Artificial Intelligence Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Artificial Intelligence Mathematics layer within the Kireth Research Semantic Library. Artificial intelligence mathematics studies the mathematical structures underlying machine learning, optimisation, probabilistic inference, neural networks, representation learning, decision systems, information theory, statistical learning, reinforcement learning, search, graph learning and computational intelligence. This parent ontology connects linear algebra, calculus, probability, statistics, optimisation, information theory, numerical methods, dynamical systems, discrete mathematics and computational complexity to AI systems.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ArtificialIntelligenceMathematics
 ENTITY_CLASS = MathematicalAIFoundationsFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Artificial intelligence mathematics is the branch of applied mathematics concerned with the mathematical structures, models, optimisation procedures and probabilistic reasoning used by artificial intelligence systems."
 SYMBOLIC_FORMS = [
  "y=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 the parent semantic framework for mathematical machine learning, neural networks, probabilistic inference, optimisation, representation learning, reinforcement learning, information theory, graph learning and AI computation."
}

SCOPE {
 INCLUDE = [
  ArtificialIntelligenceMathematics,
  AIMathematics,
  MachineLearningMathematics,
  StatisticalLearning,
  SupervisedLearning,
  UnsupervisedLearning,
  SelfSupervisedLearning,
  ReinforcementLearning,
  NeuralNetworkMathematics,
  DeepLearningMathematics,
  LinearModel,
  NonlinearModel,
  LossFunction,
  ObjectiveFunction,
  RiskFunction,
  EmpiricalRisk,
  ExpectedRisk,
  Regularisation,
  Generalisation,
  BiasVarianceTradeoff,
  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,
  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,
  NumericalStability,
  Robustness,
  UncertaintyQuantification,
  Calibration,
  ModelValidation
 ]
 EXCLUDE_AS_PRIMARY = [
  AIProductEngineeringWithoutMathematicalModel,
  PureSoftwareImplementationWithoutMathematicalStructure,
  HumanFactorsWithoutQuantitativeAIModel,
  DescriptiveAITaxonomyWithoutMathematicalReasoning
 ]
 DISAMBIGUATION_RULE = "When the central subject is the mathematical structure, learning objective, probabilistic model, optimisation process or decision rule underlying an AI system, interpret the context as artificial intelligence mathematics."
}

CORE_CONCEPTS {
 ArtificialIntelligenceMathematics = "Mathematical foundations of artificial intelligence systems."
 MachineLearningMathematics = "Mathematical methods for learning models from data."
 LossFunction = "Function quantifying prediction or decision error."
 Generalisation = "Ability of a learned model to perform beyond its training sample."
 Optimisation = "Mathematical search for parameters or decisions that minimise or maximise an objective."
 RepresentationLearning = "Learning mathematical representations useful for downstream inference or decision-making."
 ProbabilisticInference = "Reasoning about uncertain variables using probability distributions."
 ReinforcementLearning = "Sequential decision learning through rewards and environment interaction."
 InformationTheory = "Mathematical study of information, uncertainty and coding quantities used in AI objectives and representations."
 ComputationalComplexity = "Analysis of computational resource requirements for AI algorithms."
}

RELATIONS {
 ArtificialIntelligenceMathematics --is_branch_of--> AppliedMathematics
 ArtificialIntelligenceMathematics --uses--> LinearAlgebra
 ArtificialIntelligenceMathematics --uses--> Calculus
 ArtificialIntelligenceMathematics --uses--> Probability
 ArtificialIntelligenceMathematics --uses--> Statistics
 ArtificialIntelligenceMathematics --uses--> Optimisation
 ArtificialIntelligenceMathematics --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
 ReinforcementLearning --uses--> MarkovDecisionProcess
 ValueFunction --satisfies--> BellmanEquation
 GraphNeuralNetwork --uses--> GraphStructure
 Calibration --assesses--> PredictiveUncertainty
 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*(s,a)=E[r+gamma*max_a' Q*(s',a')]
 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 must define loss, risk or reward mathematically."
 A3 = "Optimisation results depend on objective geometry, constraints and numerical method."
 A4 = "Probabilistic AI models require explicit distributions and conditioning assumptions."
 A5 = "Training performance and generalisation performance must not be conflated."
 A6 = "Model capacity, regularisation and data distribution jointly affect 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)
 R5_OPTIMISATION = select(gradient,stochastic,convex,nonconvex,second_order_or_search_method)
 R6_INFERENCE = compute(predictions,posterior,decision,value_or_policy)
 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 local minima, saddle points or flat regions."
 C3 = "Gradient-based methods require differentiability or suitable subgradient structure."
 C4 = "Probabilistic predictions are only meaningful relative to their declared model assumptions."
 C5 = "Calibration and discrimination are distinct predictive properties."
 C6 = "Embedding geometry depends on the learned representation and training objective."
 C7 = "Reinforcement-learning value estimates depend on policy, environment and discount assumptions."
 C8 = "Attention scores are model-dependent and should not automatically be interpreted as causal explanations."
 C9 = "Computational complexity can dominate theoretical statistical efficiency."
 C10 = "Distribution shift can invalidate assumptions made during training."
}

