Reinforcement Learning Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Reinforcement Learning Mathematics Ontology within the Kireth Research Semantic Library. Reinforcement learning mathematics studies sequential decision-making under uncertainty using states, actions, rewards, transition dynamics, policies, value functions and optimisation. This ontology supports Markov decision processes, Bellman equations, temporal-difference learning, Monte Carlo methods, policy gradients, actor-critic methods, exploration, off-policy learning, function approximation, stochastic control, convergence, regret, multi-agent learning and relationships with AI mathematics, probability, optimisation, dynamical systems and control theory.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ReinforcementLearningMathematics
 ENTITY_CLASS = SequentialDecisionMathematicsFramework
 PRIMARY_CONTEXT = AIMathematics
 DEFINITION = "Reinforcement learning mathematics is the mathematical study of agents that learn sequential decision policies from rewards and interactions with stochastic environments."
 SYMBOLIC_FORMS = [
  "M=(S,A,P,R,gamma)",
  "V^pi(s)=E_pi[sum_t gamma^t R_t | S_0=s]",
  "Q^pi(s,a)=E_pi[sum_t gamma^t R_t | S_0=s,A_0=a]",
  "Q*(s,a)=E[R+gamma max_a' Q*(S',a')]"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for Markov decision processes, policies, value functions, Bellman equations, exploration, temporal-difference learning, policy optimisation, actor-critic methods, function approximation and sequential-decision validation."
}

SCOPE {
 INCLUDE = [
  ReinforcementLearningMathematics,
  ReinforcementLearning,
  Agent,
  Environment,
  State,
  Action,
  Reward,
  Return,
  Episode,
  Trajectory,
  Policy,
  DeterministicPolicy,
  StochasticPolicy,
  BehaviourPolicy,
  TargetPolicy,
  MarkovProperty,
  MarkovDecisionProcess,
  MDP,
  TransitionProbability,
  TransitionKernel,
  RewardFunction,
  DiscountFactor,
  FiniteHorizon,
  InfiniteHorizon,
  ValueFunction,
  StateValueFunction,
  ActionValueFunction,
  QFunction,
  AdvantageFunction,
  BellmanEquation,
  BellmanExpectationEquation,
  BellmanOptimalityEquation,
  DynamicProgramming,
  PolicyEvaluation,
  PolicyImprovement,
  PolicyIteration,
  ValueIteration,
  MonteCarloRL,
  TemporalDifferenceLearning,
  TDLearning,
  TDZero,
  NStepTD,
  EligibilityTrace,
  TDLambda,
  SARSA,
  QLearning,
  ExpectedSARSA,
  OnPolicyLearning,
  OffPolicyLearning,
  ImportanceSampling,
  Exploration,
  Exploitation,
  EpsilonGreedy,
  SoftmaxActionSelection,
  UpperConfidenceBound,
  MultiArmedBandit,
  ContextualBandit,
  Regret,
  FunctionApproximation,
  LinearFunctionApproximation,
  NonlinearFunctionApproximation,
  NeuralValueFunction,
  DeepQNetwork,
  DQN,
  ReplayBuffer,
  ExperienceReplay,
  TargetNetwork,
  PolicyGradient,
  REINFORCE,
  PolicyGradientTheorem,
  ActorCritic,
  Actor,
  Critic,
  AdvantageActorCritic,
  A2C,
  A3C,
  DeterministicPolicyGradient,
  DDPG,
  TD3,
  SoftActorCritic,
  SAC,
  ProximalPolicyOptimisation,
  PPO,
  TrustRegionPolicyOptimisation,
  TRPO,
  EntropyRegularisation,
  GeneralisedAdvantageEstimation,
  GAE,
  ModelBasedRL,
  ModelFreeRL,
  Planning,
  WorldModel,
  StateRepresentation,
  PartialObservability,
  POMDP,
  BeliefState,
  MultiAgentReinforcementLearning,
  CooperativeRL,
  CompetitiveRL,
  StochasticGame,
  RewardShaping,
  ConstraintRL,
  SafeRL,
  RiskSensitiveRL,
  DistributionalRL,
  OfflineRL,
  BatchRL,
  InverseReinforcementLearning,
  ImitationLearning,
  Convergence,
  SampleComplexity,
  NumericalStability
 ]
 EXCLUDE_AS_PRIMARY = [
  SupervisedLearningWithoutSequentialDecisionStructure,
  GeneralOptimalControlWithoutLearningFromInteraction,
  StaticOptimisationWithoutStateTransitionDynamics,
  GameTheoryWithoutLearningAgentSemantics
 ]
 DISAMBIGUATION_RULE = "When an agent learns sequential decisions from rewards under state transitions and uncertainty, interpret the context as reinforcement learning mathematics."
}

