Transformer Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Transformer Mathematics Ontology within the Kireth Research Semantic Library. Transformer mathematics concerns the mathematical structures and operations underlying transformer architectures, including token and positional embeddings, linear projections, scaled dot-product attention, multi-head attention, residual connections, normalisation, feed-forward networks, masking, sequence modelling, optimisation, numerical stability, complexity, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Transformer Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "AI Computational Mathematics / Transformer Mathematics"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/ai-computational-mathematics/transformer-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = TransformerMathematics
 ENTITY_CLASS = AttentionBasedSequenceComputationFramework
 PRIMARY_CONTEXT = AIComputationalMathematicsOntology
 DEFINITION = "Transformer mathematics is the mathematical framework describing embedding, attention, residual, normalisation and feed-forward operations in transformer neural architectures."
 SYMBOLIC_FORMS = [
  "Q=XW_Q",
  "K=XW_K",
  "V=XW_V",
  "Attention(Q,K,V)=softmax(QK^T/sqrt(d_k))V"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for attention, multi-head projections, positional information, masking, residual pathways, normalisation, feed-forward computation, training and scalable transformer execution."
}

SCOPE {
 INCLUDE = [
  TransformerMathematics,Transformer,TransformerArchitecture,
  SequenceModel,SequenceRepresentation,
  Token,TokenSequence,SequenceLength,
  VocabularyRelationship,TokenIndex,
  TokenEmbedding,EmbeddingVector,
  EmbeddingMatrix,
  PositionalEmbedding,PositionalEncoding,
  LearnedPositionalEmbedding,
  SinusoidalPositionalEncoding,
  RotaryPositionEmbeddingRelationship,
  RoPERelationship,
  RelativePositionEncodingRelationship,
  AbsolutePositionEncodingRelationship,
  InputRepresentation,
  HiddenRepresentation,
  HiddenState,
  ModelDimension,dModelRelationship,
  ProjectionDimension,
  HeadDimension,
  Query,Key,Value,
  QueryMatrix,KeyMatrix,ValueMatrix,
  QueryProjection,KeyProjection,ValueProjection,
  WeightMatrix,
  LinearProjection,
  ScaledDotProductAttention,
  DotProductAttention,
  SelfAttention,
  CrossAttention,
  MultiHeadAttention,
  AttentionHead,
  HeadCount,
  AttentionScore,
  AttentionLogit,
  SimilarityScore,
  QueryKeyProduct,
  ScalingFactor,
  SquareRootScaling,
  Softmax,
  SoftmaxNormalisation,
  AttentionWeight,
  AttentionMatrix,
  WeightedValueSum,
  CausalAttention,
  BidirectionalAttentionRelationship,
  MaskedAttention,
  AttentionMask,
  CausalMask,
  PaddingMask,
  AdditiveMaskRelationship,
  MultiplicativeMaskRelationship,
  LowerTriangularMaskRelationship,
  SequenceMask,
  DecoderMaskRelationship,
  EncoderSelfAttentionRelationship,
  DecoderSelfAttentionRelationship,
  EncoderDecoderAttentionRelationship,
  MultiQueryAttentionRelationship,
  GroupedQueryAttentionRelationship,
  GQARelationship,
  MQARelationship,
  SparseAttentionRelationship,
  LocalAttentionRelationship,
  WindowedAttentionRelationship,
  SlidingWindowAttentionRelationship,
  LinearAttentionRelationship,
  KernelAttentionRelationship,
  FlashAttentionRelationship,
  MemoryEfficientAttentionRelationship,
  ResidualConnection,
  SkipConnection,
  ResidualStream,
  AddAndNormRelationship,
  LayerNormalisation,
  RMSNormRelationship,
  PreNorm,
  PostNorm,
  Normalisation,
  Mean,
  Variance,
  Epsilon,
  FeedForwardNetwork,
  PositionwiseFeedForward,
  MLP,
  LinearLayer,
  HiddenLayer,
  IntermediateDimension,
  ExpansionRatioRelationship,
  ActivationFunction,
  ReLU,
  GELU,
  SwiGLURelationship,
  GLURelationship,
