Genetic Algorithm Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Genetic Algorithm Mathematics Ontology within the Kireth Research Semantic Library. Genetic algorithms are population-based stochastic optimisation methods inspired by evolutionary processes. This ontology supports populations, chromosomes, genes, encodings, fitness functions, selection, crossover, mutation, elitism, diversity, constraint handling, convergence, premature convergence, schema reasoning, parameter control, hybrid methods, parallel evolution, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Genetic Algorithm Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Optimisation Computation / Genetic Algorithm Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/optimisation-computation/genetic-algorithm-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = GeneticAlgorithm
 ENTITY_CLASS = PopulationBasedEvolutionaryOptimisationFramework
 PRIMARY_CONTEXT = ComputationalOptimisation
 DEFINITION = "A genetic algorithm is a stochastic population-based optimisation method that iteratively evolves encoded candidate solutions using fitness-based selection, recombination and mutation."
 SYMBOLIC_FORMS = [
  "P_(t+1)=Evolve(P_t)",
  "fitness_i=F(x_i)",
  "offspring=Crossover(parent_1,parent_2)",
  "x'=Mutation(x)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for evolutionary populations, candidate encoding, fitness evaluation, selection, crossover, mutation, diversity, elitism, constraint handling, convergence and reproducibility."
}

SCOPE {
 INCLUDE = [
  GeneticAlgorithm,GA,EvolutionaryAlgorithm,EvolutionaryOptimisation,
  Population,Individual,CandidateSolution,Chromosome,Genome,Gene,
  Allele,Genotype,Phenotype,Encoding,Representation,
  BinaryEncoding,IntegerEncoding,RealValuedEncoding,
  PermutationEncoding,TreeEncoding,MixedEncoding,
  ObjectiveFunction,FitnessFunction,FitnessValue,
  RawFitness,ScaledFitness,NormalisedFitness,
  FitnessTransformation,FitnessScaling,
  MaximisationProblem,MinimisationProblem,
  SearchSpace,SolutionSpace,FeasibleSet,
  Generation,GenerationIndex,PopulationSize,
  InitialPopulation,RandomInitialisation,HeuristicInitialisation,
  SeededInitialisation,PopulationDiversity,GeneticDiversity,
  Selection,ParentSelection,SurvivorSelection,
  FitnessProportionateSelection,RouletteWheelSelection,
  StochasticUniversalSampling,TournamentSelection,
  RankSelection,TruncationSelection,BoltzmannSelection,
  SelectionPressure,SelectionProbability,
  Crossover,Recombination,OnePointCrossover,
  TwoPointCrossover,MultiPointCrossover,UniformCrossover,
  ArithmeticCrossover,BlendCrossover,BLXAlpha,
  SimulatedBinaryCrossover,SBX,OrderCrossover,OX,
  PartiallyMappedCrossover,PMX,CycleCrossover,
  CrossoverProbability,CrossoverRate,
  Mutation,BitFlipMutation,SwapMutation,InsertionMutation,
  InversionMutation,ScrambleMutation,GaussianMutation,
  UniformMutation,PolynomialMutation,MutationProbability,
  MutationRate,AdaptiveMutation,SelfAdaptiveMutation,
  Elitism,EliteIndividual,EliteCount,GenerationalReplacement,
  SteadyStateGA,ReplacementStrategy,AgeBasedReplacement,
  Crowding,Niching,FitnessSharing,RestrictedTournamentSelection,
  DiversityPreservation,Speciation,
  Constraint,ConstraintViolation,PenaltyFunction,
  PenaltyMethod,RepairOperator,FeasibilityRule,
  Decoder,IndirectEncoding,ConstraintHandling,
  MultiobjectiveOptimisation,ParetoFront,ParetoDominance,
  NondominatedSolution,NSGA,NSGAII,SPEA2,
  CrowdingDistance,RankFront,
  Schema,SchemaTheorem,BuildingBlockHypothesis,
  HammingDistance,GenotypicDistance,PhenotypicDistance,
  Epistasis,Linkage,DeceptiveFunction,
  Exploration,Exploitation,PrematureConvergence,
  GeneticDrift,LossOfDiversity,Convergence,
  PopulationConvergence,Stagnation,Restart,
  RandomRestart,Immigration,IslandModel,
  Migration,MigrationRate,MigrationTopology,
  ParallelGeneticAlgorithm,DistributedGA,MasterSlaveGA,
  FineGrainedGA,CellularGA,
  HybridGeneticAlgorithm,MemeticAlgorithm,LocalSearchHybrid,
  SimulatedAnnealingHybrid,GradientHybrid,
  AdaptiveParameterControl,DeterministicParameterControl,
  SelfAdaptiveParameterControl,PopulationSizing,
  PopulationSizingTheory,TakeoverTime,
  SelectionIntensity,ExpectedOffspring,
  MarkovChainModel,StochasticProcess,
  RandomSeed,PseudoRandomNumberGenerator,
  Reproducibility,MultipleRuns,BestOfRuns,MeanPerformance,
  MedianPerformance,VarianceAcrossRuns,ConfidenceInterval,
  StoppingCriterion,MaximumGenerations,EvaluationLimit,
  TimeLimit,TargetFitness,NoImprovementLimit,
  FunctionEvaluationCount,FitnessEvaluationCount,
  ComputationalCost,MemoryCost,
  FitnessCaching,DuplicateDetection,
  SurrogateFitness,SurrogateAssistedGA,
  BayesianSurrogate,ResponseSurface,
  Noise,NoisyFitness,RobustOptimisation,
  DynamicOptimisation,ChangingFitnessLandscape,
  FloatingPointArithmetic,MachinePrecision,
  RoundOffError,Overflow,Underflow,
  Verification,Benchmarking,ReferenceSolution,
  SolverStatus,Success,Failure,Warning
 ]
 EXCLUDE_AS_PRIMARY = [
  DeterministicGradientOptimisationWithoutPopulationEvolution,
  PureRandomSearchWithoutSelectionAndInheritance,
  BiologicalGeneticsWithoutOptimisationContext,
  GeneticProgrammingWhenTreeProgramEvolutionIsThePrimarySpecialisation
 ]
 DISAMBIGUATION_RULE = "When a population of encoded candidate solutions evolves using selection, crossover and mutation according to fitness, interpret the context as a genetic algorithm."
}

