Markov Process Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Markov Process Ontology within the Kireth Research Semantic Library. A Markov process is a stochastic process whose conditional future evolution depends on the present state rather than the complete past, relative to the chosen state representation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Markov Process Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Applied Statistical Mathematics / Markov Processes"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/applied-statistical-mathematics/markov-process-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MarkovProcess
 ENTITY_CLASS = MemorylessStochasticProcessFramework
 PRIMARY_CONTEXT = AppliedStatisticalMathematics
 DEFINITION = "A Markov process is a stochastic process for which the conditional distribution of future states, given the present state, is independent of earlier history under the chosen state representation."
 SYMBOLIC_FORMS = [
  "P(X_(t+1)=j|X_t=i,history)=P(X_(t+1)=j|X_t=i)",
  "P^(n+m)=P^n P^m",
  "pi=pi P",
  "P(t)=exp(Qt)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for Markov states, transition probabilities and kernels, stationary distributions, recurrence, absorption, hitting times, generators and continuous-time dynamics."
}

SCOPE {
 INCLUDE = [
  MarkovProcess, MarkovProperty, StochasticProcess, State, StateSpace,
  DiscreteStateSpace, ContinuousStateSpace, MarkovChain,
  DiscreteTimeMarkovChain, ContinuousTimeMarkovChain, DTMC, CTMC,
  TransitionProbability, TransitionMatrix, TransitionKernel, MarkovKernel,
  InitialDistribution, StateDistribution, ChapmanKolmogorovEquation,
  TimeHomogeneousMarkovChain, NonhomogeneousMarkovProcess,
  StationaryDistribution, InvariantDistribution, LimitingDistribution,
  DetailedBalance, Reversibility, Irreducibility, CommunicatingStates,
  CommunicationClass, ClosedClass, Periodicity, Aperiodicity,
  Recurrence, PositiveRecurrence, NullRecurrence, Transience,
  AbsorbingState, AbsorbingMarkovChain, AbsorptionProbability,
  FundamentalMatrix, HittingTime, FirstPassageTime, ReturnTime,
  MeanRecurrenceTime, Ergodicity, Mixing, MixingTime, SpectralGap,
  GeneratorMatrix, InfinitesimalGenerator, QMatrix, HoldingTime,
  JumpChain, KolmogorovForwardEquation, KolmogorovBackwardEquation,
  BirthDeathProcess, PoissonProcess, QueueingProcess, ReliabilityProcess,
  HiddenMarkovModel, MarkovRewardProcess, MarkovDecisionProcess,
  RandomWalk, BrownianMotion, DiffusionProcess, StrongMarkovProperty,
  StoppingTime, MonteCarloSimulation, MarkovChainMonteCarlo,
  PathSimulation, ParameterEstimation
 ]
 EXCLUDE_AS_PRIMARY = [
  IndependentSequenceWithoutStateTransitionStructure,
  GeneralStochasticProcessWithoutMarkovProperty,
  DeterministicDynamicalSystemWithoutStochasticTransitions
 ]
 DISAMBIGUATION_RULE = "When stochastic evolution is conditionally determined by the present state under a specified state representation, interpret the context as a Markov process."
}

CORE_CONCEPTS {
 MarkovProcess = "Stochastic process satisfying the Markov conditional-independence property."
 MarkovProperty = "Future conditional distribution depends on the present state rather than the full past."
 StateSpace = "Set of possible states of the process."
 TransitionProbability = "Probability of moving between states over a specified interval."
 TransitionKernel = "Conditional probability kernel defining Markov evolution on general state spaces."
 StationaryDistribution = "Distribution unchanged by Markov evolution."
 Recurrence = "Property describing eventual return to a state or set."
 HittingTime = "First time a process reaches a specified state or set."
 GeneratorMatrix = "Infinitesimal transition-rate operator for a continuous-time Markov chain."
}