  GatedActivationRelationship,
  Dropout,
  AttentionDropout,
  ResidualDropoutRelationship,
  Encoder,
  Decoder,
  EncoderLayer,
  DecoderLayer,
  TransformerBlock,
  StackOfLayers,
  Depth,
  LayerCount,
  Parameter,
  TrainableParameter,
  Weight,
  Bias,
  ParameterMatrix,
  ParameterTensor,
  ForwardPropagation,
  ForwardPass,
  ComputationalGraph,
  MatrixMultiplication,
  BatchedMatrixMultiplication,
  TensorContraction,
  Transpose,
  Reshape,
  Permutation,
  Broadcasting,
  Concatenation,
  HeadConcatenation,
  OutputProjection,
  Logits,
  VocabularyLogitsRelationship,
  OutputDistribution,
  LanguageModellingRelationship,
  AutoregressiveModellingRelationship,
  NextTokenPredictionRelationship,
  SequenceToSequenceRelationship,
  ClassificationTokenRelationship,
  CLSRelationship,
  PoolingRelationship,
  LossFunction,
  CrossEntropy,
  NegativeLogLikelihood,
  LabelSmoothingRelationship,
  TrainingObjective,
  Backpropagation,
  AutomaticDifferentiation,
  ReverseModeAD,
  Gradient,
  JacobianRelationship,
  HessianRelationship,
  OptimisationForMachineLearning,
  GradientDescent,
  StochasticGradientDescent,
  Adam,
  AdamW,
  WeightDecay,
  LearningRate,
  LearningRateSchedule,
  Warmup,
  CosineDecayRelationship,
  GradientClipping,
  MixedPrecision,
  LossScaling,
  FloatingPointArithmetic,
  MachinePrecision,
  RoundOffError,
  Overflow,
  Underflow,
  NumericalStability,
  SoftmaxStability,
  LogSumExpRelationship,
  StableSoftmax,
  AttentionEntropyRelationship,
  Entropy,
  TemperatureRelationship,
  AttentionCollapseRelationship,
  HeadRedundancyRelationship,
  RepresentationCollapseRelationship,
  VanishingGradient,
  ExplodingGradient,
  Initialisation,
  XavierInitialisationRelationship,
  ResidualScalingRelationship,
  DeepTransformerRelationship,
  GradientFlow,
  OptimisationLandscapeRelationship,
  EmbeddingMathematics,
  VectorComputation,
  ComputationalMatrix,
  TensorComputation,
  DenseDataStructure,
  SparseDataStructureRelationship,
  ComputationalGraphMathematics,
  NeuralNetworkMathematics,
  SequenceLengthComplexity,
  AttentionComplexity,
  QuadraticAttentionComplexity,
  TimeComplexity,
  SpaceComplexity,
  MemoryComplexity,
  ParameterCount,
  FLOPCountRelationship,
  KVCache,
  KeyValueCache,
  CacheLength,
  AutoregressiveDecoding,
  IncrementalDecoding,
  PrefillRelationship,
  DecodingStep,
  BeamSearchRelationship,
  GreedyDecodingRelationship,
  SamplingRelationship,
  TemperatureSamplingRelationship,
  TopKSamplingRelationship,
  TopPSamplingRelationship,
  ParallelTraining,
  DataParallelism,
  TensorParallelism,
  PipelineParallelism,
  SequenceParallelismRelationship,
  DistributedTraining,
  ModelParallelism,
  GPUComputing,
  HighPerformanceComputing,
  CommunicationCost,
  AllReduceRelationship,
  ShardingRelationship,
  ParameterShardingRelationship,
  ActivationCheckpointing,
  GradientCheckpointing,
  Rematerialisation,
  MemoryTimeTradeoff,
  Verification,
  ShapeCheck,
  MaskCheck,
  SoftmaxCheck,
  AttentionWeightCheck,
  GradientCheck,
  ReferenceOutput,
  KnownAnswerTest,
  Benchmark,
  Reproducibility,
  Provenance,
  ModelVersion,
  ArchitectureVersion,
  TokeniserVersionRelationship,
  FrameworkVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  GeneralNeuralNetworkMathematicsWithoutAttentionStructure,
  RecurrentNeuralNetworksWhenRecurrenceRatherThanAttentionIsPrimary,
  EmbeddingMathematicsWhenRepresentationGeometryRatherThanTransformerComputationIsPrimary,
  LanguageModellingSemanticsWithoutTransformerMathematicalStructure
 ]
 DISAMBIGUATION_RULE = "When sequence representations are transformed through query-key-value attention, residual, normalisation and feed-forward operations, interpret the context as transformer mathematics."
}