CORE_CONCEPTS {
 ReinforcementLearningMathematics = "Mathematical foundations of reward-driven sequential decision learning."
 MarkovDecisionProcess = "Tuple defining states, actions, transitions, rewards and discounting."
 Policy = "Rule mapping states or observations to action distributions or actions."
 Return = "Discounted or undiscounted cumulative future reward."
 ValueFunction = "Expected return from a state under a policy."
 QFunction = "Expected return from a state-action pair under a policy."
 BellmanEquation = "Recursive relation linking present value to immediate reward and successor value."
 Exploration = "Action selection that gathers information about uncertain alternatives."
 PolicyGradient = "Gradient of expected return with respect to policy parameters."
 ActorCritic = "Architecture combining a policy actor with a value-estimating critic."
}

RELATIONS {
 ReinforcementLearningMathematics --is_child_of--> AIMathematics
 ReinforcementLearning --uses--> MarkovDecisionProcess
 Agent --interacts_with--> Environment
 Environment --returns--> Reward
 Policy --selects--> Action
 Action --causes--> StateTransition
 Return --aggregates--> FutureRewards
 ValueFunction --depends_on--> Policy
 QFunction --depends_on--> Policy
 BellmanEquation --relates--> ImmediateRewardAndFutureValue
 DynamicProgramming --uses--> BellmanEquation
 PolicyIteration --alternates--> PolicyEvaluationAndPolicyImprovement
 ValueIteration --uses--> BellmanOptimalityEquation
 TemporalDifferenceLearning --bootstraps_from--> ValueEstimate
 SARSA --is_type_of--> OnPolicyLearning
 QLearning --is_type_of--> OffPolicyLearning
 ImportanceSampling --corrects--> PolicyMismatch
 EpsilonGreedy --balances--> ExplorationAndExploitation
 PolicyGradient --optimises--> Policy
 REINFORCE --is_type_of--> PolicyGradientMethod
 ActorCritic --combines--> PolicyAndValueEstimation
 DQN --uses--> ExperienceReplay
 DQN --uses--> TargetNetwork
 PPO --constrains--> PolicyUpdate
 EntropyRegularisation --encourages--> Exploration
 POMDP --uses--> BeliefState
 MultiAgentReinforcementLearning --uses--> StochasticGame
}

OPERATORS {
 RETURN(R,gamma,t) := G_t=sum_{k=0}^\infty gamma^k R_(t+k+1)
 STATE_VALUE(pi,s) := V_pi(s)=E_pi[G_t|S_t=s]
 ACTION_VALUE(pi,s,a) := Q_pi(s,a)=E_pi[G_t|S_t=s,A_t=a]
 BELLMAN_EXPECTATION_V(pi,s) := V_pi(s)=sum_a pi(a|s) sum_s' P(s'|s,a)[R(s,a,s')+gamma*V_pi(s')]
 BELLMAN_OPTIMALITY_V(s) := V_star(s)=max_a sum_s' P(s'|s,a)[R+gamma*V_star(s')]
 BELLMAN_OPTIMALITY_Q(s,a) := Q_star(s,a)=sum_s' P(s'|s,a)[R+gamma*max_a' Q_star(s',a')]
 TD_ERROR(r,gamma,V_s_next,V_s) := delta=r+gamma*V_s_next-V_s
 TD_UPDATE(V_s,alpha,delta) := V_s_next=V_s+alpha*delta
 SARSA_UPDATE(Q,alpha,r,gamma,Q_next) := Q_next_current=Q+alpha*(r+gamma*Q_next-Q)
 Q_LEARNING_UPDATE(Q,alpha,r,gamma,maxQnext) := Q_new=Q+alpha*(r+gamma*maxQnext-Q)
 POLICY_GRADIENT(theta) := grad J(theta)=E[grad_theta log pi_theta(a|s)*Q_pi(s,a)]
 REINFORCE_UPDATE(theta,alpha,G) := theta_next=theta+alpha*G*grad_theta log pi_theta(a|s)
 ADVANTAGE(Q,V) := A(s,a)=Q(s,a)-V(s)
 ENTROPY(pi) := H(pi(.|s))=-sum_a pi(a|s)log pi(a|s)
}