SEMANTIC_EQUIVALENCE {
 "ArtificialIntelligenceMathematics" <=> "AIMathematicalFoundations"
 "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_DEFINITION = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 OPTIMISATION_REASONING = 0.99
 PROBABILISTIC_REASONING = 0.98
 REPRESENTATION_REASONING = 0.98
 GENERALISATION_REASONING = 0.98
 RL_REASONING = 0.98
 COMPLEXITY_REASONING = 0.98
 VALIDATION_REASONING = 0.99
 POLICY = "Reduce confidence when task definition, data distribution, objective, model family, optimisation method, uncertainty assumptions, environment dynamics or validation evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, ArtificialIntelligenceMathematicsReasoning,
  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 {
  "artificial intelligence mathematics" -> ArtificialIntelligenceMathematics
  "AI mathematics" -> ArtificialIntelligenceMathematics
  "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 ArtificialIntelligenceMathematics."
 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 calculus, linear algebra and backpropagation reasoning."
 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 = ArtificialIntelligenceMathematics
 EXPAND(ArtificialIntelligenceMathematics) -> [
  MachineLearningMathematics, StatisticalLearning, SupervisedLearning,
  UnsupervisedLearning, SelfSupervisedLearning, ReinforcementLearning,
  NeuralNetworkMathematics, DeepLearningMathematics, LinearModel,
  NonlinearModel, LossFunction, ObjectiveFunction, EmpiricalRisk,
  ExpectedRisk, Regularisation, Generalisation, BiasVarianceTradeoff,
  Optimisation, GradientDescent, StochasticGradientDescent, Momentum,
  AdaptiveOptimisation, ConvexOptimisation, NonConvexOptimisation,
  LinearAlgebra, Vector, Matrix, Tensor, Eigenvalue,
  SingularValueDecomposition, Calculus, Gradient, Jacobian, Hessian,
  ChainRule, Backpropagation, Probability, ConditionalProbability,
  BayesRule, RandomVariable, ProbabilityDistribution, Expectation,
  Variance, Covariance, 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, Search, HeuristicSearch, ComputationalComplexity,
  NumericalStability, Robustness, UncertaintyQuantification,
  Calibration, ModelValidation
 ]
 STOP_CONDITION = "Stop when AI task, mathematical model, objective, optimisation, inference, uncertainty, generalisation and validation are sufficiently resolved."
}

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

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible AI model families, learning paradigms 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 semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Artificial intelligence mathematics provides the mathematical foundations for learning, inference, optimisation, representation and decision-making in AI systems."
 LEARNING_SUMMARY = "Machine learning mathematics formalises empirical risk, generalisation, regularisation and model fitting from data."
 NEURAL_SUMMARY = "Neural-network mathematics combines linear algebra, nonlinear functions, calculus and backpropagation."
 PROBABILITY_SUMMARY = "Probabilistic and Bayesian methods represent uncertainty, likelihoods, priors, posteriors and predictive distributions."
 INFORMATION_SUMMARY = "Entropy, cross-entropy, KL divergence and mutual information quantify uncertainty and representation quality."
 RL_SUMMARY = "Reinforcement-learning mathematics models sequential decisions using states, actions, rewards, policies and Bellman equations."
 COMPLEXITY_SUMMARY = "Computational and sample complexity determine the practical and theoretical cost of AI methods."
 VALIDATION_SUMMARY = "AI mathematical models should be checked for numerical stability, generalisation, calibration, robustness and reproducibility."
}

FLOW {
 INPUT
 -> detect_artificial_intelligence_mathematics_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_applied_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 Artificial Intelligence Mathematics layer defines mathematical foundations for AI systems. It supports machine and statistical learning, supervised, unsupervised, self-supervised and reinforcement learning, neural networks and deep learning, 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 complexity, numerical stability, robustness, uncertainty, calibration and model validation."
}

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

}

END_FLOW