CORE_CONCEPTS {
 GeneticAlgorithm = "Population-based stochastic optimisation using evolutionary operators."
 Population = "Collection of candidate solutions evolved together."
 Chromosome = "Encoded representation of a candidate solution."
 FitnessFunction = "Function mapping a candidate solution to a selection-relevant quality measure."
 Selection = "Mechanism choosing individuals for reproduction or survival."
 Crossover = "Operator recombining information from two or more parents."
 Mutation = "Stochastic operator introducing variation into candidate representations."
 Elitism = "Preservation of selected high-quality individuals across generations."
}

RELATIONS {
 GeneticAlgorithm --is_child_of--> ComputationalOptimisation
 GeneticAlgorithm --is_type_of--> EvolutionaryAlgorithm
 Population --contains--> Individual
 Individual --has--> Chromosome
 Chromosome --contains--> Gene
 Genotype --maps_to--> Phenotype
 FitnessFunction --evaluates--> Individual
 Selection --uses--> FitnessValue
 SelectionPressure --influences--> DiversityLoss
 Crossover --combines--> ParentGenotypes
 Mutation --modifies--> OffspringGenotype
 Elitism --preserves--> EliteIndividual
 DiversityPreservation --reduces--> PrematureConvergence
 PenaltyMethod --handles--> ConstraintViolation
 RepairOperator --restores--> Feasibility
 NSGAII --targets--> MultiobjectiveOptimisation
 IslandModel --uses--> Migration
 MemeticAlgorithm --combines--> GeneticAlgorithmAndLocalSearch
 RandomSeed --supports--> Reproducibility
 Verification --checks--> GeneticAlgorithmResult
}

OPERATORS {
 INITIALISE(N,representation) := population_of_N_candidates
 FITNESS(x) := F(x)
 SELECT(P,F) := parent_set_according_to_selection_rule
 CROSSOVER(p1,p2,pc) := offspring_if_random_threshold_below_pc
 MUTATE(x,pm) := mutated_candidate_according_to_mutation_rule
 ELITIST_REPLACEMENT(P,offspring,e) := preserve_best_e_and_fill_remaining_slots
 PENALISED_FITNESS(x,lambda) := raw_fitness(x)-lambda*constraint_violation(x)
 HAMMING_DISTANCE(x,y) := number_of_differing_positions
 DIVERSITY(P) := aggregate_pairwise_or_population_variation_measure
}