CORE_CONCEPTS {
 TransformerMathematics = "Mathematics of attention-based transformer architectures."
 Query = "Projected representation used to request information from keys."
 Key = "Projected representation used to determine compatibility with queries."
 Value = "Projected representation combined according to attention weights."
 ScaledDotProductAttention = "Attention mechanism based on scaled query-key dot products followed by weighted value aggregation."
 MultiHeadAttention = "Parallel attention heads operating in distinct projected subspaces."
 PositionalEncoding = "Representation injecting sequence-order information."
 ResidualConnection = "Additive skip pathway preserving and transforming hidden representations."
 FeedForwardNetwork = "Positionwise nonlinear transformation applied independently across sequence positions."
}

RELATIONS {
 TransformerMathematics --is_child_of--> AIComputationalMathematicsOntology
 TransformerMathematics --is_type_of--> NeuralNetworkMathematics
 Transformer --contains--> TransformerBlocks
 TransformerBlock --contains--> AttentionAndFeedForwardSubLayers
 TokenEmbedding --maps--> TokensToVectors
 PositionalEncoding --adds--> SequencePositionInformation
 QueryProjection --produces--> QueryMatrix
 KeyProjection --produces--> KeyMatrix
 ValueProjection --produces--> ValueMatrix
 ScaledDotProductAttention --uses--> QueryKeyValue
 Softmax --normalises--> AttentionScores
 MultiHeadAttention --combines--> AttentionHeads
 ResidualConnection --adds--> InputAndSublayerOutput
 LayerNormalisation --normalises--> HiddenRepresentation
 FeedForwardNetwork --transforms--> PositionwiseRepresentations
 Backpropagation --computes--> TransformerParameterGradients
 Verification --checks--> TransformerComputation
}

OPERATORS {
 QUERY(X,WQ) := X*WQ
 KEY(X,WK) := X*WK
 VALUE(X,WV) := X*WV
 ATTENTION_SCORES(Q,K) := Q*transpose(K)/sqrt(d_k)
 MASK(scores,M) := scores + M
 ATTENTION_WEIGHTS(Q,K,M) := softmax(MASK(ATTENTION_SCORES(Q,K),M))
 ATTENTION(Q,K,V,M) := ATTENTION_WEIGHTS(Q,K,M)*V
 MULTIHEAD(X) := concatenate(head_1(X),...,head_h(X))*W_O
 FFN(x) := activation(x*W_1+b_1)*W_2+b_2
}

AXIOMS {
 A1 = "Transformer attention computes weighted combinations of value vectors from query-key compatibility."
 A2 = "Scaled dot-product attention divides query-key products by the square root of key dimension to moderate logit magnitude."
 A3 = "Attention weights are normalised across the attended dimension."
 A4 = "Causal masking prevents access to future sequence positions in autoregressive attention."
 A5 = "Multi-head attention uses multiple learned projections rather than a single shared attention geometry."
 A6 = "Residual and normalisation placement affects optimisation and numerical behaviour."
 A7 = "Standard dense self-attention has quadratic sequence-length interaction cost."
 A8 = "Transformer results require explicit shape, projection, masking, normalisation, precision, optimisation and validation semantics."
}

REASONING {
 R1_CONTEXT = identify(sequence_length,model_dimension,head_count,head_dimension,masking,architecture,task,hardware)
 R2_EMBED = construct_token_and_positional_representations)
 R3_PROJECT = compute_query_key_value_linear_projections)
 R4_SCORE = compute_scaled_query_key_compatibility_scores)
 R5_MASK = apply_causal_padding_or_task_specific_attention_mask)
 R6_NORMALISE = apply_stable_softmax_to_scores)
 R7_AGGREGATE = compute_weighted_value_sum_for_each_head)
 R8_COMBINE = concatenate_heads_and_apply_output_projection)
 R9_BLOCK = apply_residual_normalisation_and_feed_forward_transformations)
 R10_VALIDATE = check_shapes_masks_numerics_gradients_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Query, key and value dimensions must be compatible within each attention head."
 C2 = "Incorrect masks can leak future or padded information."
 C3 = "Large attention logits can saturate softmax."
 C4 = "Quadratic attention can become memory intensive at long sequence lengths."
 C5 = "Mixed precision can destabilise softmax or normalisation without numerical safeguards."
 C6 = "Attention weights are not automatically equivalent to causal or semantic explanations."
 C7 = "KV-cache memory grows with decoded context length."
 C8 = "Deep post-norm and pre-norm transformers have different optimisation behaviour."
 C9 = "Head redundancy can reduce effective representational diversity."
 C10 = "Equivalent tensor rearrangements can have different memory and execution costs."
}