AXIOMS {
 A1 = "Reinforcement learning requires explicit state, action, reward and transition semantics."
 A2 = "The Markov assumption must be stated or relaxed through partial-observability models."
 A3 = "Value functions are policy-dependent unless explicitly optimal."
 A4 = "Bootstrapping introduces bias-variance trade-offs."
 A5 = "Exploration and exploitation are mathematically distinct objectives."
 A6 = "Off-policy learning requires accounting for behaviour-target policy mismatch."
 A7 = "Function approximation changes convergence properties relative to tabular methods."
 A8 = "Reinforcement-learning results require statistical, numerical and policy-evaluation validation."
}

REASONING {
 R1_CONTEXT = identify(agent,environment,state,action,reward,horizon,observability)
 R2_MDP = construct(states,actions,transition_kernel,reward_function,discount_factor)
 R3_OBJECTIVE = define(expected_return,risk_sensitive_return,constraint_or_regret)
 R4_METHOD_CLASS = classify(dynamic_programming,MonteCarlo,TD,value_based,policy_based,actor_critic,model_based)
 R5_VALUE = estimate(V,Q,advantage)
 R6_POLICY = derive_or_optimise(policy,exploration_strategy)
 R7_OFF_POLICY = if_required apply importance_sampling_replay_or_correction
 R8_FUNCTION_APPROXIMATION = select(tabular,linear,neural_or_distributional representation)
 R9_CONVERGENCE = analyse(stability,sample_complexity,bias,variance,policy_improvement)
 R10_VALIDATION = evaluate(return,regret,robustness,safety,generalisation,reproducibility)
}

CONSTRAINTS {
 C1 = "High episodic return does not guarantee robust policy performance."
 C2 = "Q-learning convergence guarantees require assumptions that may fail with nonlinear function approximation."
 C3 = "Off-policy bootstrapping with function approximation can be unstable."
 C4 = "Reward misspecification can produce undesired behaviour."
 C5 = "Discount factor changes the effective planning horizon."
 C6 = "Policy-gradient estimators can have high variance."
 C7 = "Replay buffers alter the training data distribution."
 C8 = "Partial observability can invalidate ordinary MDP state assumptions."
 C9 = "Offline RL is vulnerable to distributional shift outside the dataset support."
 C10 = "Multi-agent environments are non-stationary from the perspective of a learning agent."
}

SEMANTIC_EQUIVALENCE {
 "ReinforcementLearningMathematics" <=> "MathematicsOfReinforcementLearning"
 "MarkovDecisionProcess" <=> "MDP"
 "ActionValueFunction" <=> "QFunction"
 "TemporalDifferenceLearning" <=> "TDLearning"
 "ProximalPolicyOptimisation" <=> "PPO"
 "SoftActorCritic" <=> "SAC"
 "GeneralisedAdvantageEstimation" <=> "GAE"
}

VALIDATION {
 V1 = "State, action and reward semantics are explicit."
 V2 = "Transition assumptions and observability are documented."
 V3 = "Discount factor and horizon are explicit."
 V4 = "Behaviour and target policies are distinguished where relevant."
 V5 = "Exploration strategy is documented."
 V6 = "Value or policy estimates include uncertainty where feasible."
 V7 = "Training and evaluation environments are separated where possible."
 V8 = "Convergence or stopping criteria are explicit."
 V9 = "Safety or constraint violations are monitored where relevant."
 V10 = "Policy performance is evaluated across multiple random seeds or independent runs where feasible."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 RL_MATHEMATICS_DEFINITION = 0.99
 MDP_REASONING = 0.99
 BELLMAN_REASONING = 0.99
 TD_REASONING = 0.99
 POLICY_GRADIENT_REASONING = 0.98
 ACTOR_CRITIC_REASONING = 0.98
 OFF_POLICY_REASONING = 0.97
 FUNCTION_APPROXIMATION_REASONING = 0.97
 MULTI_AGENT_REASONING = 0.97
 POLICY = "Reduce confidence when state representation, reward design, transition dynamics, observability, discounting, exploration, off-policy correction, function approximation or evaluation protocol are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ReinforcementLearningMathematicsReasoning,
  MDPReasoning,
  BellmanEquationReasoning,
  ValueFunctionReasoning,
  QFunctionReasoning,
  DynamicProgrammingReasoning,
  MonteCarloRLReasoning,
  TemporalDifferenceReasoning,
  SARSARReasoning,
  QLearningReasoning,
  ExplorationReasoning,
  PolicyGradientReasoning,
  REINFORCEReasoning,
  ActorCriticReasoning,
  DQNReasoning,
  PPOReasoning,
  SACReasoning,
  ModelBasedRLReasoning,
  POMDPReasoning,
  OfflineRLReasoning,
  MultiAgentRLReasoning,
  SafeRLReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "reinforcement learning mathematics" -> ReinforcementLearningMathematics
  "mathematics of reinforcement learning" -> ReinforcementLearningMathematics
  "Markov decision process" -> MarkovDecisionProcess
  "MDP" -> MarkovDecisionProcess
  "Q function" -> QFunction
  "temporal difference learning" -> TemporalDifferenceLearning
  "TD learning" -> TemporalDifferenceLearning
  "PPO" -> ProximalPolicyOptimisation
  "SAC" -> SoftActorCritic
  "GAE" -> GeneralisedAdvantageEstimation
 }
}

