,A) := test_symbolic_equivalence(expr1,expr2,A)
 SERIALIZE(expr) := stable_symbolic_representation(expr)
}

AXIOMS {
 A1 = "A computer algebra system represents mathematical expressions symbolically."
 A2 = "Exact and approximate computation are distinct execution modes."
 A3 = "Simplification depends on domain, assumptions and rewrite policy."
 A4 = "Equivalent expressions need not share the same internal form."
 A5 = "A CAS may return unevaluated expressions when no supported exact algorithm applies."
 A6 = "User-defined rules can alter transformation behaviour."
 A7 = "Versioned algorithms and rule sets can change output form while preserving mathematical equivalence."
 A8 = "CAS results require explicit assumptions, algorithms, versions and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(expression_type,domain,assumptions,target_operation,exactness_goal,output_form)
 R2_PARSE = parse_input_to_internal_expression)
 R3_NORMALISE = canonicalise_internal_representation)
 R4_ASSUMPTIONS = attach_domain_and_branch_context)
 R5_DISPATCH = choose(algebra,polynomial,solver,differentiation,integration,series,matrix_or_numeric_backend)
 R6_EXECUTE = apply_backend_algorithm_and_rewrite_rules)
 R7_SIMPLIFY = reduce_output_under_assumptions)
 R8_FALLBACK = if_exact_method_unavailable preserve_unevaluated_or_use_declared_numeric_fallback)
 R9_VERIFY = check_by_identity_substitution_inverse_operation_or_reference_result)
 R10_REPRODUCE = document_CAS_kernel_rules_algorithms_libraries_and_versions)
}

CONSTRAINTS {
 C1 = "Different CAS platforms can return syntactically different but equivalent results."
 C2 = "Aggressive simplification can be invalid without sufficient assumptions."
 C3 = "User-defined rewrite rules can create nontermination."
 C4 = "Expression swell can dominate exact computation cost."
 C5 = "Numeric fallback can lose exact mathematical structure."
 C6 = "Branch handling can differ across systems and functions."
 C7 = "An unevaluated expression is not equivalent to computational failure."
 C8 = "Solver completeness varies by equation class and domain."
 C9 = "Serialization must preserve symbolic structure and assumptions."
 C10 = "Reproducibility depends on software, algorithm and rule-set versions."
}

SEMANTIC_EQUIVALENCE {
 "ComputerAlgebraSystem" <=> "CAS"
 "AbstractSyntaxTree" <=> "AST"
 "SymbolicKernel" <=> "SymbolicEngine"
 "Canonicalisation" <=> "Canonicalization"
 "SymbolicNumericComputation" <=> "HybridSymbolicNumericComputation"
}

VALIDATION {
 V1 = "CAS and kernel versions are documented."
 V2 = "Expression domain and assumptions are documented."
 V3 = "Input parsing preserves intended mathematical structure."
 V4 = "Simplification preserves equivalence under assumptions."
 V5 = "Symbolic solutions are substituted back where feasible."
 V6 = "Differentiation and integration results are checked by inverse operations where feasible."
 V7 = "Exact and numerical execution modes are clearly distinguished."
 V8 = "User-defined rules and extensions are documented."
 V9 = "Regression tests cover known identities and representative algorithms."
 V10 = "Results are reproducible under documented CAS, kernel, rule-set and library versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CAS_DEFINITION = 0.99
 EXPRESSION_REPRESENTATION_REASONING = 0.99
 SIMPLIFICATION_ENGINE_REASONING = 0.99
 SOLVER_REASONING = 0.98
 INTEGRATION_REASONING = 0.98
 REWRITE_REASONING = 0.99
 REPRODUCIBILITY_REASONING = 0.99
 POLICY = "Reduce confidence when CAS version, assumptions, rewrite policy, solver backend, numeric fallback or user-defined rules are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,CASReasoning,
  SymbolicEngineReasoning,ExpressionRepresentationReasoning,
  ParserReasoning,SimplificationEngineReasoning,
  RewriteEngineReasoning,AssumptionEngineReasoning,
  ExactArithmeticReasoning,PolynomialBackendReasoning,
  SolverBackendReasoning,DifferentiationBackendReasoning,
  IntegrationBackendReasoning,SymbolicNumericReasoning,
  EvaluationStrategyReasoning,VerificationReasoning,
  ReproducibilityReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "computer algebra system" -> ComputerAlgebraSystem
  "CAS" -> ComputerAlgebraSystem
  "AST" -> AbstractSyntaxTree
  "symbolic kernel" -> SymbolicEngine
  "canonicalization" -> Canonicalisation
 }
}

EXECUTION_RULES {
 E1 = "If software manipulates mathematical expressions symbolically, activate ComputerAlgebraSystemsOntology."
 E2 = "Record CAS version, kernel version and assumptions before interpreting exact output."
 E3 = "Parse and canonicalise expressions before backend dispatch."
 E4 = "Apply simplification only under valid domain and branch assumptions."
 E5 = "Preserve exact computation unless numerical fallback is explicitly requested."
 E6 = "Return unevaluated forms when exact algorithms do not support a case."
 E7 = "Treat user-defined rewrite rules as part of the computational environment."
 E8 = "Verify symbolic outputs with identities, substitution or inverse operations where feasible."
 E9 = "Document rule-set, algorithm and library versions for reproducibility."
}

RECURSION {
 ENTRY = ComputerAlgebraSystem
 EXPAND(ComputerAlgebraSystem) -> [
  CAS,SymbolicEngine,SymbolicKernel,SymbolicComputation,SymbolicMathematics,
  SymbolicAlgebra,Expression,ExpressionTree,AST,Parser,InternalRepresentation,
  CanonicalForm,SimplificationEngine,RewriteEngine,PatternMatching,
  AssumptionEngine,ExactArithmetic,BigInteger,RationalArithmetic,AlgebraicNumber,
  PolynomialArithmetic,PolynomialFactorisation,GroebnerBasis,RationalFunction,
  SymbolicEquationSolving,SymbolicDifferentiation,SymbolicIntegration,
  SymbolicLimit,SeriesExpansion,SymbolicSummation,SymbolicLinearAlgebra,
  SpecialFunction,UnevaluatedExpression,SymbolicNumericComputation,
  NumericApproximation,SolverBackend,AlgebraBackend,IntegrationBackend,
  DifferentiationBackend,PolynomialBackend,MatrixBackend,NumericBackend,
  UserDefinedFunction,UserDefinedRewriteRule,API,ExpressionSerialization,
  Caching,Memoisation,EvaluationStrategy,ExpressionSwell,
  Verification,RegressionTest,Reproducibility,Provenance
 ]
 EXPAND(ComputerAlgebraSystemFamilies) -> [
  ExpressionRepresentationAndParsing,
  SimplificationAndRewriteSystems,
  ExactArithmeticAndAlgebraBackends,
  SymbolicCalculusBackends,
  SymbolicSolvingAndPolynomialSystems,
  SymbolicNumericExecution,
  ExtensibilityAndEvaluationControl,
  VerificationVersioningAndReproducibility
 ]
 STOP_CONDITION = "Stop when representation, assumptions, backend, execution mode, verification and version metadata are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Symbolic Computation"
 RELATED_ONTOLOGIES = [
  "Symbolic Computation",
  "Symbolic Mathematics Ontology",
  "Symbolic Algebra Ontology",
  "Symbolic Differentiation Ontology",
  "Symbolic Integration Ontology",
  "Symbolic Manipulation Ontology",
  "Computational Algebra Ontology",
  "Polynomial Ontology",
  "Equation Ontology",
  "Computational Matrix Ontology",
  "Tensor Computation Ontology",
  "Numerical Methods Ontology 2",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Computer Algebra Systems -> Symbolic Computation for parent semantics",
  "Computer Algebra Systems -> Symbolic Mathematics for expression semantics",
  "Computer Algebra Systems -> Symbolic Algebra for algebraic backend semantics",
  "Computer Algebra Systems -> Symbolic Differentiation and Integration for calculus backends",
  "Computer Algebra Systems -> Symbolic Manipulation for rewrite engines",
  "Computer Algebra Systems -> Computational Algebra for exact algebraic algorithms",
  "Computer Algebra Systems -> Polynomial and Equation ontologies for solver domains",
  "Computer Algebra Systems -> Computational Matrix and Tensor ontologies for structured symbolic objects",
  "Computer Algebra Systems -> Numerical Methods for symbolic-numeric fallback relationships",
  "Computer Algebra Systems -> Computational Complexity for expression-growth and algorithm-cost semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible CAS capabilities, backends or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve symbolic-engine-versus-user-interface distinction.",
  "Preserve exact-versus-numeric execution distinction.",
  "Preserve expression-semantics-versus-internal-representation distinction.",
  "Preserve built-in-versus-user-defined rewrite distinction.",
  "Preserve unevaluated-expression-versus-error distinction.",
  "Preserve mathematical-equivalence-versus-output-form distinction.",
  "Preserve machine-readable CAS semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A computer algebra system is software for representing and manipulating mathematical expressions symbolically using exact algorithms, rewrite systems and symbolic-numeric methods."
 ENGINE_SUMMARY = "Symbolic kernels parse, represent, simplify and transform exact mathematical expressions."
 BACKEND_SUMMARY = "Algebra, polynomial, solver, differentiation, integration, matrix and numeric backends implement specialised algorithms."
 ASSUMPTION_SUMMARY = "Domain and branch assumptions govern valid simplification and solution semantics."
 EXTENSION_SUMMARY = "User-defined functions, rewrite rules, APIs and serialization extend CAS behaviour."
 VALIDATION_SUMMARY = "CAS results should be checked through inverse operations, substitution, regression tests, exact identities and reproducible version metadata."
}

FLOW {
 INPUT
 -> detect_computer_algebra_system_context
 -> identify_expression_domain_assumptions_target_operation_exactness_goal_and_output_form
 -> parse_input_into_internal_symbolic_representation
 -> canonicalise_expression_structure
 -> attach_domain_branch_and_assumption_context
 -> dispatch_to_algebra_polynomial_solver_differentiation_integration_series_matrix_or_numeric_backend
 -> execute_algorithm_and_rewrite_rules
 -> simplify_output_under_declared_assumptions
 -> if_exact_method_unavailable_preserve_unevaluated_form_or_use_declared_numeric_fallback
 -> verify_by_identity_substitution_inverse_operation_or_reference_result
 -> document_CAS_kernel_rule_set_algorithm_library_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-COMPUTER-ALGEBRA-SYSTEMS-ONTOLOGY-2-V1"
}

}

END_FLOW