SEMANTIC_EQUIVALENCE {
 "GraphNeuralNetwork" <=> "GNN"
 "MultiQueryAttention" <=> "MQA"
 "GroupedQueryAttention" <=> "GQA"
 "KeyValueCache" <=> "KVCache"
 "MultilayerPerceptron" <=> "MLP"
}

VALIDATION {
 V1 = "Sequence length, model dimension, head count and head dimensions are documented."
 V2 = "Q, K, V and output projection shapes are validated."
 V3 = "Attention mask semantics are documented and tested."
 V4 = "Softmax is computed using numerically stable semantics."
 V5 = "Attention weights sum to one across the declared attention axis."
 V6 = "Residual and normalisation ordering is documented."
 V7 = "Forward outputs are compared against trusted reference implementations where feasible."
 V8 = "Backpropagated gradients are checked where feasible."
 V9 = "Mixed precision, KV-cache and distributed execution settings are documented where used."
 V10 = "Results are reproducible under documented model, architecture, framework and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRANSFORMER_MATHEMATICS_DEFINITION = 0.99
 ATTENTION_REASONING = 0.99
 MULTIHEAD_REASONING = 0.99
 MASK_REASONING = 0.99
 POSITIONAL_REASONING = 0.98
 NUMERICAL_STABILITY_REASONING = 0.98
 DISTRIBUTED_TRANSFORMER_REASONING = 0.98
 POLICY = "Reduce confidence when tensor shapes, masking axis, positional semantics, normalisation order, precision or distributed partitioning are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,TransformerMathematicsReasoning,
  AttentionReasoning,QueryKeyValueReasoning,
  ScaledDotProductAttentionReasoning,
  MultiHeadAttentionReasoning,MaskReasoning,
  PositionalEncodingReasoning,ResidualReasoning,
  LayerNormalisationReasoning,FeedForwardReasoning,
  KVCacheReasoning,AutoregressiveReasoning,
  ComplexityReasoning,MixedPrecisionReasoning,
  DistributedTransformerReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "transformer mathematics" -> TransformerMathematics
  "multi-head attention" -> MultiHeadAttention
  "MQA" -> MultiQueryAttention
  "GQA" -> GroupedQueryAttention
  "KV cache" -> KeyValueCache
  "MLP" -> MultilayerPerceptron
 }
}

EXECUTION_RULES {
 E1 = "If sequence representations are transformed through Q-K-V attention blocks, activate TransformerMathematicsOntology."
 E2 = "Validate model, head and projection dimensions before attention computation."
 E3 = "Scale query-key dot products by sqrt(d_k) before softmax in standard scaled attention."
 E4 = "Apply causal, padding or task-specific masks before softmax."
 E5 = "Use numerically stable softmax and normalisation."
 E6 = "Preserve residual and normalisation ordering as part of architecture semantics."
 E7 = "Account for quadratic sequence-length cost in dense self-attention."
 E8 = "Track KV-cache semantics separately for prefill and autoregressive decoding."
 E9 = "Verify shapes, masks, gradients, numerical stability and reproducibility."
}

