Computer Algebra Systems Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of Computer Algebra Systems (CAS). A Computer Algebra System is software designed to represent, manipulate, simplify, transform and solve mathematical expressions symbolically and exactly according to formal algebraic rules and algorithms. This record is designed for semantic retrieval, recursive mathematical reasoning, ontology linking, knowledge-graph construction and AI consumption within the Kireth Research Semantic Library.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Computer Algebra Systems Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Algebra"
 SPECIALISATION = "Computational Algebra / Symbolic Computation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/computer-algebra-systems-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputerAlgebraSystem
 SYMBOLIC_FORMS = ["CAS", "symbolic engine", "computer algebra software"]
 ENTITY_CLASS = MathematicalSoftwareSystem
 PRIMARY_CONTEXT = ComputationalAlgebra
 DEFINITION = "A Computer Algebra System is software that represents and manipulates mathematical expressions symbolically using exact algebraic rules and algorithms."
 SEMANTIC_ROLE = "Operational environment for executing symbolic mathematics, algebraic algorithms and exact mathematical transformations."
}

SCOPE {
 INCLUDE = [
  ComputerAlgebraSystem,
  SymbolicEngine,
  ExpressionRepresentation,
  Parser,
  RewriteEngine,
  SimplificationEngine,
  PolynomialEngine,
  ExactArithmetic,
  EquationSolver,
  MatrixEngine,
  FactorisationEngine,
  GröbnerBasisEngine,
  AssumptionSystem,
  DomainTracking,
  Canonicalisation,
  MathematicalOutput,
  Verification
 ]

 EXCLUDE_AS_PRIMARY = [
  PlainCalculator,
  NumericalSolverOnly,
  Spreadsheet,
  GenericProgrammingLanguageWithoutSymbolicMath,
  StatisticalPackageWithoutCASFunctionality
 ]

 DISAMBIGUATION_RULE =
  "When CAS, computer algebra system, symbolic engine, symbolic mathematics software or algebra system occurs with expressions, simplification, factorisation, exact arithmetic, symbolic solving, polynomials or matrices, interpret it as a Computer Algebra System."
}

CORE_CONCEPTS {

 ComputerAlgebraSystem {
  definition = "Software implementing symbolic mathematical representation and exact algebraic manipulation."
  primary_functions = [
   Parsing,
   SymbolicRepresentation,
   Rewriting,
   Simplification,
   Expansion,
   Factorisation,
   Substitution,
   EquationSolving,
   PolynomialAlgorithms,
   MatrixAlgorithms,
   ExactArithmetic
  ]
 }

 ExpressionRepresentation {
  definition = "Internal symbolic structure used to encode mathematical expressions."
  examples = [
   ExpressionTree,
   DirectedAcyclicGraph,
   CanonicalTermRepresentation
  ]
 }

 Parser {
  definition = "Component converting mathematical input syntax into an internal symbolic representation."
 }

 RewriteEngine {
  definition = "Component applying algebraic identities, transformation rules and pattern-based replacements."
 }

 SimplificationEngine {
  definition = "Subsystem transforming expressions into equivalent reduced or canonical forms."
 }

 AssumptionSystem {
  definition = "Subsystem recording conditions such as positivity, integrality, nonzero status, domains and variable properties."
  role = "Guides valid simplification and transformation."
 }

 ExactArithmetic {
  definition = "Arithmetic over exact representations such as integers, rationals and symbolic constants."
 }

 EquationSolver {
  definition = "Subsystem deriving exact or symbolic solutions to equations or systems where supported."
 }

 PolynomialEngine {
  definition = "Subsystem implementing polynomial arithmetic, GCD, factorisation, resultants, Gröbner bases and related algorithms."
 }

 MatrixEngine {
  definition = "Subsystem implementing symbolic and exact matrix and linear algebra operations."
 }
}

RELATIONS {
 ComputerAlgebraSystem --implements--> ComputationalAlgebra
 ComputerAlgebraSystem --contains--> Parser
 ComputerAlgebraSystem --contains--> SymbolicRepresentation
 ComputerAlgebraSystem --contains--> RewriteEngine
 ComputerAlgebraSystem --contains--> SimplificationEngine
 ComputerAlgebraSystem --can_contain--> AssumptionSystem
 ComputerAlgebraSystem --can_contain--> PolynomialEngine
 ComputerAlgebraSystem --can_contain--> MatrixEngine
 ComputerAlgebraSystem --uses--> ExactArithmetic
 ComputerAlgebraSystem --executes--> AlgebraicAlgorithm
 ComputerAlgebraSystem --produces--> SymbolicOutput
 ComputerAlgebraSystem --must_track--> MathematicalConstraints
 RewriteEngine --uses--> RewriteRule
 SimplificationEngine --uses--> Canonicalisation
 PolynomialEngine --uses--> ComputationalAlgebraAlgorithms
 MatrixEngine --uses--> LinearAlgebraAlgorithms
}