AXIOMS {
 A1 = "Genetic algorithms operate on populations rather than a single incumbent solution."
 A2 = "Selection biases reproduction toward fitter individuals according to the chosen selection mechanism."
 A3 = "Crossover recombines inherited information while mutation introduces new variation."
 A4 = "Excessive selection pressure can reduce diversity and cause premature convergence."
 A5 = "Mutation rates that are too low can reduce exploration; excessively high rates can destroy useful structure."
 A6 = "Representation and genetic operators must be compatible with the optimisation problem."
 A7 = "Single-run outcomes are insufficient to characterise stochastic algorithm performance robustly."
 A8 = "Genetic-algorithm results require explicit population, operator, stopping and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(objective,representation,constraints,search_space,noise,accuracy_goal,computational_budget)
 R2_ENCODING = select(binary,integer,real,permutation,tree_or_mixed_representation)
 R3_INITIALISE = create_population_using_random_heuristic_or_seeded_strategy
 R4_EVALUATE = compute_fitness_and_constraint_status_for_population
 R5_SELECT = choose_parents_according_to_selection_rule_and_pressure
 R6_RECOMBINE = apply_crossover_with_declared_probability
 R7_MUTATE = apply_mutation_with_declared_rate_or_adaptive_policy
 R8_REPLACE = apply_elitist_generational_or_steady_state_survivor_strategy
 R9_MONITOR = track(best_fitness,mean_fitness,diversity,stagnation_and_constraint_violation)
 R10_VALIDATE = compare_multiple_runs,benchmarks,parameter_sensitivity_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Poor encoding can make useful genetic variation difficult to express."
 C2 = "Incompatible crossover or mutation operators can generate invalid candidates."
 C3 = "High selection pressure can cause premature convergence."
 C4 = "Low diversity can reduce the probability of escaping local optima."
 C5 = "Penalty functions can distort search if penalty magnitudes are poorly tuned."
 C6 = "Population sizes that are too small can lose useful genetic material rapidly."
 C7 = "Very large populations increase evaluation cost."
 C8 = "Noisy fitness can mislead selection and replacement."
 C9 = "A single best-of-run result does not quantify expected algorithm performance."
 C10 = "Resource-limit termination is not equivalent to convergence."
}

SEMANTIC_EQUIVALENCE {
 "GeneticAlgorithm" <=> "GA"
 "EvolutionaryOptimisation" <=> "EvolutionaryAlgorithmInOptimisationContext"
 "RouletteWheelSelection" <=> "FitnessProportionateSelection"
 "SimulatedBinaryCrossover" <=> "SBX"
 "PartiallyMappedCrossover" <=> "PMX"
 "OrderCrossover" <=> "OX"
 "MemeticAlgorithm" <=> "HybridGeneticAlgorithmWithLocalSearch"
}

VALIDATION {
 V1 = "Objective function and optimisation direction are explicit."
 V2 = "Representation and genotype-phenotype mapping are documented."
 V3 = "Population size and initialisation strategy are documented."
 V4 = "Selection mechanism and selection pressure are documented."
 V5 = "Crossover and mutation operators and probabilities are documented."
 V6 = "Replacement and elitism strategy are documented."
 V7 = "Constraint-handling strategy is documented where required."
 V8 = "Population diversity and convergence history are monitored where useful."
 V9 = "Multiple independent runs are compared for stochastic robustness."
 V10 = "Results are reproducible under documented random and algorithm settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 GENETIC_ALGORITHM_DEFINITION = 0.99
 SELECTION_REASONING = 0.99
 CROSSOVER_REASONING = 0.99
 MUTATION_REASONING = 0.99
 DIVERSITY_REASONING = 0.98
 CONSTRAINT_REASONING = 0.98
 MULTIOBJECTIVE_REASONING = 0.98
 STOCHASTIC_PERFORMANCE_REASONING = 0.97
 POLICY = "Reduce confidence when encoding, operator compatibility, population size, selection pressure, mutation rate, constraint handling or run-to-run variability are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,GeneticAlgorithmReasoning,
  PopulationReasoning,EncodingReasoning,FitnessReasoning,
  SelectionReasoning,CrossoverReasoning,MutationReasoning,
  ElitismReasoning,DiversityReasoning,
  PrematureConvergenceReasoning,ConstraintHandlingReasoning,
  SchemaReasoning,MultiobjectiveGAReasoning,
  IslandModelReasoning,ParallelGAReasoning,
  MemeticAlgorithmReasoning,ParameterControlReasoning,
  StochasticPerformanceReasoning,ReproducibilityReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "genetic algorithm" -> GeneticAlgorithm
  "GA" -> GeneticAlgorithm
  "roulette wheel selection" -> FitnessProportionateSelection
  "SBX" -> SimulatedBinaryCrossover
  "PMX" -> PartiallyMappedCrossover
  "OX" -> OrderCrossover
  "memetic algorithm" -> MemeticAlgorithm
 }
}

