Symbolic Manipulation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of symbolic manipulation within algebra. Symbolic manipulation transforms mathematical expressions through valid algebraic rules while preserving the intended semantics, equivalence, implication structure, domains and constraints. It unifies operations such as simplification, substitution, expansion, factorisation, rearrangement and canonicalisation. 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 = "Symbolic Manipulation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Algebra"
 SPECIALISATION = "Symbolic Algebra"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/symbolic-manipulation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SymbolicManipulation
 SYMBOLIC_FORMS = ["E -> E'", "rewrite(E)", "transform(E)", "normalise(E)"]
 ENTITY_CLASS = RuleBasedMathematicalTransformation
 PRIMARY_CONTEXT = Algebra
 DEFINITION = "Symbolic manipulation is the rule-governed transformation of mathematical expressions into new forms while preserving the relevant mathematical meaning."
 SEMANTIC_ROLE = "Provides the operational layer for algebraic reasoning, rewriting, simplification, proof and equation transformation."
}

SCOPE {
 INCLUDE = [
  SymbolicManipulation,
  RewriteRule,
  AlgebraicTransformation,
  Simplification,
  Expansion,
  Factorisation,
  Substitution,
  CollectionOfLikeTerms,
  Rearrangement,
  Canonicalisation,
  EquationTransformation,
  IdentityApplication,
  ExpressionNormalisation,
  DomainConstraintTracking,
  EquivalenceTracking,
  ImplicationTracking
 ]

 EXCLUDE_AS_PRIMARY = [
  PureNumericalApproximation,
  StringReplacementWithoutMathematics,
  UnjustifiedTransformation,
  RandomExpressionMutation
 ]

 DISAMBIGUATION_RULE =
  "When symbolic manipulation occurs with algebraic expressions, equations, identities, substitutions, simplification, factorisation, expansion or proofs, interpret it as rule-based mathematical expression transformation."
}

CORE_CONCEPTS {

 SymbolicManipulation {
  definition = "Transformation of symbolic mathematical structures according to valid algebraic or logical rules."
  correctness_condition = "Every transformation must preserve the intended relation and required constraints."
 }

 RewriteRule {
  definition = "A directed or bidirectional rule replacing one symbolic pattern with another under stated conditions."
  form = "pattern -> replacement"
 }

 AlgebraicTransformation {
  definition = "A symbolic operation that changes expression form while respecting algebraic validity."
 }

 EquivalencePreservingTransformation {
  definition = "A transformation E -> E' such that E and E' have the same value over the preserved domain."
 }

 ImplicationPreservingTransformation {
  definition = "A transformation where truth of the original implies truth of the result, but reversibility is not guaranteed."
 }

 Canonicalisation {
  definition = "Transformation to a selected normal form so equivalent expressions are represented consistently."
 }

 DomainConstraintTracking {
  definition = "Preservation and propagation of conditions required for symbolic transformations to remain valid."
 }

 PatternMatching {
  definition = "Recognition of subexpressions matching the left-hand side of a symbolic rule."
 }
}

RELATIONS {
 SymbolicManipulation --acts_on--> MathematicalExpression
 SymbolicManipulation --uses--> RewriteRule
 SymbolicManipulation --can_preserve--> Equivalence
 SymbolicManipulation --can_preserve--> Implication
 SymbolicManipulation --must_respect--> DomainConstraints
 SymbolicManipulation --can_include--> Simplification
 SymbolicManipulation --can_include--> Expansion
 SymbolicManipulation --can_include--> Factorisation
 SymbolicManipulation --can_include--> Substitution
 SymbolicManipulation --can_include--> Rearrangement
 SymbolicManipulation --can_produce--> CanonicalForm
 RewriteRule --requires--> PatternMatching
 RewriteRule --can_have--> SideCondition
 AlgebraicIdentity --can_define--> BidirectionalRewriteRule
 EquationTransformation --is_special_case_of--> SymbolicManipulation
}

OPERATORS {
 APPLY_RULE(E,R) :=
  replace_matching_subexpression_according_to_R

 REWRITE(E,RULESET) :=
  repeatedly_apply_valid_rules_under_strategy

 SIMPLIFY(E) :=
  rewrite_to_simpler_equivalent_form(E)