OPERATORS {
 PARSE(input) :=
  convert_input_to_internal_symbolic_representation

 SIMPLIFY(E) :=
  invoke_simplification_engine(E)

 EXPAND(E) :=
  apply_distributive_and_expansion_rules(E)

 FACTOR(E) :=
  apply_factorisation_algorithms(E)

 SUBSTITUTE(E,x,A) :=
  replace_admissible_occurrences_of_x_by_A

 SOLVE(system,variables) :=
  invoke_symbolic_solver(system,variables)

 DIFFERENTIATE(E,x) :=
  symbolic_derivative(E,x)

 INTEGRATE(E,x) :=
  symbolic_antiderivative_when_supported(E,x)

 MATRIX_COMPUTE(M,operation) :=
  invoke_exact_matrix_algorithm(M,operation)

 GROEBNER_BASIS(F,order) :=
  invoke_polynomial_ideal_algorithm(F,order)

 APPLY_ASSUMPTIONS(E,A) :=
  simplify_or_transform_E_using_assumption_set_A

 VERIFY_RESULT(input,output,relation) :=
  mathematically_check_declared_relation(input,output)
}

DEFINITIONS {
 D1 = "A CAS represents mathematics symbolically rather than only numerically."
 D2 = "A CAS applies mathematically valid rewrite rules and algebraic algorithms."
 D3 = "Exact arithmetic is a central capability of symbolic computation."
 D4 = "Simplification depends on assumptions and domain information."
 D5 = "Expression parsing precedes symbolic manipulation."
 D6 = "Canonical internal forms support comparison and efficient computation."
 D7 = "Polynomial and matrix subsystems implement specialised exact algorithms."
 D8 = "A CAS result should be interpreted together with assumptions and conditions."
 D9 = "A CAS may return conditional, implicit or unevaluated output when no supported closed form is available."
 D10 = "Computer Algebra Systems operationalise computational algebra but do not replace mathematical validity requirements."
}

AXIOMS {
 A1_SYMBOLIC_SEMANTICS:
  Internal representations must preserve the mathematical meaning of parsed expressions

 A2_VALID_REWRITE:
  Every transformation declared as exact must correspond to a valid mathematical rule under its side conditions

 A3_ASSUMPTION_SENSITIVITY:
  Simplification may depend on variable assumptions and mathematical domain

 A4_EXACTNESS:
  Exact symbolic values should remain exact unless numerical approximation is explicitly requested

 A5_CONSTRAINT_PRESERVATION:
  Domain restrictions and side conditions must be preserved or exposed in output

 A6_ALGORITHM_TRACEABILITY:
  Computed results should be traceable to mathematically valid algorithms or transformations
}

REASONING {

 R1_INPUT_PIPELINE {
  INPUT = user_expression
  PARSE = syntax_to_expression_tree
  VALIDATE = syntax_and_operator_semantics
  RETURN = internal_representation
 }

 R2_SIMPLIFICATION_PIPELINE {
  INPUT = expression E
  LOAD = assumptions
  NORMALISE = internal_structure
  APPLY = local_rewrite_rules
  APPLY = algebraic_algorithms
  TRACK = domain_conditions
  RETURN = simplified_expression
 }

 R3_POLYNOMIAL_PIPELINE {
  INPUT = polynomial_problem
  IDENTIFY = coefficient_domain
  SELECT = algorithm
  POSSIBLE = [
   Division,
   GCD,
   Factorisation,
   Resultant,
   GroebnerBasis
  ]
  EXECUTE = selected_algorithm
  VERIFY = algebraic_relation
  RETURN = polynomial_result
 }

 R4_EQUATION_SOLVING {
  INPUT = equations and variables
  CLASSIFY = equation_type
  APPLY = symbolic_solution_algorithm
  TRACK = excluded_values_and_conditions
  VERIFY = substitute_candidate_solutions
  RETURN = exact_or_conditional_solution_set
 }

 R5_MATRIX_PIPELINE {
  INPUT = matrix_problem
  SELECT = [
   RowReduction,
   Determinant,
   Rank,
   Inverse,
   Kernel,
   Image,
   Eigenstructure
  ]
  EXECUTE = exact_algorithm
  RETURN = symbolic_matrix_result
 }