EXECUTION_RULES {
 E1 = "If optimisation evolves a population using fitness-based selection, crossover and mutation, activate GeneticAlgorithmMathematicsOntology."
 E2 = "Choose a representation compatible with the structure of the optimisation variables."
 E3 = "Define selection, crossover, mutation and replacement rules explicitly."
 E4 = "Preserve or restore feasibility when genetic operators can violate constraints."
 E5 = "Monitor diversity to detect premature convergence."
 E6 = "Use elitism carefully to preserve strong solutions without eliminating variation."
 E7 = "For multiobjective problems, use Pareto-ranking and diversity mechanisms appropriate to the chosen algorithm."
 E8 = "Assess stochastic performance over multiple independent runs."
 E9 = "Validate against benchmarks, parameter sensitivity and reproducibility controls."
}

RECURSION {
 ENTRY = GeneticAlgorithm
 EXPAND(GeneticAlgorithm) -> [
  GA,EvolutionaryAlgorithm,EvolutionaryOptimisation,Population,
  Individual,CandidateSolution,Chromosome,Genome,Gene,Allele,
  Genotype,Phenotype,Encoding,BinaryEncoding,IntegerEncoding,
  RealValuedEncoding,PermutationEncoding,TreeEncoding,MixedEncoding,
  ObjectiveFunction,FitnessFunction,FitnessValue,FitnessScaling,
  SearchSpace,FeasibleSet,Generation,PopulationSize,InitialPopulation,
  RandomInitialisation,HeuristicInitialisation,PopulationDiversity,
  Selection,ParentSelection,SurvivorSelection,
  FitnessProportionateSelection,RouletteWheelSelection,
  TournamentSelection,RankSelection,StochasticUniversalSampling,
  SelectionPressure,Crossover,OnePointCrossover,TwoPointCrossover,
  UniformCrossover,ArithmeticCrossover,BlendCrossover,
  SimulatedBinaryCrossover,OrderCrossover,PartiallyMappedCrossover,
  CycleCrossover,CrossoverProbability,Mutation,BitFlipMutation,
  SwapMutation,InsertionMutation,InversionMutation,GaussianMutation,
  PolynomialMutation,MutationProbability,AdaptiveMutation,
  Elitism,EliteIndividual,GenerationalReplacement,SteadyStateGA,
  Crowding,Niching,FitnessSharing,DiversityPreservation,
  ConstraintViolation,PenaltyMethod,RepairOperator,FeasibilityRule,
  MultiobjectiveOptimisation,ParetoFront,ParetoDominance,
  NSGAII,SPEA2,CrowdingDistance,Schema,SchemaTheorem,
  BuildingBlockHypothesis,HammingDistance,Epistasis,Exploration,
  Exploitation,PrematureConvergence,GeneticDrift,Stagnation,
  Restart,Immigration,IslandModel,Migration,ParallelGeneticAlgorithm,
  DistributedGA,CellularGA,HybridGeneticAlgorithm,MemeticAlgorithm,
  AdaptiveParameterControl,PopulationSizing,RandomSeed,
  MultipleRuns,StoppingCriterion,MaximumGenerations,EvaluationLimit,
  TargetFitness,NoImprovementLimit,FitnessCaching,
  SurrogateAssistedGA,NoisyFitness,DynamicOptimisation,
  FloatingPointArithmetic,Verification,Benchmarking,
  ReferenceSolution,Reproducibility
 ]
 EXPAND(GeneticAlgorithmFamilies) -> [
  BinaryGeneticAlgorithms,
  RealCodedGeneticAlgorithms,
  PermutationGeneticAlgorithms,
  MultiobjectiveGeneticAlgorithms,
  ConstrainedGeneticAlgorithms,
  AdaptiveGeneticAlgorithms,
  MemeticAlgorithms,
  ParallelAndIslandGeneticAlgorithms
 ]
 STOP_CONDITION = "Stop when representation, population, fitness, selection, crossover, mutation, replacement, diversity, constraints, stopping and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Optimisation Ontology"
 RELATED_ONTOLOGIES = [
  "Computational Optimisation Ontology",
  "Optimisation Computation",
  "Nonlinear Optimisation Solver Ontology",
  "Simulated Annealing Ontology",
  "Stochastic Gradient Descent Ontology",
  "Monte Carlo Simulation Ontology",
  "Markov Process Ontology",
  "Random Process Ontology",
  "Probability and Statistics Ontologies",
  "Numerical Convergence Ontology",
  "Scientific Computing Ontology 2",
  "Artificial Intelligence Mathematics"
 ]
 ROUTE_RULES = [
  "Genetic Algorithm -> Computational Optimisation for parent semantics",
  "Genetic Algorithm -> Nonlinear Optimisation Solver for nonconvex objective context",
  "Genetic Algorithm -> Simulated Annealing for stochastic global-search comparison",
  "Genetic Algorithm -> Monte Carlo Simulation for random sampling relationships",
  "Genetic Algorithm -> Random Process for population-evolution stochasticity",
  "Genetic Algorithm -> Probability and Statistics for selection and variation probabilities",
  "Genetic Algorithm -> Numerical Convergence for population convergence concepts",
  "Genetic Algorithm -> Scientific Computing for parallel population evaluation",
  "Genetic Algorithm -> Artificial Intelligence Mathematics for evolutionary learning and search"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible genetic operators, population models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve genotype-versus-phenotype distinction.",
  "Preserve population-versus-individual distinction.",
  "Preserve selection-versus-crossover-versus-mutation distinction.",
  "Preserve current-population-versus-best-solution distinction.",
  "Preserve exploration-versus-exploitation distinction.",
  "Preserve single-run-versus-multiple-run performance distinction.",
  "Preserve stochastic-search-versus-deterministic-objective distinction.",
  "Preserve machine-readable genetic-algorithm semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A genetic algorithm optimises by evolving a population of encoded candidate solutions through fitness-based selection, crossover and mutation."
 REPRESENTATION_SUMMARY = "Binary, integer, real-valued, permutation and mixed encodings map optimisation variables into chromosomes."
 SELECTION_SUMMARY = "Selection controls reproductive bias and selection pressure."
 VARIATION_SUMMARY = "Crossover recombines inherited information while mutation introduces new variation."
 DIVERSITY_SUMMARY = "Population diversity helps preserve exploration and reduce premature convergence."
 MULTIOBJECTIVE_SUMMARY = "Pareto-ranking, nondomination and crowding mechanisms support multiobjective genetic algorithms."
 VALIDATION_SUMMARY = "Genetic-algorithm results should be checked through multiple runs, diversity histories, operator settings, benchmark comparison and reproducibility."
}