RELATIONS {
 MarkovProcess --is_child_of--> AppliedStatisticalMathematics
 MarkovProcess --is_type_of--> StochasticProcess
 MarkovProcess --satisfies--> MarkovProperty
 MarkovChain --is_type_of--> MarkovProcess
 TransitionMatrix --defines--> DiscreteStateEvolution
 TransitionKernel --defines--> GeneralMarkovEvolution
 ChapmanKolmogorovEquation --composes--> TransitionProbabilities
 StationaryDistribution --is_invariant_under--> TransitionKernel
 DetailedBalance --supports--> Reversibility
 Irreducibility --connects--> StateSpace
 AbsorbingState --has--> SelfTransitionProbabilityOne
 HittingTime --measures--> FirstEntry
 MixingTime --measures--> ApproachToStationarity
 SpectralGap --influences--> MixingRate
 GeneratorMatrix --defines--> ContinuousTimeTransitionRates
 KolmogorovForwardEquation --evolves--> StateProbabilities
 BirthDeathProcess --is_type_of--> ContinuousTimeMarkovChain
 MarkovDecisionProcess --extends--> MarkovProcessWithActions
 MarkovChainMonteCarlo --constructs--> MarkovChain
}

OPERATORS {
 MARKOV_PROPERTY(X,i,j,t) := P(X_(t+1)=j|X_t=i,history)=P(X_(t+1)=j|X_t=i)
 ONE_STEP_TRANSITION(P,i,j) := P_ij=P(X_(t+1)=j|X_t=i)
 N_STEP_TRANSITION(P,n) := P_n=P^n
 STATE_EVOLUTION(mu,P,n) := mu_n=mu_0*P^n
 CHAPMAN_KOLMOGOROV(P,n,m) := P^(n+m)=P^n*P^m
 STATIONARY_DISTRIBUTION(pi,P) := pi=pi*P
 DETAILED_BALANCE(pi,P,i,j) := pi_i*P_ij=pi_j*P_ji
 HITTING_TIME(A) := tau_A=inf{t>=0:X_t in A}
 CTMC_GENERATOR(Q) := q_ii=-sum_(j!=i)q_ij
 CTMC_TRANSITION(Q,t) := P(t)=exp(Q*t)
 FORWARD_EQUATION(p,Q) := dp(t)/dt=p(t)*Q
}

AXIOMS {
 A1 = "The Markov property is defined relative to a chosen state representation."
 A2 = "Transition probabilities must be nonnegative and normalised."
 A3 = "Time-homogeneous chains use transition laws independent of absolute time."
 A4 = "Chapman-Kolmogorov consistency governs composition of transitions."
 A5 = "Stationarity does not automatically imply uniqueness or convergence from every initial state."
 A6 = "Irreducibility and aperiodicity materially affect long-run behaviour."
 A7 = "CTMC generators require nonnegative off-diagonal rates and row sums of zero."
 A8 = "Markov-process conclusions require state, transition and long-run validation."
}

REASONING {
 R1_CONTEXT = identify(state_space,time_index,state_definition,observations,transition_mechanism)
 R2_CLASS = classify(discrete_or_continuous_time,discrete_or_continuous_state,homogeneous_or_nonhomogeneous)
 R3_TRANSITION = construct(transition_matrix,kernel_or_generator)
 R4_STRUCTURE = analyse(communication_classes,irreducibility,periodicity,absorption)
 R5_DISTRIBUTION = propagate(initial_distribution)
 R6_LONG_RUN = solve(stationary_distribution,recurrence,mixing,limiting_behaviour)
 R7_PASSAGE = compute(hitting_times,return_times,absorption_probabilities)
 R8_CONTINUOUS_TIME = if_required analyse(generator,holding_times,forward_backward_equations)
 R9_INFERENCE = if_data estimate_transition_parameters_and_uncertainty
 R10_VALIDATION = check(normalisation,Markov_assumption,stationarity_claims,simulation_consistency)
}

