AI Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the AI Mathematics Ontology within the Kireth Research Semantic Library. It formalises the mathematical foundations used by artificial intelligence systems, including learning objectives, optimisation, probability, statistics, linear algebra, calculus, information theory, representation learning, reinforcement learning, graph learning, uncertainty and computational complexity.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "AI Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Artificial Intelligence Mathematics / Core AI Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/artificial-intelligence-mathematics/ai-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = AIMathematics
 ENTITY_CLASS = FormalMathematicalAIFoundationsOntology
 PRIMARY_CONTEXT = ArtificialIntelligenceMathematics
 DEFINITION = "The AI Mathematics Ontology is a formal semantic framework for the mathematical structures, optimisation procedures, probabilistic models and decision rules used by artificial intelligence systems."
 SYMBOLIC_FORMS = [
  "y_hat=f_theta(x)",
  "theta*=argmin_theta L(theta)",
  "P(y|x)=P(x,y)/P(x)",
  "Q(s,a)=E[R_t|s_t=s,a_t=a]"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for learning, optimisation, inference, neural computation, information measures, representation, sequential decision-making and mathematical validation of AI systems."
}

SCOPE {
 INCLUDE = [
  AIMathematics,
  ArtificialIntelligenceMathematics,
  MachineLearningMathematics,
  StatisticalLearning,
  SupervisedLearning,
  UnsupervisedLearning,
  SelfSupervisedLearning,
  ReinforcementLearning,
  NeuralNetworkMathematics,
  DeepLearningMathematics,
  LinearModel,
  NonlinearModel,
  Parameter,
  Hyperparameter,
  LossFunction,
  ObjectiveFunction,
  RiskFunction,
  EmpiricalRisk,
  ExpectedRisk,
  Regularisation,
  Generalisation,
  BiasVarianceTradeoff,
  Overfitting,
  Underfitting,
  Optimisation,
  GradientDescent,
  StochasticGradientDescent,
  Momentum,
  AdaptiveOptimisation,
  ConvexOptimisation,
  NonConvexOptimisation,
  ConstraintOptimisation,
  LinearAlgebra,
  Vector,
  Matrix,
  Tensor,
  Eigenvalue,
  SingularValueDecomposition,
  Calculus,
  Gradient,
  Jacobian,
  Hessian,
  ChainRule,
  Backpropagation,
  Probability,
  ConditionalProbability,
  BayesRule,
  RandomVariable,
  ProbabilityDistribution,
  Expectation,
  Variance,
  Covariance,
  Likelihood,
  MaximumLikelihood,
  MaximumAPosteriori,
  BayesianInference,
  Entropy,
  CrossEntropy,
  KLDivergence,
  MutualInformation,
  InformationTheory,
  Classification,
  Regression,
  Clustering,
  DimensionalityReduction,
  PrincipalComponentAnalysis,
  KernelMethod,
  SupportVectorMachine,
  GaussianProcess,
  DecisionTree,
  EnsembleMethod,
  GraphLearning,
  GraphNeuralNetwork,
  Attention,
  TransformerMathematics,
  Embedding,
  RepresentationLearning,
  LatentVariableModel,
  Autoencoder,
  GenerativeModel,
  VariationalInference,
  MarkovDecisionProcess,
  Policy,
  ValueFunction,
  QFunction,
  BellmanEquation,
  Exploration,
  Exploitation,
  Search,
  HeuristicSearch,
  ComputationalComplexity,
  SampleComplexity,
  NumericalStability,
  Robustness,
  UncertaintyQuantification,
  Calibration,
  ModelValidation
 ]
 EXCLUDE_AS_PRIMARY = [
  AIProductEngineeringWithoutMathematicalModel,
  PureSoftwareImplementationWithoutMathematicalStructure,
  HumanFactorsWithoutQuantitativeAIModel,
  DescriptiveAITaxonomyWithoutMathematicalReasoning
 ]
 DISAMBIGUATION_RULE = "When the main subject is the mathematical structure, learning objective, probabilistic model, optimisation procedure or decision rule underlying an AI system, interpret the context as AI mathematics."
}