EXECUTION_RULES {
 E1 = "If an agent learns sequential decisions from rewards, activate ReinforcementLearningMathematicsOntology."
 E2 = "If transitions and rewards are known, consider dynamic programming."
 E3 = "If complete returns are sampled without bootstrapping, route to Monte Carlo RL."
 E4 = "If bootstrapped value updates are used, route to temporal-difference learning."
 E5 = "If learning action values directly, route to SARSA, Q-learning or related value-based methods."
 E6 = "If directly optimising a stochastic policy, route to policy-gradient methods."
 E7 = "If combining policy and value estimation, route to actor-critic methods."
 E8 = "If observations are incomplete, route to POMDP or belief-state reasoning."
 E9 = "Any RL method should be validated for stability, sample efficiency, policy performance, robustness and safety."
}

RECURSION {
 ENTRY = ReinforcementLearningMathematics
 EXPAND(ReinforcementLearningMathematics) -> [
  ReinforcementLearning, Agent, Environment, State, Action, Reward, Return,
  Episode, Trajectory, Policy, DeterministicPolicy, StochasticPolicy,
  BehaviourPolicy, TargetPolicy, MarkovProperty, MarkovDecisionProcess,
  TransitionProbability, TransitionKernel, RewardFunction, DiscountFactor,
  FiniteHorizon, InfiniteHorizon, ValueFunction, StateValueFunction,
  ActionValueFunction, QFunction, AdvantageFunction, BellmanEquation,
  BellmanExpectationEquation, BellmanOptimalityEquation, DynamicProgramming,
  PolicyEvaluation, PolicyImprovement, PolicyIteration, ValueIteration,
  MonteCarloRL, TemporalDifferenceLearning, TDZero, NStepTD, EligibilityTrace,
  TDLambda, SARSA, QLearning, ExpectedSARSA, OnPolicyLearning,
  OffPolicyLearning, ImportanceSampling, Exploration, Exploitation,
  EpsilonGreedy, SoftmaxActionSelection, UpperConfidenceBound,
  MultiArmedBandit, ContextualBandit, Regret, FunctionApproximation,
  LinearFunctionApproximation, NonlinearFunctionApproximation,
  DeepQNetwork, ExperienceReplay, TargetNetwork, PolicyGradient,
  REINFORCE, PolicyGradientTheorem, ActorCritic, AdvantageActorCritic,
  A2C, A3C, DeterministicPolicyGradient, DDPG, TD3, SoftActorCritic,
  ProximalPolicyOptimisation, TrustRegionPolicyOptimisation,
  EntropyRegularisation, GeneralisedAdvantageEstimation, ModelBasedRL,
  ModelFreeRL, Planning, WorldModel, PartialObservability, POMDP,
  BeliefState, MultiAgentReinforcementLearning, StochasticGame,
  RewardShaping, ConstraintRL, SafeRL, RiskSensitiveRL, DistributionalRL,
  OfflineRL, BatchRL, InverseReinforcementLearning, ImitationLearning,
  Convergence, SampleComplexity, NumericalStability
 ]
 EXPAND(MethodFamilies) -> [
  DynamicProgrammingMethods,
  MonteCarloMethods,
  TemporalDifferenceMethods,
  ValueBasedMethods,
  PolicyGradientMethods,
  ActorCriticMethods,
  ModelBasedMethods,
  MultiAgentMethods
 ]
 STOP_CONDITION = "Stop when state-action-reward semantics, transition dynamics, value or policy method, exploration, approximation, convergence and evaluation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Mathematics Ontology"
 IMMEDIATE_CONTEXTS = [
  "AI Mathematics Ontology",
  "Optimisation for AI Ontology",
  "Probability & Statistics Ontologies",
  "Control Theory Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "AI Mathematics Ontology",
  "Artificial Intelligence Mathematics",
  "Optimisation for AI Ontology",
  "Gradient Optimisation Ontology",
  "Loss Function Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Dynamical Systems Ontologies",
  "Control Theory Ontologies",
  "Game Theory Ontologies",
  "Markov Chain Ontologies",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Reinforcement Learning Mathematics -> AI Mathematics Ontology for parent semantics",
  "Reinforcement Learning Mathematics -> Optimisation for AI for policy and value optimisation",
  "Reinforcement Learning Mathematics -> Gradient Optimisation for policy-gradient training",
  "Reinforcement Learning Mathematics -> Loss Function Ontology for Bellman and value losses",
  "Reinforcement Learning Mathematics -> Probability & Statistics for stochastic returns and estimation",
  "Reinforcement Learning Mathematics -> Stochastic Processes for Markov dynamics",
  "Reinforcement Learning Mathematics -> Dynamical Systems for iterative learning dynamics",
  "Reinforcement Learning Mathematics -> Control Theory for sequential control semantics",
  "Reinforcement Learning Mathematics -> Game Theory for multi-agent strategic interaction",
  "Reinforcement Learning Mathematics -> Computational Complexity for sample and computational cost"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible RL algorithms, environments or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve state-action-reward semantics.",
  "Preserve policy-versus-value distinction.",
  "Preserve model-based-versus-model-free distinction.",
  "Preserve on-policy-versus-off-policy distinction.",
  "Preserve exploration-versus-exploitation distinction.",
  "Preserve tabular-versus-function-approximation distinction.",
  "Preserve partial-observability and multi-agent semantics.",
  "Preserve machine-readable reinforcement-learning mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Reinforcement learning mathematics models how agents learn sequential decisions from rewards in stochastic environments."
 MDP_SUMMARY = "Markov decision processes define states, actions, transition probabilities, rewards and discounting."
 VALUE_SUMMARY = "Value and action-value functions represent expected future return under policies."
 BELLMAN_SUMMARY = "Bellman equations recursively connect current values to immediate rewards and successor values."
 TD_SUMMARY = "Temporal-difference methods combine sampling and bootstrapping to update value estimates."
 POLICY_SUMMARY = "Policy-gradient and actor-critic methods optimise parameterised policies directly."
 EXPLORATION_SUMMARY = "Exploration strategies balance information gathering against exploitation of current knowledge."
 VALIDATION_SUMMARY = "RL systems should be checked for convergence, stability, sample efficiency, generalisation, robustness and safety."
}

FLOW {
 INPUT
 -> detect_reinforcement_learning_mathematics_context
 -> identify_agent_environment_states_actions_rewards_horizon_observability_and_constraints
 -> construct_MDP_POMDP_bandit_or_stochastic_game_model
 -> define_return_discounting_and_learning_objective
 -> choose_dynamic_programming_MonteCarlo_TD_value_based_policy_gradient_actor_critic_or_model_based_method
 -> estimate_values_advantages_models_or_policies
 -> apply_exploration_and_if_required_off_policy_correction
 -> update_value_function_policy_or_environment_model
 -> monitor_return_regret_Bellman_error_gradient_norm_and_constraint violations
 -> assess_convergence_sample_complexity_stability_and_distribution shift
 -> validate_policy_across_independent_evaluation_runs
 -> route_to_AI_mathematics_optimisation_for_AI_gradient_optimisation_loss_function_probability_statistics_stochastic_processes_dynamical_systems_control_theory_game_theory_or_computational_complexity_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Reinforcement Learning Mathematics Ontology defines mathematical sequential decision learning. It supports agents, environments, states, actions, rewards, returns and trajectories, Markov decision processes and transition kernels, deterministic and stochastic policies, state and action values, Q-functions and advantages, Bellman expectation and optimality equations, dynamic programming, policy and value iteration, Monte Carlo reinforcement learning, temporal-difference learning, eligibility traces, SARSA, Q-learning and Expected SARSA, on-policy and off-policy learning, importance sampling, exploration and multi-armed bandits, function approximation, deep Q-networks, replay buffers and target networks, policy gradients, REINFORCE, actor-critic methods, A2C, A3C, DDPG, TD3, SAC, PPO and TRPO, entropy regularisation and GAE, model-based and model-free RL, planning and world models, POMDPs and belief states, multi-agent RL and stochastic games, safe, constrained, risk-sensitive, distributional and offline RL, inverse reinforcement learning, imitation learning, convergence, sample complexity and numerical stability."
}

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

}

END_FLOW