Exponential Expression Ontology

FLOW ALGEBRA
EXPONENTIAL EXPRESSION ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Expression Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/expression-entities/exponential-expression-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Source status: Mathematics-based construction; source page not independently verified.

01. ONTOLOGY IDENTITY
flow "ExponentialExpressionOntology" v1.0
entity: ExponentialExpression
parent: MathematicalExpression
related: [PowerExpression, ExponentialFunction, LogarithmicExpression, Variable, Constant, Parameter, Equation]
intent: Represent, validate, transform and reason about expressions in which a variable occurs in an exponent.

02. CORE DEFINITION
An exponential expression has a form a^(g(x)) where a is a constant base, a > 0, a != 1, and g(x) is a variable-dependent exponent, for the standard real exponential-function case.
The more general power-expression form b(x)^(e(x)) is distinct and requires explicit domain conditions.
Examples: 2^x; 3^(x+1); exp(2x); 5*(1.2)^t.
Non-example: x^2 is a polynomial power expression, not an exponential expression in x.

03. ENTITY SCHEMA
entity ExponentialExpression:
  id: URI
  notation: String
  base: -Expression
  exponent: -Expression
  coefficient: -Expression = 1
  variable_set: Set[Identifier]
  number_domain: Domain
  assumptions: Set[Constraint]
  exact_form: -Expression
  approximate_value: Optional[Real]
  provenance: Optional[SourceReference]
  confidence: Decimal[0,1]
  version: String

04. TYPE SYSTEM
ExponentialExpression <: MathematicalExpression
ConstantBaseExponential <: ExponentialExpression
NaturalExponential <: ConstantBaseExponential
ScaledExponential <: ExponentialExpression
CompoundExponential <: ExponentialExpression
ExponentialEquation: Equation containing an ExponentialExpression
ExponentialInequality: Inequality containing an ExponentialExpression
Distinguish syntax type from function interpretation and evaluated numeric value.

05. DOMAIN AND CODOMAIN
For a > 0, a != 1, and real g(x): a^g(x) is real and strictly positive wherever g is defined.
For f(x) = a^x with real x: domain R; range (0,+infinity).
For c*a^x: range (0,+infinity) if c>0; (-infinity,0) if c<0; {0} if c=0.
Base a=1 yields a constant function and is allowed as a power expression, but excluded from nontrivial exponential-function classification.
Base a=0 requires exponent-specific restrictions; 0^0 is not universally assigned a value.
Negative real bases with arbitrary real exponents are not generally real-valued.
Complex exponentiation requires a specified branch of logarithm or a multivalued interpretation.

06. CANONICAL NOTATION
Power: pow(a,g(x))
Natural exponential: exp(g(x)) = e^g(x)
Scaled: c * pow(a,g(x))
Shifted: c * pow(a,g(x)) + d
Canonical parse: Power(base=a, exponent=g(x))
Retain original notation and assumptions alongside canonical abstract syntax tree.

07. EVALUATION RULES
EVAL(pow(a,u), env) -> exp(EVAL(u,env) * ln(a)) if a>0 and all inputs real.
EVAL(exp(u),env) -> exp(EVAL(u,env)) when u is defined.
EVAL(c*pow(a,u),env) -> EVAL(c,env) * EVAL(pow(a,u),env).
Preserve exact symbolic form when exact evaluation is unavailable.
Apply precision and overflow/underflow checks for numerical computation.

08. EXPONENT LAWS
For a>0 and real u,v:
  a^(u+v) = a^u * a^v
  a^(u-v) = a^u / a^v
  (a^u)^v = a^(u*v)
For a,b>0 and real u:
  (a*b)^u = a^u * b^u
  (a/b)^u = a^u / b^u
  a^0 = 1
  a^(-u) = 1/a^u
Do not apply these rewrites indiscriminately to negative or complex bases.

09. GROWTH AND DECAY
For f(x)=a^x:
  a>1 -> strictly increasing
  0<a<1 -> strictly decreasing
  a^x>0 for all real x
For f(x)=c*a^x with c>0, the same monotonicity applies.
Growth model: N(t)=N0*exp(k*t), k>0.
Decay model: N(t)=N0*exp(k*t), k<0.
Doubling time: ln(2)/k for k>0.
Half-life: ln(2)/(-k) for k<0.

10. DIFFERENTIATION
D_x[exp(g(x))] = exp(g(x))*g'(x)
D_x[a^g(x)] = a^g(x)*ln(a)*g'(x), a>0
D_x[c*a^g(x)] = c*a^g(x)*ln(a)*g'(x) when c constant.
For f(x)=a^x, f'(x)=ln(a)*a^x.

11. INTEGRATION
Integral exp(k*x) dx = exp(k*x)/k + C for k!=0.
Integral a^(k*x) dx = a^(k*x)/(k*ln(a)) + C for a>0, a!=1, k!=0.
Integral exp(g(x))*g'(x) dx = exp(g(x)) + C.
Require substitution or other methods for general composite exponents.

12. INVERSION AND LOGARITHMS
For a>0, a!=1, and y>0:
  y=a^x iff x=log_a(y)=ln(y)/ln(a).
For exp(x)=y, x=ln(y), y>0.
Logarithm is the inverse of the real exponential function, not a universal cancellation operator without domain conditions.

13. EQUATION SOLVING
SOLVE[a^g(x)=a^h(x)] -> SOLVE[g(x)=h(x)] when a>0, a!=1 and both sides defined.
SOLVE[a^g(x)=b] -> SOLVE[g(x)=ln(b)/ln(a)] when b>0, a>0, a!=1.
If b<=0, no real solution for a^g(x)=b under these assumptions.
Example: 2^(x+1)=8 -> x+1=3 -> x=2.