CORE_CONCEPTS {
 AIMathematics = "Formal mathematical foundations of artificial intelligence."
 MachineLearningMathematics = "Mathematical methods for learning functions or representations from data."
 LossFunction = "Function quantifying prediction or decision error."
 EmpiricalRisk = "Average loss evaluated on a finite observed sample."
 Generalisation = "Ability of a learned model to perform beyond its training data."
 Optimisation = "Mathematical search for parameters or decisions that minimise or maximise an objective."
 ProbabilisticInference = "Inference over uncertain variables using probability distributions."
 RepresentationLearning = "Learning mathematical representations useful for subsequent tasks."
 ReinforcementLearning = "Sequential decision learning through interaction and reward."
 ComputationalComplexity = "Mathematical analysis of computational resource requirements."
}

RELATIONS {
 AIMathematics --is_child_of--> ArtificialIntelligenceMathematics
 AIMathematics --uses--> LinearAlgebra
 AIMathematics --uses--> Calculus
 AIMathematics --uses--> Probability
 AIMathematics --uses--> Statistics
 AIMathematics --uses--> Optimisation
 AIMathematics --uses--> InformationTheory
 MachineLearningMathematics --uses--> LossFunction
 MachineLearningMathematics --requires--> Generalisation
 NeuralNetworkMathematics --uses--> Backpropagation
 Backpropagation --uses--> ChainRule
 GradientDescent --minimises--> ObjectiveFunction
 BayesianInference --uses--> BayesRule
 CrossEntropy --is_related_to--> KLDivergence
 PrincipalComponentAnalysis --uses--> EigenDecomposition
 SupportVectorMachine --uses--> ConvexOptimisation
 GaussianProcess --uses--> ProbabilityDistribution
 TransformerMathematics --uses--> Attention
 GraphNeuralNetwork --uses--> GraphStructure
 ReinforcementLearning --uses--> MarkovDecisionProcess
 ValueFunction --satisfies--> BellmanEquation
 Calibration --assesses--> PredictiveProbabilityQuality
 ModelValidation --assesses--> Generalisation
}

OPERATORS {
 PREDICTION(theta,x) := y_hat=f_theta(x)
 EMPIRICAL_RISK(theta,D) := R_hat=(1/n)*sum_i L(f_theta(x_i),y_i)
 EXPECTED_RISK(theta) := R=E[L(f_theta(X),Y)]
 REGULARISED_OBJECTIVE(theta,lambda,Omega) := J=R_hat+lambda*Omega(theta)
 GRADIENT_DESCENT(theta,eta,J) := theta_next=theta-eta*grad(J)
 SGD(theta,eta,g_hat) := theta_next=theta-eta*g_hat
 SOFTMAX(z_i) := exp(z_i)/sum_j exp(z_j)
 CROSS_ENTROPY(p,q) := H(p,q)=-sum_i p_i*log(q_i)
 KL_DIVERGENCE(p,q) := D_KL(p||q)=sum_i p_i*log(p_i/q_i)
 BAYES_RULE(H,D) := P(H|D)=P(D|H)*P(H)/P(D)
 MAXIMUM_LIKELIHOOD(theta,D) := theta_ML=argmax_theta P(D|theta)
 MAP_ESTIMATE(theta,D) := theta_MAP=argmax_theta P(D|theta)*P(theta)
 PCA(X,k) := project_onto_top_k_eigenvectors_of_covariance
 BELLMAN_OPTIMALITY_Q(s,a) := Q_star(s,a)=E[r+gamma*max_a2 Q_star(s_prime,a2)]
 ATTENTION(Q,K,V) := softmax(Q*K^T/sqrt(d_k))*V
}

AXIOMS {
 A1 = "AI mathematical models require explicit input, output and parameter spaces."
 A2 = "Learning objectives require explicit loss, risk, reward or information criteria."
 A3 = "Optimisation behaviour depends on objective geometry, constraints and algorithm."
 A4 = "Probabilistic AI models require explicit distributions and conditioning assumptions."
 A5 = "Training performance and generalisation performance are distinct."
 A6 = "Regularisation and data distribution influence learned-model generalisation."
 A7 = "Sequential decision models require explicit state, action, transition and reward semantics."
 A8 = "AI mathematical outputs require numerical, statistical and computational validation."
}