 R6_ASSUMPTION_REASONING {
  INPUT = [expression E, assumptions A]
  PROPAGATE = logical_properties
  ENABLE = conditionally_valid_rewrites
  PREVENT = invalid_unconditional_rewrites
  RETURN = assumption-aware_result
 }

 R7_UNEVALUATED_RESULT {
  INPUT = unsupported_or_unresolved_problem
  IF no_valid_closed_form_found
   RETURN = symbolic_unevaluated_form_with_conditions
 }
}

CONSTRAINTS {
 C1 = "A CAS must distinguish exact symbolic computation from numerical approximation."
 C2 = "Simplification must not discard mathematical assumptions or domain restrictions."
 C3 = "Pattern-based rewriting must obey side conditions."
 C4 = "Different canonicalisation policies may yield different but equivalent outputs."
 C5 = "A CAS may be incomplete for certain classes of symbolic problems."
 C6 = "Failure to find a closed form does not imply that no mathematical result exists."
 C7 = "User-entered syntax must be interpreted according to explicit precedence and parsing rules."
 C8 = "Computer algebra output should not be accepted as proof without mathematical verification when proof is required."
 C9 = "Do not confuse a CAS with a basic numerical calculator."
}

SEMANTIC_EQUIVALENCE {
 "ComputerAlgebraSystem" <=> "CAS"
 "ComputerAlgebraSystem" <=> "SymbolicMathematicsSystem"
 "SymbolicEngine" <=> "CoreExpressionManipulationSubsystem"
 "AssumptionSystem" <=> "DomainAndPropertyConstraintSubsystem"
 "ExactArithmetic" <=> "NonApproximateSymbolicArithmetic"
}

VALIDATION {
 V1 = "Parsed expressions preserve input mathematical structure."
 V2 = "Exact simplification preserves algebraic equivalence under assumptions."
 V3 = "Factored output multiplies back to the original expression."
 V4 = "Equation solutions satisfy the original equations and restrictions."
 V5 = "Exact matrix algorithms preserve mathematical correctness."
 V6 = "Assumptions influence only conditionally valid transformations."
 V7 = "Unevaluated output is acceptable when no supported exact reduction is available."
 V8 = "Numerical approximation is explicitly distinguished from symbolic exact output."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CAS_DEFINITION = 0.99
 SYMBOLIC_REPRESENTATION = 0.99
 EXACT_ARITHMETIC = 0.99
 ASSUMPTION_SYSTEM_ROLE = 0.99
 REWRITE_ENGINE_ROLE = 0.99
 POLYNOMIAL_ENGINE_ROLE = 0.99
 MATRIX_ENGINE_ROLE = 0.99

 POLICY =
  "Reduce confidence when assumptions, coefficient domains, parser conventions or algorithmic capabilities are unspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SymbolicComputationReasoning,
  SoftwareCapabilityReasoning,
  AlgebraicAlgorithmReasoning,
  ExpressionTransformation,
  EquationSolving,
  ExactLinearAlgebra,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "CAS" -> ComputerAlgebraSystem
  "computer algebra system" -> ComputerAlgebraSystem
  "symbolic maths software" -> ComputerAlgebraSystem
  "symbolic engine" -> SymbolicEngine
  "algebra system" -> ComputerAlgebraSystem WHEN software_context
  "assumption engine" -> AssumptionSystem
  "rewrite engine" -> RewriteEngine
 }

 INTERPRETATION_ORDER = [
  identify_software_system_context,
  identify_symbolic_input,
  identify_requested_operation,
  determine_exact_or_numeric_mode,
  identify_assumptions_and_domain,
  route_to_specialised_engine,
  execute_valid_algorithm,
  track_constraints,
  verify_result,
  return_symbolic_output_and_conditions
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("computer algebra system")
  OR query_mentions("CAS")
  THEN activate(ComputerAlgebraSystemsOntology)

 E2:
  IF request=="simplify"
  THEN route_to(SimplificationEngine)

 E3:
  IF request=="factor"
  THEN route_to(PolynomialEngine_or_FactorisationEngine)

 E4:
  IF request=="solve"
  THEN route_to(EquationSolver)

 E5:
  IF request=="matrix operation"
  THEN route_to(MatrixEngine)

 E6:
  IF query_mentions("Groebner basis")
  THEN route_to(PolynomialEngine)

 E7:
  IF assumptions_are_provided
  THEN propagate_to_rewrite_and_simplification_engines