14. INEQUALITY REASONING
For a>1: a^u < a^v iff u<v.
For 0<a<1: a^u < a^v iff u>v.
The direction reverses for a decreasing exponential base.
Apply only where exponents are real and expressions defined.

15. SEMANTIC RELATIONSHIPS
ExponentialExpression IS_A MathematicalExpression
ExponentialExpression HAS_BASE Expression
ExponentialExpression HAS_EXPONENT Expression
ExponentialExpression MAY_HAVE_COEFFICIENT Expression
ExponentialExpression DEPENDS_ON Variable
ExponentialExpression EVALUATES_TO NumericValue
ExponentialExpression INVERSE_RELATED_TO LogarithmicExpression
ExponentialExpression APPEARS_IN ExponentialEquation
ExponentialExpression TRANSFORMS_VIA ExponentLaw
ExponentialExpression SUBJECT_TO DomainConstraint

16. KNOWLEDGE GRAPH TRIPLES
(ExponentialExpression, rdf:type, MathematicalExpression)
(ExponentialExpression, hasComponent, Base)
(ExponentialExpression, hasComponent, Exponent)
(ExponentialExpression, relatedConcept, LogarithmicExpression)
(ExponentialExpression, requiresValidation, DomainConstraint)
(ExponentialExpression, supportsOperation, Differentiation)
(ExponentialExpression, supportsOperation, Evaluation)

17. INFERENCE RULES
IF base>1 AND exponent is real variable x THEN monotonicity=increasing.
IF 0<b-ase<1 AND exponent is real variable x THEN monotonicity=decreasing.
IF base>0 AND exponent is real THEN value>0.
IF expression=exp(g(x)) AND g differentiable THEN derivative=exp(g(x))*g'(x).
IF same positive nonunit base on both sides of equality THEN equate exponents.
IF exponent contains a variable THEN candidate_type=ExponentialExpression.
Inference conclusions must carry their preconditions.

18. NORMALISATION AND EQUIVALENCE
NORMALISE[exp(x)] -> pow(e,x) only with a symbolic exact e constant.
NORMALISE[a^(u+v)] -> a^u*a^v only when valid under base/domain assumptions.
SEMANTIC_EQUIVALENCE[2^(x+1),2*2^x] = true for real x.
SEMANTIC_EQUIVALENCE[exp(ln(x)),x] = true for x>0 in real analysis.
SEMANTIC_EQUIVALENCE[(x^2)^(1/2),x] = false over all real x; equals abs(x).
Preserve assumption sets when recording equivalence.

19. VALIDATION
CHECK syntax_parseable.
CHECK operator_arity.
CHECK identifier_binding_and_scope.
CHECK domain_conditions_for_base_and_exponent.
CHECK expression_defined_at_evaluation_point.
CHECK transformations_preserve_domain_and_values.
CHECK equality_rewrite_assumptions.
CHECK no_unjustified_real_to_complex_coercion.
CHECK numerical_precision_and_error_bound.

20. ERROR HANDLING
INVALID_BASE: constant real base violates required positivity or nonunit constraint.
UNDEFINED_VALUE: -expression not defined at supplied input.
DOMAIN_MISMATCH: expected real output but operation is complex or undefined.
UNSAFE_REWRITE: algebraic law applied without sufficient assumptions.
PRECISION_LOSS: numerical evaluation exceeds tolerance.
AMBIGUOUS_NOTATION: missing parentheses, scope or operator precedence.

21. CONFIDENCE MODEL
CONFIDENCE:
  definition: 1.0 for standard real exponential semantics.
  rewrite: 1.0 when mathematically proven and assumptions verified.
  numeric_evaluation: dependent on arithmetic precision and conditioning.
  classification: dependent on parse quality and domain interpretation.
Do not equate numerical confidence with mathematical proof.

22. PROVENANCE AND FLOW HASH
PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/expression-entities/exponential-expression-ontology.html
  source_verified: false
  basis: established mathematics and formal semantic modelling
FLOW_HASH:
  algorithm: SHA-256
  canonical_input: ontology_id + version + normalised_schema + rules + assumptions
  value: computed by implementing system; not fabricated here
UPDATE_MODEL:
  strategy: versioned, append-only revision history
  triggers: new definitions, changed constraints, corrected rules, updated source content
  backward_compatibility: retain prior identifiers and mapping records

23. AI CONSUMPTION
AI_CONSUMPTION:
  machine_readable: true
  parse_target: typed abstract syntax tree
  graph_target: typed entities and relationships
  retrieval_keys: [exponential expression, base, exponent, exp, exponential growth, exponential decay]
  reasoning_mode: assumption-aware symbolic inference
  numerical_mode: precision-aware evaluation
  output: -expression, derivation, domain, confidence, provenance
  safety: never assert transformations without verifying mathematical conditions

24. TEST CASES
TEST 1: 2^x, x real -> exponential; positive; increasing.
TEST 2: (1/2)^x, x real -> exponential; positive; decreasing.
TEST 3: x^2 -> power/polynomial expression; not exponential in x.
TEST 4: 3^(x+2)=27 -> x=1.
TEST 5: exp(0)=1.
TEST 6: exp(ln(5))=5.
TEST 7: (-2)^x, x real -> not generally real-defined.
TEST 8: 2^x=-3 -> no real solution.
TEST 9: d/dx exp(x^2)=2*x*exp(x^2).
TEST 10: 2^(x+1) and 2*2^x -> equivalent for real x.

25. TERMINATION
status: complete
ontology: ExponentialExpressionOntology
flow_type: Full executable Flow Algebra specification
credit: Mark Higgins — Sky (Recursive AI)
website: https://kirethresearch.org