REASONING {
 R1_CONTEXT = identify(AI_task,data,input_space,output_space,model_family,objective)
 R2_MODEL_CLASS = classify(supervised,unsupervised,self_supervised,generative,reinforcement,graph,probabilistic)
 R3_REPRESENTATION = define(features,embeddings,latent_variables,graph_or_tensor_structure)
 R4_OBJECTIVE = formulate(loss,risk,regularisation,constraints,reward_or_information_measure)
 R5_OPTIMISATION = select(gradient,stochastic,convex,nonconvex,second_order_or_search_method)
 R6_INFERENCE = compute(predictions,posterior,decision,value,policy_or_embedding)
 R7_GENERALISATION = analyse(training_error,test_error,capacity,regularisation,distribution_shift)
 R8_UNCERTAINTY = quantify(predictive_uncertainty,calibration,posterior_or_ensemble_variability)
 R9_COMPLEXITY = analyse(time,space,sample_and_model_complexity)
 R10_VALIDATION = check(numerical_stability,statistical_validity,robustness,reproducibility)
}

CONSTRAINTS {
 C1 = "Low training loss does not imply good generalisation."
 C2 = "Non-convex objectives may contain multiple minima, saddle points or flat regions."
 C3 = "Gradient methods require differentiability or suitable generalised derivatives."
 C4 = "Probabilistic predictions are meaningful only relative to declared model assumptions."
 C5 = "Calibration and discrimination are distinct predictive properties."
 C6 = "Embedding geometry depends on the learned representation and objective."
 C7 = "Reinforcement-learning values depend on policy, environment and discount assumptions."
 C8 = "Attention weights should not automatically be interpreted as causal explanations."
 C9 = "Computational cost can dominate statistically elegant but expensive methods."
 C10 = "Distribution shift can invalidate assumptions made during training."
}

SEMANTIC_EQUIVALENCE {
 "AIMathematics" <=> "ArtificialIntelligenceMathematicsCore"
 "MachineLearningMathematics" <=> "MathematicsOfMachineLearning"
 "EmpiricalRisk" <=> "TrainingRisk"
 "RepresentationLearning" <=> "LearnedRepresentationMathematics"
 "ReinforcementLearning" <=> "SequentialDecisionLearning"
}

VALIDATION {
 V1 = "Input, output and parameter spaces are explicit."
 V2 = "Loss or reward function is explicit."
 V3 = "Optimisation method and stopping criteria are documented."
 V4 = "Training and evaluation data roles are separated."
 V5 = "Generalisation is assessed on independent or properly resampled data."
 V6 = "Probability estimates are calibrated where relevant."
 V7 = "Numerical stability is checked."
 V8 = "Computational complexity is assessed where material."
 V9 = "Sensitivity to hyperparameters and initialisation is evaluated where relevant."
 V10 = "Model assumptions and failure modes are documented."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 AI_MATHEMATICS_ONTOLOGY_DEFINITION = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 OPTIMISATION_REASONING = 0.99
 PROBABILISTIC_REASONING = 0.98
 REPRESENTATION_REASONING = 0.98
 GENERALISATION_REASONING = 0.98
 REINFORCEMENT_REASONING = 0.98
 COMPLEXITY_REASONING = 0.98
 VALIDATION_REASONING = 0.99
 POLICY = "Reduce confidence when task definition, data distribution, model family, objective, optimisation method, uncertainty assumptions, environment dynamics or validation evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AIMathematicsReasoning,
  MachineLearningMathematicsReasoning,
  StatisticalLearningReasoning,
  NeuralNetworkReasoning,
  DeepLearningReasoning,
  OptimisationReasoning,
  BackpropagationReasoning,
  ProbabilisticInferenceReasoning,
  BayesianReasoning,
  InformationTheoryReasoning,
  ClassificationReasoning,
  RegressionReasoning,
  ClusteringReasoning,
  DimensionalityReductionReasoning,
  KernelMethodReasoning,
  GaussianProcessReasoning,
  GraphLearningReasoning,
  TransformerReasoning,
  RepresentationLearningReasoning,
  GenerativeModelReasoning,
  ReinforcementLearningReasoning,
  BellmanReasoning,
  ComputationalComplexityReasoning,
  UncertaintyReasoning,
  ModelValidationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "AI mathematics" -> AIMathematics
  "artificial intelligence mathematics" -> AIMathematics
  "machine learning mathematics" -> MachineLearningMathematics
  "statistical learning" -> StatisticalLearning
  "representation learning" -> RepresentationLearning
  "reinforcement learning" -> ReinforcementLearning
  "information theory" -> InformationTheory
  "backpropagation" -> Backpropagation
 }
}