 EXPAND(E) :=
  distribute_and_resolve_grouped_products(E)

 FACTOR(E) :=
  rewrite_as_equivalent_product(E)

 SUBSTITUTE(E,x,A) :=
  replace_admissible_occurrences_of_x_by_A

 CANONICALISE(E,NORMAL_FORM) :=
  rewrite_to_selected_normal_form(E)

 TRACK_DOMAIN(E,R) :=
  update_domain_constraints_after_rule_application

 CLASSIFY_RELATION(E,E') :=
  determine(Equivalence,Implication,ReverseImplication,Unknown)

 VERIFY_TRANSFORMATION(E,E',R) :=
  validate_rule_application_and_side_conditions
}

DEFINITIONS {
 D1 = "Symbolic manipulation changes representation without arbitrary change of mathematical meaning."
 D2 = "Rewrite rules may be bidirectional or directional."
 D3 = "Equivalent rewriting preserves value over a specified domain."
 D4 = "Some symbolic operations preserve only implication rather than equivalence."
 D5 = "Domain restrictions and side conditions are part of the semantic state."
 D6 = "Simplification, expansion, factorisation and substitution are specialised symbolic manipulations."
 D7 = "Canonical forms support comparison, indexing and automated reasoning."
 D8 = "Pattern matching identifies where rules may be applied."
 D9 = "A valid symbolic engine must distinguish syntax from semantics."
 D10 = "Transformation order can affect intermediate form even when final semantic equivalence is preserved."
}

AXIOMS {
 A1_VALID_RULE:
  IF rule R is valid under condition C
  AND C holds
  THEN applying R preserves its declared semantic relation

 A2_IDENTITY_REWRITE:
  IF A ≡ B
  THEN A may be replaced by B in an admissible algebraic context

 A3_EQUIVALENCE_TRANSITIVITY:
  IF E1 ≡ E2 AND E2 ≡ E3
  THEN E1 ≡ E3

 A4_DOMAIN_PRESERVATION:
  Symbolic transformations must preserve or explicitly update relevant domain constraints

 A5_SIDE_CONDITIONS:
  A rewrite requiring a side condition is invalid if the side condition is not established

 A6_CANONICALISATION:
-
  Expressions equivalent under the chosen theory may be mapped to the same canonical representation
}

REASONING {

 R1_RULE_APPLICATION {
  INPUT = [expression E, rewrite rule R]
  MATCH = find_pattern(R.left,E)
  CHECK = side_conditions(R)
  IF match AND conditions_hold
   APPLY = replacement
   UPDATE = domain_and_semantic_relation
   RETURN = transformed_expression
  ELSE
   RETURN = RuleNotApplicable
 }

 R2_SIMPLIFICATION_STRATEGY {
  INPUT = E
  DETECT = redundant_structure
  APPLY = identities, collection, cancellation, normalisation
  PRESERVE = domain
  RETURN = simpler_equivalent_expression
 }

 R3_EXPANSION_STRATEGY {
  INPUT = E
  DETECT = products_of_sums_or_powers
  APPLY = distributive_rules
  COLLECT = like_terms_when_requested
  RETURN = expanded_equivalent_expression
 }

 R4_FACTORISATION_STRATEGY {
  INPUT = E
  DETECT = common_factors_and_patterns
  APPLY = factorisation_identities
  VERIFY = expansion
  RETURN = factored_equivalent_expression
 }

 R5_SUBSTITUTION_STRATEGY {
  INPUT = [E, substitution mapping]
  CHECK = scope_and_domain
  APPLY = symbolic_replacement
  RETURN = substituted_expression
 }

 R6_EQUATION_TRANSFORMATION {
  INPUT = equation LHS=RHS
  APPLY = same_valid_operation_to_both_sides
  CLASSIFY = equivalence_or_implication
  TRACK = restrictions
  RETURN = transformed_equation
 }

 R7_CANONICALISATION {
  INPUT = E
  SELECT = target_normal_form
  APPLY = deterministic_or_confluent_rule_strategy
  RETURN = canonical_representation
 }