CONSTRAINTS {
 C1 = "A process can appear non-Markovian if the chosen state omits relevant information."
 C2 = "A stationary distribution need not be a limiting distribution."
 C3 = "Reducible chains can possess multiple stationary distributions."
 C4 = "Periodic chains can fail to converge pointwise despite having a stationary distribution."
 C5 = "Continuous-state Markov processes require kernel semantics."
 C6 = "Estimated transitions can be unreliable for rarely observed events."
 C7 = "CTMC generator entries are rates, not one-step probabilities."
 C8 = "MCMC dependence is intentional and must not be interpreted as IID sampling."
 C9 = "Simulation accuracy does not correct a misspecified state representation."
}

VALIDATION {
 V1 = "State space and state representation are explicit."
 V2 = "Time index is classified as discrete or continuous."
 V3 = "Transition probabilities or rates satisfy validity constraints."
 V4 = "Markov assumption is assessed where possible."
 V5 = "Communication classes and irreducibility are assessed."
 V6 = "Periodicity is checked before claiming convergence."
 V7 = "Stationary distributions are verified by the transition law."
 V8 = "Generator matrices satisfy rate constraints."
 V9 = "Estimated transition models include uncertainty where appropriate."
 V10 = "Simulated paths reproduce theoretical properties within tolerance."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MARKOV_PROCESS_DEFINITION = 0.99
 TRANSITION_REASONING = 0.99
 STATIONARY_REASONING = 0.99
 RECURRENCE_REASONING = 0.98
 CTMC_REASONING = 0.99
 ERGODIC_REASONING = 0.98
 POLICY = "Reduce confidence when state representation, transition law, time homogeneity, periodicity, generator rates or observational coverage are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, MarkovProcessReasoning,
  MarkovPropertyReasoning, MarkovChainReasoning, TransitionMatrixReasoning,
  TransitionKernelReasoning, ChapmanKolmogorovReasoning,
  StationaryDistributionReasoning, RecurrenceReasoning,
  AbsorptionReasoning, HittingTimeReasoning, ErgodicityReasoning,
  MixingReasoning, CTMCReasoning, GeneratorReasoning,
  BirthDeathReasoning, MCMCRelationshipReasoning, PathSimulationReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "Markov process" -> MarkovProcess
  "Markovian process" -> MarkovProcess
  "Markov chain" -> MarkovChain
  "transition kernel" -> TransitionKernel
  "stationary distribution" -> StationaryDistribution
  "CTMC" -> ContinuousTimeMarkovChain
  "DTMC" -> DiscreteTimeMarkovChain
  "Q matrix" -> GeneratorMatrix
 }
}

EXECUTION_RULES {
 E1 = "If future stochastic evolution depends conditionally on the present state, activate MarkovProcessOntology."
 E2 = "If time and state are discrete, use discrete-time Markov-chain reasoning."
 E3 = "If time is continuous with discrete states, use CTMC generator reasoning."
 E4 = "If state space is general or continuous, use transition-kernel semantics."
 E5 = "For long-run behaviour analyse irreducibility, recurrence, periodicity and invariant distributions."
 E6 = "For absorption compute absorption probabilities and expected passage times."
 E7 = "For CTMCs distinguish transition probabilities from instantaneous rates."
 E8 = "For estimated transitions quantify uncertainty."
 E9 = "Validate state sufficiency, transition validity and claimed long-run behaviour."
}

RECURSION {
 ENTRY = MarkovProcess
 EXPAND(MarkovProcess) -> [
  MarkovProperty, StochasticProcess, State, StateSpace, MarkovChain,
  DiscreteTimeMarkovChain, ContinuousTimeMarkovChain, TransitionProbability,
  TransitionMatrix, TransitionKernel, InitialDistribution, StateDistribution,
  ChapmanKolmogorovEquation, StationaryDistribution, DetailedBalance,
  Reversibility, Irreducibility, CommunicatingStates, Periodicity,
  Aperiodicity, Recurrence, Transience, AbsorbingState, AbsorptionProbability,
  HittingTime, ReturnTime, Ergodicity, Mixing, MixingTime, SpectralGap,
  GeneratorMatrix, HoldingTime, JumpChain, KolmogorovForwardEquation,
  KolmogorovBackwardEquation, BirthDeathProcess, PoissonProcess,
  HiddenMarkovModel, MarkovRewardProcess, MarkovDecisionProcess,
  RandomWalk, BrownianMotion, DiffusionProcess, StrongMarkovProperty,
  StoppingTime, MarkovChainMonteCarlo, PathSimulation
 ]
 STOP_CONDITION = "Stop when state representation, transition law, communication structure, long-run behaviour and passage properties are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Statistical Mathematics"
 RELATED_ONTOLOGIES = [
  "Applied Statistical Mathematics",
  "Time Series Analysis Ontology",
  "Monte Carlo Simulation Ontology",
  "Bayesian Statistics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Matrix Ontologies",
  "Spectral Theory Ontologies",
  "Queueing Theory Ontologies",
  "Reliability Mathematics Ontologies",
  "Reinforcement Learning Mathematics Ontology"
 ]
 ROUTE_RULES = [
  "Markov Process -> Applied Statistical Mathematics for parent semantics",
  "Markov Process -> Time Series Analysis for temporal observations",
  "Markov Process -> Monte Carlo Simulation for path simulation",
  "Markov Process -> Bayesian Statistics for transition inference",
  "Markov Process -> Probability & Statistics for stochastic foundations",
  "Markov Process -> Matrix and Spectral Theory for finite chains",
  "Markov Process -> Queueing and Reliability Mathematics for state transitions",
  "Markov Process -> Reinforcement Learning for MDP extensions"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Markov process classes or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve present-state conditional-independence semantics.",
  "Preserve discrete-time-versus-continuous-time distinction.",
  "Preserve probability-versus-rate distinction.",
  "Preserve stationary-versus-limiting distribution distinction.",
  "Preserve recurrent-versus-transient distinction.",
  "Preserve state-representation semantics.",
  "Preserve machine-readable Markov-process semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A Markov process is a stochastic process whose future conditional distribution depends on the present state rather than the complete past."
 TRANSITION_SUMMARY = "Transition matrices and kernels specify probabilistic state evolution."
 STATIONARY_SUMMARY = "Stationary distributions remain invariant under the transition law."
 STRUCTURE_SUMMARY = "Irreducibility, recurrence and periodicity determine important long-run properties."
 PASSAGE_SUMMARY = "Hitting, return and absorption times describe when states or sets are reached."
 CTMC_SUMMARY = "Continuous-time Markov chains use generator matrices and holding-time dynamics."
 VALIDATION_SUMMARY = "Markov models should be checked for state sufficiency, transition validity, periodicity and long-run consistency."
}

FLOW {
 INPUT
 -> detect_Markov_process_context
 -> identify_state_space_state_representation_time_index_and_transition_mechanism
 -> classify_discrete_or_continuous_time_and_state
 -> construct_transition_matrix_kernel_or_generator
 -> verify_probability_or_rate_constraints
 -> analyse_communication_classes_irreducibility_periodicity_recurrence_and_absorption
 -> propagate_initial_distribution
 -> compute_stationary_or_limiting_behaviour_if_applicable
 -> compute_hitting_return_or_absorption_quantities_if_required
 -> if_continuous_time_analyse_generator_and_Kolmogorov_equations
 -> validate_state_sufficiency_transition_structure_and_simulation_consistency
 -> route_to_applied_statistical_mathematics_time_series_Monte_Carlo_Bayesian_statistics_probability_stochastic_process_matrix_spectral_queueing_reliability_or_reinforcement_learning_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Markov Process Ontology defines stochastic processes with present-state conditional dependence. It supports discrete- and continuous-time Markov chains, state spaces, transition matrices and kernels, Chapman-Kolmogorov equations, stationary and limiting distributions, detailed balance and reversibility, communicating classes, irreducibility, periodicity, recurrence and transience, absorbing states and absorption probabilities, hitting and return times, ergodicity and mixing, spectral gaps, CTMC generators and holding times, Kolmogorov equations, birth-death and Poisson processes, queueing and reliability processes, hidden Markov processes, Markov reward and decision processes, random walks, diffusion relationships, stopping times, MCMC relationships and path simulation."
}

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

}

END_FLOW