RECURSION {
 ENTRY = TransformerMathematics
 EXPAND(TransformerMathematics) -> [
  Transformer,SequenceModel,TokenEmbedding,PositionalEmbedding,
  PositionalEncoding,EmbeddingMatrix,Query,Key,Value,
  QueryMatrix,KeyMatrix,ValueMatrix,LinearProjection,
  ScaledDotProductAttention,SelfAttention,CrossAttention,
  MultiHeadAttention,AttentionHead,AttentionScore,Softmax,
  AttentionWeight,AttentionMatrix,CausalAttention,AttentionMask,
  CausalMask,PaddingMask,MultiQueryAttention,GroupedQueryAttention,
  ResidualConnection,LayerNormalisation,RMSNormRelationship,
  PreNorm,PostNorm,FeedForwardNetwork,MLP,GELU,SwiGLURelationship,
  Encoder,Decoder,TransformerBlock,ForwardPropagation,
  MatrixMultiplication,TensorContraction,Backpropagation,
  AutomaticDifferentiation,OptimisationForMachineLearning,
  AdamW,LearningRate,Warmup,GradientClipping,
  MixedPrecision,NumericalStability,EmbeddingMathematics,
  TensorComputation,ComputationalGraphMathematics,
  AttentionComplexity,KVCache,AutoregressiveDecoding,
  ParallelTraining,TensorParallelism,PipelineParallelism,
  GPUComputing,Verification,Reproducibility,Provenance
 ]
 EXPAND(TransformerMathematicsFamilies) -> [
  TokenAndPositionalRepresentation,
  QueryKeyValueProjection,
  ScaledDotProductAndMultiHeadAttention,
  AttentionMaskingAndAutoregressiveSemantics,
  ResidualNormalisationAndFeedForwardBlocks,
  TrainingOptimisationAndGradientFlow,
  KVCacheComplexityAndScalableExecution,
  NumericalValidationAndReproducibility
 ]
 STOP_CONDITION = "Stop when embeddings, attention projections, masking, block structure, optimisation, complexity and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Computational Mathematics Ontology"
 RELATED_ONTOLOGIES = [
  "AI Computational Mathematics Ontology",
  "Neural Network Mathematics Ontology 2",
  "Automatic Differentiation Ontology",
  "Backpropagation Mathematics Ontology",
  "Optimisation for Machine Learning Ontology",
  "Gradient Flow Ontology",
  "Embedding Mathematics Ontology",
  "Computational Graph Mathematics Ontology",
  "Vector Computation Ontology",
  "Computational Matrix Ontology",
  "Tensor Computation Ontology",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology",
  "Floating Point Arithmetic Ontology",
  "Numerical Stability Ontology 2"
 ]
 ROUTE_RULES = [
  "Transformer Mathematics -> AI Computational Mathematics Ontology for parent semantics",
  "Transformer Mathematics -> Neural Network Mathematics for general layered-model semantics",
  "Transformer Mathematics -> Automatic Differentiation and Backpropagation for gradient computation",
  "Transformer Mathematics -> Optimisation for Machine Learning and Gradient Flow for training dynamics",
  "Transformer Mathematics -> Embedding Mathematics for token and positional representation semantics",
  "Transformer Mathematics -> Computational Graph Mathematics for execution graph semantics",
  "Transformer Mathematics -> Vector, Matrix and Tensor ontologies for Q-K-V and attention computation",
  "Transformer Mathematics -> HPC, Parallel, Distributed and GPU Computing for scalable execution",
  "Transformer Mathematics -> Floating Point Arithmetic and Numerical Stability for finite-precision softmax, normalisation and training behaviour"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible attention mechanisms, positional methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve query-versus-key-versus-value distinction.",
  "Preserve attention-score-versus-attention-weight distinction.",
  "Preserve self-attention-versus-cross-attention distinction.",
  "Preserve masked-versus-unmasked attention distinction.",
  "Preserve attention-head-versus-concatenated-multi-head-output distinction.",
  "Preserve positional-representation-versus-content-embedding distinction.",
  "Preserve prefill-versus-autoregressive-decoding distinction.",
  "Preserve machine-readable transformer-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Transformer mathematics describes sequence computation using embeddings, query-key-value attention, residual pathways, normalisation and feed-forward transformations."
 ATTENTION_SUMMARY = "Scaled dot-product attention converts query-key compatibility into normalised weights used to aggregate value vectors."
 MULTIHEAD_SUMMARY = "Multi-head attention applies several learned projected attention operations in parallel and combines their outputs."
 POSITION_SUMMARY = "Positional representations supply sequence-order information not inherently present in permutation-symmetric attention."
 BLOCK_SUMMARY = "Residual connections, normalisation and positionwise feed-forward networks form the core transformer block around attention."
 SCALE_SUMMARY = "Sequence length, attention complexity, KV-cache growth, tensor parallelism and mixed precision strongly influence practical execution."
 VALIDATION_SUMMARY = "Transformer computations should be checked for tensor shapes, mask correctness, softmax normalisation, gradient consistency, numerical stability and reproducible architecture settings."
}

FLOW {
 INPUT
 -> detect_transformer_mathematics_context
 -> identify_sequence_length_model_dimension_head_count_head_dimension_positional_method_masking_architecture_task_and_hardware
 -> construct_token_and_positional_representations
 -> compute_query_key_value_projections
 -> validate_projection_and_head_shapes
 -> compute_scaled_query_key_attention_scores
 -> apply_causal_padding_or_task_specific_masks
 -> apply_numerically_stable_softmax
 -> aggregate_value_vectors_per_head
 -> concatenate_heads_and_apply_output_projection
 -> apply_residual_normalisation_and_feed_forward_block_operations
 -> if_training_compute_loss_gradients_and_optimiser_updates
 -> if_autoregressive_manage_KV_cache_prefill_and_incremental_decoding
 -> analyse_attention_time_memory_and_distributed_execution_cost
 -> verify_shapes_masks_attention_weights_outputs_gradients_and_numerical_stability
 -> document_model_architecture_precision_framework_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

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

}

END_FLOW