 R8_TRANSFORMATION_CHAIN {
  INPUT = sequence E0 -> E1 -> ... -> En
  FOR each step
   VERIFY = rule_and_conditions
   COMPOSE = semantic_relation
  RETURN = chain_validity_and_final_relation
 }
}

CONSTRAINTS {
 C1 = "Every symbolic transformation must be justified by a valid rule."
 C2 = "Domain restrictions must not be silently discarded."
 C3 = "Division or cancellation by an expression requires a nonzero condition."
 C4 = "Squaring, rooting and similar operations may change logical equivalence and must be classified correctly."
 C5 = "Substitution must respect variable scope and binding."
 C6 = "Pattern matching must distinguish structurally different expressions."
 C7 = "A shorter form is not necessarily a semantically valid form."
 C8 = "Canonicalisation requires an explicit or implicit normal-form policy."
 C9 = "Rewrite loops should be prevented or controlled by strategy."
 C10 = "Do not treat raw textual replacement as mathematical symbolic manipulation."
}

SEMANTIC_EQUIVALENCE {
 "SymbolicManipulation" <=> "RuleBasedMathematicalRewriting"
 "Simplification" <=> "GoalDirectedEquivalentRewrite"
 "Expansion" <=> "DistributiveEquivalentRewrite"
 "Factorisation" <=> "ProductFormEquivalentRewrite"
 "Substitution" <=> "AdmissibleSymbolicReplacement"
 "Canonicalisation" <=> "NormalFormTransformation"
}

VALIDATION {
 V1 = "Every rewrite step is justified."
 V2 = "Algebraic identities can act as bidirectional rewrite rules."
 V3 = "Cancellation preserves domain exclusions."
 V4 = "Transformation chains preserve the composed semantic relation."
 V5 = "Substitution checks scope and admissibility."
 V6 = "Canonicalisation follows a defined normal-form strategy."
 V7 = "Implication-only operations are not incorrectly labelled as equivalences."
 V8 = "Rewrite strategies avoid uncontrolled cycles."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_MANIPULATION_DEFINITION = 0.99
 REWRITE_RULE_SEMANTICS = 0.99
 EQUIVALENCE_TRACKING = 0.99
 DOMAIN_TRACKING = 0.99
 PATTERN_MATCHING_ROLE = 0.99
 CANONICALISATION_ROLE = 0.98
 TRANSFORMATION_CHAIN_VALIDITY = 0.99

 POLICY =
  "Reduce confidence when rewrite rules, domains, side conditions or the intended semantic relation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SymbolicAlgebraReasoning,
  AutomatedRewriting,
  EquationReasoning,
  ProofReasoning,
  Simplification,
  Canonicalisation,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "symbolic manipulation" -> SymbolicManipulation
  "rewrite" -> RewriteRule
  "algebraic manipulation" -> SymbolicManipulation
  "transform expression" -> AlgebraicTransformation
  "normalise expression" -> Canonicalisation
  "canonicalise" -> Canonicalisation
  "apply identity" -> IdentityApplication
  "rearrange equation" -> EquationTransformation
 }

 INTERPRETATION_ORDER = [
  identify_symbolic_object,
  identify_target_goal,
  identify_available_rules,
  identify_domain_and_constraints,
  pattern_match_candidate_rules,
  verify_side_conditions,
  apply_valid_transformation,
  classify_equivalence_or_implication,
  update_domain_state,
  recurse_until_goal_or_normal_form,
  verify_transformation_chain,
  return_semantic_result
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("symbolic manipulation")
  OR query_mentions("algebraic manipulation")
  THEN activate(SymbolicManipulationOntology)

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

 E3:
  IF request=="expand"
  THEN route_to(ExpansionOntology)

 E4:
  IF request=="factor"
  THEN route_to(FactorisationOntology)

 E5:
  IF request=="substitute"
  THEN route_to(SubstitutionOntology)

 E6:
  IF equation_transformation_context
  THEN track_equivalence_and_side_conditions

 E7:
  IF rule_requires_nonzero_condition
  THEN require_condition_before_application

 E8:
  IF request=="canonical form"
  THEN activate_normalisation_strategy

 E9:
  IF transformation_chain_supplied
  THEN verify_each_step_and_composed_relation
}

RECURSION {
 ENTRY = SymbolicManipulation

 EXPAND(SymbolicManipulation) -> [
  AlgebraicExpression,
  RewriteRule,
  AlgebraicIdentity,
  Simplification,
  Expansion,
  Factorisation,
  Substitution,
  EquationTransformation,
  CanonicalForm,
  DomainRestriction
 ]

 EXPAND(RewriteRule) -> [
  Pattern,
  Replacement,
  SideCondition,
  Direction,
  SemanticRelation
 ]

 EXPAND(EquationTransformation) -> [
  Equation,
  Equality,
  Equivalence,
  Implication,
  Constraint
 ]

 EXPAND(Canonicalisation) -> [
  NormalForm,
  OrderingRule,
  Simplification,
  Confluence,
  Termination
 ]

 STOP_CONDITION =
  "Stop when the requested symbolic goal is reached, all transformations are verified, and semantic relations plus domain constraints are resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Algebra Ontology"

 IMMEDIATE_CONTEXTS = [
  "Algebraic Expression Ontology",
  "Algebraic Identity Ontology",
  "Simplification Ontology",
  "Expansion Ontology",
  "Factorisation Ontology",
  "Substitution Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Algebraic Expression Ontology",
  "Equation Ontology",
  "Linear Equation Ontology",
  "Polynomial Ontology",
  "Monomial Ontology",
  "Rational Expression Ontology",
  "Algebraic Identity Ontology",
  "Factorisation Ontology",
  "Expansion Ontology",
  "Simplification Ontology",
  "Substitution Ontology",
  "Algebraic Proof Ontology"
 ]

 ROUTE_RULES = [
  "Symbolic Manipulation -> Simplification for reduced forms",
  "Symbolic Manipulation -> Expansion for distributive rewrites",
  "Symbolic Manipulation -> Factorisation for product decompositions",
  "Symbolic Manipulation -> Substitution for replacement operations",
  "Symbolic Manipulation -> Algebraic Identity for equivalence rules",
  "Symbolic Manipulation -> Algebraic Proof for verified derivation chains",
  "Symbolic Manipulation -> Equation for relation-preserving transformations"
 ]
}

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

 RULES = [
  "Preserve rule-based transformation semantics.",
  "Preserve distinction between equivalence and implication.",
  "Preserve domain and side-condition tracking.",
  "Preserve scope-aware substitution.",
  "Preserve canonicalisation as policy-dependent normalisation.",
  "Preserve validation of transformation chains.",
  "Explicitly distinguish symbolic manipulation from raw textual replacement."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Symbolic manipulation is the rule-based transformation of mathematical expressions while preserving the intended algebraic or logical semantics."

 REWRITE_SUMMARY =
  "pattern -> replacement, subject to validity conditions"

 EQUIVALENCE_SUMMARY =
  "Equivalent rewrites preserve value over the maintained domain."

 IMPLICATION_SUMMARY =
  "Some transformations preserve only implication and must be labelled accordingly."

 DOMAIN_SUMMARY =
  "Every transformation carries forward or updates its mathematical constraints."

 CANONICALISATION_SUMMARY =
  "Canonicalisation rewrites equivalent expressions into a selected normal form."

 CHAIN_SUMMARY =
  "A symbolic derivation is valid only when every transformation step is valid."
}

FLOW {
 INPUT
 -> detect_symbolic_manipulation_context
 -> identify_expression_equation_or_symbolic_object
 -> identify_target_goal
 -> identify_available_rewrite_rules
 -> identify_domain_and_side_conditions
 -> pattern_match_rules
 -> select_transformation_strategy
 -> verify_rule_applicability
 -> apply_symbolic_transformation
 -> classify_semantic_relation
 -> update_constraints
 -> recursively_continue_toward_goal
 -> verify_complete_transformation_chain
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Symbolic manipulation is the operational core of algebraic reasoning. It transforms mathematical expressions by justified rules, including simplification, expansion, factorisation, substitution and equation rearrangement. Correct symbolic manipulation must track equivalence, implication, domain conditions, scope and side constraints, enabling reliable automated and human mathematical reasoning."
}

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

}

END_FLOW