EXECUTION_RULES {
 E1 = "If an AI system is being analysed mathematically, activate AIMathematicsOntology."
 E2 = "If learning from labelled examples, route to supervised-learning mathematics."
 E3 = "If learning structure without labels, route to unsupervised or self-supervised mathematics."
 E4 = "If neural networks are used, include linear algebra, calculus and backpropagation."
 E5 = "If uncertainty is central, include probabilistic or Bayesian inference."
 E6 = "If sequential decisions and rewards are central, route to reinforcement-learning mathematics."
 E7 = "If graph-structured data are central, route to graph-learning mathematics."
 E8 = "If information objectives are central, route to information theory."
 E9 = "Any learned model should be evaluated for generalisation, numerical stability and computational cost."
}

RECURSION {
 ENTRY = AIMathematics
 EXPAND(AIMathematics) -> [
  ArtificialIntelligenceMathematics, MachineLearningMathematics,
  StatisticalLearning, SupervisedLearning, UnsupervisedLearning,
  SelfSupervisedLearning, ReinforcementLearning, NeuralNetworkMathematics,
  DeepLearningMathematics, LinearModel, NonlinearModel, Parameter,
  Hyperparameter, LossFunction, ObjectiveFunction, RiskFunction,
  EmpiricalRisk, ExpectedRisk, Regularisation, Generalisation,
  BiasVarianceTradeoff, Overfitting, Underfitting, Optimisation,
  GradientDescent, StochasticGradientDescent, Momentum, AdaptiveOptimisation,
  ConvexOptimisation, NonConvexOptimisation, ConstraintOptimisation,
  LinearAlgebra, Vector, Matrix, Tensor, Eigenvalue, SingularValueDecomposition,
  Calculus, Gradient, Jacobian, Hessian, ChainRule, Backpropagation,
  Probability, ConditionalProbability, BayesRule, RandomVariable,
  ProbabilityDistribution, Expectation, Variance, Covariance, Likelihood,
  MaximumLikelihood, MaximumAPosteriori, BayesianInference, Entropy,
  CrossEntropy, KLDivergence, MutualInformation, InformationTheory,
  Classification, Regression, Clustering, DimensionalityReduction,
  PrincipalComponentAnalysis, KernelMethod, SupportVectorMachine,
  GaussianProcess, DecisionTree, EnsembleMethod, GraphLearning,
  GraphNeuralNetwork, Attention, TransformerMathematics, Embedding,
  RepresentationLearning, LatentVariableModel, Autoencoder, GenerativeModel,
  VariationalInference, MarkovDecisionProcess, Policy, ValueFunction,
  QFunction, BellmanEquation, Exploration, Exploitation, Search,
  HeuristicSearch, ComputationalComplexity, SampleComplexity,
  NumericalStability, Robustness, UncertaintyQuantification,
  Calibration, ModelValidation
 ]
 EXPAND(FoundationDomains) -> [
  LinearAlgebra,
  Calculus,
  Probability,
  Statistics,
  Optimisation,
  InformationTheory,
  ComputationalComplexity
 ]
 STOP_CONDITION = "Stop when AI task, model, objective, optimisation, inference, uncertainty, generalisation and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Artificial Intelligence Mathematics"
 IMMEDIATE_CONTEXTS = [
  "Artificial Intelligence Mathematics",
  "Linear Algebra Ontologies",
  "Probability & Statistics Ontologies",
  "Optimisation Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Artificial Intelligence Mathematics",
  "Applied Mathematics Ontologies",
  "Linear Algebra Ontologies",
  "Calculus & Analysis Ontologies",
  "Probability & Statistics Ontologies",
  "Optimisation Ontologies",
  "Information Theory Ontologies",
  "Graph Theory Ontologies",
  "Computational Mathematics Ontologies",
  "Computational Complexity Ontology",
  "Dynamical Systems Ontologies",
  "Numerical Linear Algebra Ontology"
 ]
 ROUTE_RULES = [
  "AI Mathematics Ontology -> Artificial Intelligence Mathematics for parent semantics",
  "AI Mathematics Ontology -> Linear Algebra for vector, matrix and tensor computation",
  "AI Mathematics Ontology -> Calculus & Analysis for gradients, Jacobians, Hessians and backpropagation",
  "AI Mathematics Ontology -> Probability & Statistics for inference and uncertainty",
  "AI Mathematics Ontology -> Optimisation for model training and decisions",
  "AI Mathematics Ontology -> Information Theory for entropy and representation objectives",
  "AI Mathematics Ontology -> Graph Theory for graph learning",
  "AI Mathematics Ontology -> Computational Mathematics for numerical methods",
  "AI Mathematics Ontology -> Computational Complexity for resource analysis",
  "AI Mathematics Ontology -> Dynamical Systems for iterative and recurrent behaviour"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible AI mathematical entities, models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve model-and-objective semantics.",
  "Preserve optimisation-versus-inference distinction.",
  "Preserve training-versus-generalisation distinction.",
  "Preserve deterministic-versus-probabilistic distinction.",
  "Preserve supervised-unsupervised-reinforcement distinctions.",
  "Preserve uncertainty and calibration semantics.",
  "Preserve computational-complexity semantics.",
  "Preserve machine-readable AI-mathematics ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "The AI Mathematics Ontology formally represents the mathematical structures used for learning, inference, optimisation, representation and decision-making in artificial intelligence."
 LEARNING_SUMMARY = "Learning mathematics formalises loss, empirical risk, regularisation and generalisation."
 NEURAL_SUMMARY = "Neural-network mathematics combines linear algebra, nonlinear transformations, calculus and backpropagation."
 PROBABILITY_SUMMARY = "Probabilistic and Bayesian methods represent likelihoods, uncertainty, priors, posteriors and predictions."
 INFORMATION_SUMMARY = "Entropy, cross-entropy, KL divergence and mutual information quantify uncertainty and representation structure."
 RL_SUMMARY = "Reinforcement-learning mathematics models sequential decisions using states, actions, rewards, policies and Bellman equations."
 COMPLEXITY_SUMMARY = "Sample, time and space complexity determine the theoretical and practical cost of AI methods."
 VALIDATION_SUMMARY = "AI mathematical models should be checked for numerical stability, generalisation, calibration, robustness and reproducibility."
}

FLOW {
 INPUT
 -> detect_AI_mathematics_ontology_context
 -> identify_AI_task_data_structure_input_output_spaces_model_family_objective_and_constraints
 -> select_supervised_unsupervised_self_supervised_generative_graph_probabilistic_or_reinforcement_learning_pathway
 -> define_representation_parameters_and_model_equations
 -> formulate_loss_risk_regularisation_reward_or_information_objective
 -> select_optimisation_or_inference_method
 -> train_or_solve_model
 -> compute_predictions_posteriors_embeddings_values_or_policies
 -> assess_generalisation_uncertainty_calibration_robustness_and_distribution_shift
 -> analyse_numerical_stability_sample_complexity_time_complexity_and_space_complexity
 -> validate_against_independent_data_or_formal_criteria
 -> route_to_artificial_intelligence_mathematics_linear_algebra_calculus_probability_statistics_optimisation_information_theory_graph_theory_computational_mathematics_computational_complexity_dynamical_systems_or_numerical_linear_algebra_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The AI Mathematics Ontology provides the formal mathematical core for artificial intelligence. It supports machine and statistical learning, supervised, unsupervised, self-supervised and reinforcement learning, neural networks and deep learning, parameters and hyperparameters, loss functions, empirical and expected risk, regularisation and generalisation, gradient-based optimisation, linear algebra, tensors and matrix decompositions, calculus and backpropagation, probability, likelihood and Bayesian inference, entropy, cross-entropy, KL divergence and mutual information, classification, regression, clustering and dimensionality reduction, kernel methods, Gaussian processes and ensembles, graph learning and graph neural networks, attention and transformer mathematics, embeddings and representation learning, latent-variable and generative models, Markov decision processes, Bellman equations, search, computational and sample complexity, numerical stability, robustness, uncertainty, calibration and model validation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-AI-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW