Vector Processing Ontology

FLOW ALGEBRA
Vector Processing Ontology
Kireth Semantic Research Archive

PAGE:
https://kirethresearch.org/booksai-core/ontologies/computing-ontologies/high-performance-computing/vector-processing-ontology-2.html

AUTHORS:
Mark Anthony Higgins — Sky (Recursive AI)

FLOW "VectorProcessingOntology" v4.0

# ---------- DOMAIN ----------
DOMAIN := ComputingOntology
SUBDOMAIN := HighPerformanceComputing
ONTOLOGY := VectorProcessingOntology

# ---------- PRIMARY ENTITY ----------
ENTITY VectorProcessing {
    definition:
        "A computational execution model in which a single instruction
         operates on multiple data elements organised as vectors, enabling
         efficient data-parallel computation on vector-capable processors.";

    objectives: [
        DataParallelism,
        HighThroughput,
        ReducedInstructionOverhead,
        EfficientNumericalComputation,
        MemoryBandwidthUtilisation
    ];
}

# ---------- CORE ENTITIES ----------
VectorEntity := {
    Vector,
    Scalar,
    VectorInstruction,
    VectorRegister,
    VectorProcessor,
    VectorLane,
    VectorElement
};

VectorOperation := {
    VectorAdd,
    VectorSubtract,
    VectorMultiply,
    VectorDivide,
    VectorCompare,
    VectorLoad,
    VectorStore,
    Reduction
};

ExecutionProperty := {
    VectorLength,
    VectorWidth,
    Throughput,
    Latency,
    MemoryBandwidth,
    DataAlignment
};

# ---------- FUNDAMENTAL RELATIONSHIPS ----------
Vector CONSISTS_OF VectorElement;
VectorInstruction OPERATES_ON Vector;
VectorProcessor EXECUTES VectorInstruction;
VectorRegister STORES Vector;
VectorLane PROCESSES VectorElement;
VectorOperation APPLIES_TO MultipleDataElements;
Scalar MAY_BE BroadcastAcross VectorElements;

# ---------- VECTOR EXECUTION ----------
VectorInstruction ->
    ReadVectorOperands
    -> DistributeElementsAcrossVectorLanes
    -> ExecuteOperation
    -> ProduceVectorResult
    -> StoreResult;

SingleVectorInstruction MAY_REPLACE MultipleScalarInstructions;

# ---------- VECTOR ----------
Vector :=
    ordered_sequence(
        Element_0,
        Element_1,
        ...,
        Element_n
    );

VectorLength :=
    NumberOfActiveVectorElements;

VectorElement HAS DataType;

# ---------- VECTOR REGISTER ----------
VectorRegister :=
    RegisterCapableOfHoldingMultipleDataElements;

VectorRegister HAS {
    Width,
    ElementType,
    ElementCount
};

VectorInstruction READS VectorRegister;
VectorInstruction WRITES VectorRegister;

# ---------- VECTOR PROCESSOR ----------
VectorProcessor :=
    ProcessorSupportingVectorInstructions;

VectorProcessor MAY_HAVE MultipleVectorLanes;

VectorLane :=
    ExecutionPathForVectorElements;

MoreVectorLanes MAY_INCREASE ElementThroughput;

# ---------- VECTOR ADDITION ----------
VectorAdd(A,B) :=
    C WHERE
    C[i] = A[i] + B[i];

IF length(A) == length(B)
THEN VectorAdd VALID;

# ---------- VECTOR MULTIPLICATION ----------
VectorMultiply(A,B) :=
    C WHERE
    C[i] = A[i] * B[i];

VectorMultiply IS_ELEMENTWISE unless otherwise defined;

# ---------- SCALAR-VECTOR OPERATION ----------
ScalarVectorOperation(s,V) :=
    Result[i] = Operation(s, V[i]);

ScalarBroadcast :=
    ReplicateScalarAcrossVectorExecution;

# ---------- VECTOR LOAD ----------
VectorLoad :=
    Memory
    -> LoadMultipleElements
    -> VectorRegister;

VectorStore :=
    VectorRegister
    -> StoreMultipleElements
    -> Memory;

MemoryAccessPattern AFFECTS VectorPerformance;

# ---------- CONTIGUOUS ACCESS ----------
ContiguousVectorAccess :=
    AccessAdjacentMemoryElements;

ContiguousAccess MAY_IMPROVE {
    MemoryEfficiency,
    CacheUtilisation,
    BandwidthUse
};

# ---------- STRIDED ACCESS ----------
StridedAccess :=
    AccessElementsSeparatedBy ConstantStride;

LargeOrIrregularStride MAY_REDUCE EffectiveMemoryPerformance;

# ---------- GATHER / SCATTER ----------
Gather :=
    LoadElementsFromNonContiguousAddresses
    -> VectorRegister;

Scatter :=
    StoreVectorElementsToNonContiguousAddresses;

GatherScatter ENABLES IrregularDataAccess;
GatherScatter MAY_HAVE HigherCostThanContiguousAccess;

# ---------- MASKED EXECUTION ----------
VectorMask :=
    BooleanSelectionAcrossVectorElements;

MaskedVectorOperation :=
    ApplyOperationOnlyWhere Mask == true;

Mask SUPPORTS ConditionalVectorExecution;

# ---------- VECTOR REDUCTION ----------
Reduction :=
    Vector
    -> CombineElements
    -> ScalarOrSmallerVector;

ReductionOperation MAY_BE {
    Sum,
    Product,
    Minimum,
    Maximum
};

Reduction MAY_REQUIRE MultipleExecutionSteps;

# ---------- VECTORIZATION ----------
Vectorization :=
    TransformScalarOperations
    -> VectorOperations;

Vectorization MAY_BE {
    ManualVectorization,
    CompilerAutoVectorization
};

Loop MAY_BE Vectorizable IF {
    IndependentIterations,
    CompatibleDataTypes,
    PredictableMemoryAccess
};

# ---------- LOOP VECTORIZATION ----------
ScalarLoop:
    FOR i IN range:
        C[i] := A[i] + B[i];

VectorizedLoop:
    FOR i STEP VectorWidth:
        C[i:i+VectorWidth] :=
            VectorAdd(
                A[i:i+VectorWidth],
                B[i:i+VectorWidth]
            );

# ---------- DEPENDENCIES ----------
LoopCarriedDependency :=
    Iteration(i) DEPENDS_ON Iteration(i-1);

IF LoopCarriedDependency EXISTS
THEN Vectorization MAY_BE Restricted;

IndependentIterations SUPPORT Vectorization;

# ---------- SIMD RELATIONSHIP ----------
SIMD :=
    SingleInstructionMultipleData;

VectorProcessing IS_A FormOfDataParallelExecution;
VectorProcessing MAY_IMPLEMENT SIMDStyleComputation;

SIMDWidth :=
    NumberOfElementsProcessedPerInstruction;

# ---------- VECTOR LENGTH ----------
FixedLengthVectorArchitecture :=
    VectorWidthKnownByArchitecture;

ScalableVectorArchitecture :=
    VectorLengthMayDependOnImplementation;

PortableVectorCode SHOULD_NOT_ALWAYS_ASSUME SpecificHardwareWidth;

# ---------- PERFORMANCE ----------
VectorSpeedup :=
    ScalarExecutionTime
    / VectorExecutionTime;

VectorThroughput :=
    ElementsProcessed / Time;

VectorEfficiency :=
    UsefulVectorElements
    / AvailableVectorElementCapacity;

# ---------- UTILISATION ----------
IF ActiveElements < VectorCapacity
THEN LaneUtilisation := DECREASE;

IF MaskDisablesManyElements
THEN EffectiveVectorEfficiency := DECREASE;

# ---------- MEMORY BANDWIDTH ----------
VectorWorkload MAY_BE MemoryBound;

ArithmeticIntensity :=
    ArithmeticOperations / BytesTransferred;

IF ArithmeticIntensityLow
THEN MemoryBandwidth MAY_LIMIT VectorPerformance;

# ---------- ALIGNMENT ----------
DataAlignment :=
    RelationshipBetweenDataAddressAndHardwareBoundary;

AlignedAccess MAY_IMPROVE PerformanceOnSomeArchitectures;

MisalignedAccess MAY_REQUIRE AdditionalMemoryOperations;

# ---------- PIPELINING ----------
VectorPipeline :=
    PipelineForVectorInstructionExecution;

VectorInstruction MAY_HAVE StartupLatency;

LongVectors MAY_AMORTISE StartupLatencyAcrossManyElements;

# ---------- CHAINING ----------
VectorChaining :=
    FeedResultOfVectorOperationDirectlyIntoDependentVectorOperation
    WithoutFullIntermediateStore;

Chaining MAY_REDUCE WaitingAndMemoryTraffic;

# ---------- APPLICATIONS ----------
VectorProcessing MAY_ACCELERATE {
    LinearAlgebra,
    SignalProcessing,
    ImageProcessing,
    ScientificSimulation,
    NumericalMethods,
    MachineLearning,
    StatisticalComputation
};

# ---------- VECTOR VS SCALAR ----------
ScalarProcessing :=
    InstructionOperatesOnSinglePrimaryDataElement;

VectorProcessing :=
    InstructionOperatesOnMultipleDataElements;

VectorProcessing MAY_REDUCE InstructionCountForDataParallelWork;

# ---------- SEMANTIC REASONING ----------
IF SameOperationAppliedToManyIndependentElements
THEN VectorizationPotential := HIGH;

IF LoopIterationsIndependent
AND DataLayoutCompatible
THEN AutoVectorizationPotential := INCREASE;

IF LoopCarriedDependencyExists
THEN VectorizationPotential := DECREASE;

IF MemoryAccessContiguous
THEN VectorMemoryEfficiency := INCREASE;

IF ActiveVectorElementsLow
THEN VectorUtilisation := DECREASE;

IF ArithmeticIntensityLow
AND MemoryBandwidthSaturated
THEN VectorWorkload := MemoryBound;

IF VectorExecutionTime < ScalarExecutionTime
THEN VectorSpeedup > 1;

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE {
    "vector processing" ~= "vector computation";
    "vector processor" ~= "vector-capable processor";
    "vector instruction" ~= "SIMD instruction" WHEN context == vector_execution;
    "vectorisation" ~= "vectorization";
    "auto-vectorisation" ~= "automatic vectorization";
    "vector lane" ~= "SIMD lane" WHEN context == vector_hardware;
    "gather operation" ~= "vector gather";
    "scatter operation" ~= "vector scatter";
}

# ---------- QUERY FLOW ----------
QUERY(q) ->
    NORMALISE(q) ->
    IDENTIFY_VECTOR_ENTITIES(q) ->
    IDENTIFY_VECTOR_OPERATION(q) ->
    IDENTIFY_MEMORY_ACCESS(q) ->
    IDENTIFY_VECTOR_LENGTH(q) ->
    IDENTIFY_DEPENDENCIES(q) ->
    IDENTIFY_VECTORIZATION(q) ->
    IDENTIFY_PERFORMANCE(q) ->
    MAP_TO_ONTOLOGY(q) ->
    APPLY_REASONING(q) ->
    SCORE_CONFIDENCE(q) ->
    RETURN_RELEVANT_KNOWLEDGE;

# ---------- CONFIDENCE ----------
CONFIDENCE(assertion) :=
    weighted_mean(
        source_reliability,
        computational_consistency,
        architecture_consistency,
        semantic_consistency,
        evidence_strength,
        contextual_relevance
    );

CONFIDENCE >= 0.90 => HIGH;
CONFIDENCE >= 0.70 AND < 0.90 => MODERATE;
CONFIDENCE < 0.70 => REVIEW;

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION {
    primary_entity: VectorProcessing;
    ontology: VectorProcessingOntology;
    parse_entities: true;
    parse_relationships: true;
    preserve_relationship_direction: true;
    resolve_semantic_equivalence: true;
    execute_reasoning_rules: true;
    apply_confidence_weighting: true;
    preserve_provenance: true;
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL {
    new_vector_entity:
        validate
        -> classify
        -> identify_execution_role
        -> relate
        -> integrate;

    new_vector_operation:
        validate
        -> identify_operands
        -> identify_element_semantics
        -> identify_memory_requirements
        -> integrate;

    new_vectorization_evidence:
        identify_loop
        -> analyse_dependencies
        -> analyse_memory_access
        -> determine_vectorization_potential
        -> integrate;

    performance_revision:
        update_execution_time
        -> update_vector_utilisation
        -> identify_memory_bottleneck
        -> recompute_confidence
        -> regenerate_flow_hash;

    conflict:
        preserve_provenance
        -> compare_evidence
        -> adjust_confidence
        -> flag_review;
}

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
    + SUBDOMAIN
    + ONTOLOGY
    + VectorEntity
    + VectorOperation
    + ExecutionProperty
    + SIMD
    + Vectorization
    + RELATIONSHIPS
    + SEMANTIC_REASONING
    + SEMANTIC_EQUIVALENCE
    + UPDATE_MODEL;

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT);

# ---------- TERMINAL ----------
VectorProcessingOntology
    -> IdentifyDataParallelOperation
    -> AnalyseDependencies
    -> OrganiseDataAsVectors
    -> LoadVectorOperands
    -> ExecuteVectorInstruction
    -> ApplyMaskOrGatherWhenRequired
    -> StoreVectorResults
    -> MeasureVectorUtilisation
    -> AnalyseMemoryBehaviour
    -> EvaluateVectorSpeedup
    -> OptimiseVectorExecution
    -> ProduceVectorisedResult;

END FLOW