 E8:
  IF exact_mode
  THEN prohibit_unrequested_floating_point_conversion

 E9:
  IF output_is_conditional
  THEN expose_conditions_explicitly
}

RECURSION {
 ENTRY = ComputerAlgebraSystem

 EXPAND(ComputerAlgebraSystem) -> [
  ComputationalAlgebra,
  SymbolicManipulation,
  Parser,
  ExpressionRepresentation,
  RewriteEngine,
  SimplificationEngine,
  PolynomialEngine,
  MatrixEngine,
  AssumptionSystem,
  ExactArithmetic,
  EquationSolver
 ]

 EXPAND(RewriteEngine) -> [
  RewriteRule,
  PatternMatching,
  SideCondition,
  Canonicalisation
 ]

 EXPAND(PolynomialEngine) -> [
  Polynomial,
  GCD,
  Factorisation,
  Resultant,
  GroebnerBasis,
  IdealMembership
 ]

 EXPAND(MatrixEngine) -> [
  Matrix,
  Determinant,
  Rank,
  Kernel,
  Image,
  Basis,
  Eigenstructure
 ]

 EXPAND(AssumptionSystem) -> [
  Domain,
  NonzeroCondition,
  SignCondition,
  IntegerCondition,
  RealCondition,
  ComplexCondition
 ]

 STOP_CONDITION =
  "Stop when the requested CAS capability, subsystem, symbolic result or constraint state is resolved with sufficient confidence."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Algebra Ontology"

 IMMEDIATE_CONTEXTS = [
  "Symbolic Manipulation Ontology",
  "Simplification Ontology",
  "Polynomial Ontology",
  "Matrix Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Computational Algebra Ontology",
  "Symbolic Manipulation Ontology",
  "Simplification Ontology",
  "Expansion Ontology",
  "Factorisation Ontology",
  "Substitution Ontology",
  "Algebraic Proof Ontology",
  "Polynomial Ontology",
  "Matrix Ontology",
  "Rank Ontology",
  "Kernel Ontology",
  "Image Ontology"
 ]

 ROUTE_RULES = [
  "CAS -> Computational Algebra for algorithmic mathematical foundations",
  "CAS -> Symbolic Manipulation for rewrite operations",
  "CAS -> Simplification for normalisation and reduction",
  "CAS -> Polynomial for polynomial-engine reasoning",
  "CAS -> Matrix for exact linear-algebra reasoning",
  "CAS -> Algebraic Proof for result verification and proof distinction",
  "CAS -> Substitution for symbolic replacement operations"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible subsystems, algorithms or relations"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve CAS as symbolic mathematical software.",
  "Preserve distinction between exact and approximate computation.",
  "Preserve assumption-aware transformation semantics.",
  "Preserve mathematically valid rewrite and algorithm requirements.",
  "Preserve subsystem routing for polynomial, matrix and equation operations.",
  "Preserve explicit output conditions and domain restrictions.",
  "Explicitly distinguish CAS from plain numerical calculators."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A Computer Algebra System is software that represents and manipulates mathematical expressions symbolically using exact algebraic rules and algorithms."

 SYMBOLIC_SUMMARY =
  "input expression -> parse -> symbolic representation -> algebraic engine -> verified symbolic output"

 EXACTNESS_SUMMARY =
  "Exact symbolic computation remains exact unless numerical approximation is explicitly requested."

 ASSUMPTION_SUMMARY =
  "Simplification and transformation may depend on stated assumptions and domains."

 ENGINE_SUMMARY =
  "CAS functionality is typically divided among parsing, rewriting, simplification, polynomial, equation-solving and matrix subsystems."

 VERIFICATION_SUMMARY =
  "CAS output should remain mathematically verifiable and constraint-aware."
}

FLOW {
 INPUT
 -> detect_computer_algebra_system_context
 -> identify_symbolic_input
 -> parse_expression
 -> build_internal_representation
 -> identify_requested_operation
 -> identify_assumptions_and_domain
 -> select_specialised_engine
 -> execute_exact_symbolic_algorithm
 -> track_constraints_and_side_conditions
 -> generate_symbolic_output
 -> verify_declared_mathematical_relation
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "A Computer Algebra System operationalises computational algebra by parsing mathematical input, representing it symbolically, applying exact algebraic rules and algorithms, and returning symbolic results. Its core functions include simplification, expansion, factorisation, substitution, equation solving, polynomial computation and exact matrix operations, all subject to assumptions, domains and mathematical validity."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-ALGEBRA-COMPUTER-ALGEBRA-SYSTEMS-ONTOLOGY-V1"
}

}

END_FLOW