FLOW {
 INPUT
 -> detect_genetic_algorithm_context
 -> identify_objective_search_space_representation_constraints_noise_accuracy_goal_and_budget
 -> choose_encoding_and_initial_population
 -> evaluate_fitness_and_constraint_status
 -> select_parents
 -> apply_crossover
 -> apply_mutation
 -> repair_or_penalise_infeasible_offspring_if_required
 -> apply_elitist_generational_or_steady_state_replacement
 -> update_population_and_best_solution
 -> monitor_best_mean_fitness_diversity_stagnation_and_constraint_violation
 -> adapt_population_operator_rates_restart_or_migration_if_required
 -> stop_on_generation_evaluation_time_target_or_no_improvement_criterion
 -> assess_multiple_run_statistics_and_solution_quality
 -> validate_against_reference_or_benchmark_solution_and_reproducibility
 -> route_to_related_optimisation_simulated_annealing_Monte_Carlo_random_process_probability_convergence_scientific_computing_and_AI_mathematics_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Genetic Algorithm Mathematics Ontology defines stochastic population-based optimisation using evolutionary operators. It supports populations, chromosomes, genes and genotype-phenotype mappings, binary, integer, real-valued and permutation encodings, fitness scaling, roulette-wheel, tournament, rank and stochastic-universal selection, one-point, two-point, uniform, arithmetic, blend, SBX, OX and PMX crossover, bit-flip, swap, insertion, inversion, Gaussian and polynomial mutation, elitism, steady-state replacement, diversity preservation, crowding and niching, penalty and repair constraint handling, Pareto ranking and NSGA-II-style multiobjective search, schema and building-block reasoning, island models, migration, memetic hybrids, adaptive parameters, stochastic run analysis, verification, benchmarking and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-GENETIC-ALGORITHM-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW