The Fidelity Framework faces a fascinating challenge: how do we identify opportunities for massive parallelism hidden within sequential-looking F# code? The answer lies in an elegant application of graph coloring to our Program Semantic Graph (PSG), using bidirectional zippers to traverse and analyze control flow and data flow patterns. This approach, inspired by insights from Ramsey graph theory, enables automatic discovery of where async and continuation-based code can be transformed into interaction nets for parallel execution.
This blog explores how compound graph construction principles from Ramsey theory guide our compilation strategy, helping us identify “purely parallel” regions that can leverage MLIR’s interaction net dialect for dramatic performance improvements across diverse hardware.
The Parallelization Discovery Problem
Consider typical F# async code that appears inherently sequential:
let processDataPipeline data = async {
let! normalized = normalizeAsync data
let! validated = validateAsync normalized
let! transformed = transformAsync validated
let! results = analyzeAsync transformed
return results
}
To the naked eye, this looks like it must execute sequentially. But what if normalizeAsync could process array elements independently? What if validation rules don’t interact? The compiler needs to discover these hidden parallelization opportunities automatically.
Graph Coloring as Parallelization Inference
The key insight is that graph coloring on our PSG reveals parallelization patterns. Nodes that can be assigned the same color can execute simultaneously:
// PSG nodes with their data and control dependencies
type PSGNode = {
Id: NodeId
Operation: Computation
DataDependencies: Edge list // What data this needs
ControlDependencies: Edge list // What must complete first
Coeffects: ContextRequirement // From coeffect analysis
}
// Graph coloring assigns parallel execution groups
type ParallelizationColoring = Map<NodeId, Color>
Bidirectional Zippers: The Analysis Engine
Our bidirectional zipper traversal enables sophisticated pattern matching across both control flow and data flow:
// Zipper provides context-aware graph traversal
type PSGZipper = {
Focus: PSGNode
Path: ZipperContext list // Where we came from
Future: ZipperContext list // Where we can go
}
// Traverse to find parallelizable regions
let findParallelRegions (zipper: PSGZipper) =
match zipper.Focus with
| AsyncMap operation ->
// Check if map operation has independent iterations
if hasIndependentElements zipper then
Some (ParallelRegion (InteractionNet operation))
else None
| AsyncSequence ops ->
// Check if sequence operations can be pipelined
checkPipelineParallelism zipper ops
From Async to Interaction Nets
The magic happens when we identify that async operations are actually data-parallel:
// Original async code
let processImages images = async {
let! results =
images |> Array.map (fun img -> async {
let! processed = enhanceImage img
return processed
}) |> Async.Sequential // Looks sequential!
return results
}
// PSG analysis reveals hidden parallelism
let analyzeForParallelism psg =
// Zipper traversal finds that each image processing is independent
let zipper = PSGZipper.create psg
zipper
|> findPattern (fun node ->
match node with
| AsyncMap where allIterationsIndependent ->
// This can become an interaction net!
true
| _ -> false)
The Ramsey Connection: Compound Patterns
Ramsey graph theory provides crucial insights about compound structures. Just as Ramsey’s compound graph construction shows how to build larger graphs from smaller ones while preserving properties, we build parallel execution strategies from simpler patterns:
Linear Patterns Compound
// Simple patterns that compound into parallel regions
type ParallelPattern =
| IndependentMap of itemCount: int
| ReductionTree of depth: int
| StencilGrid of dimensions: int * int
// Compound patterns following Ramsey principles
let compoundPatterns patterns =
match patterns with
| [IndependentMap n; ReductionTree d] ->
// Map-reduce pattern - fully parallelizable!
MapReduceNet(n, d)
| [StencilGrid(w,h); IndependentMap n] ->
// Stencil with post-processing
StencilProcessingNet(w, h, n)
The key Ramsey insight: if smaller subgraphs have certain properties (like independence), the compound graph preserves these properties in predictable ways.
MLIR Transformation Strategy
Once we identify parallel regions through graph coloring, we transform them:
// Original: DCont dialect (sequential continuations)
dcont.func @processData(%data: tensor<1024xf32>) {
%0 = dcont.async @normalize(%data)
dcont.suspend
%1 = dcont.async @validate(%0)
dcont.suspend
%2 = dcont.async @transform(%1)
dcont.suspend
dcont.return %2
}
// After graph coloring analysis: Inet dialect (parallel)
inet.func @processData(%data: tensor<1024xf32>) {
// Graph coloring revealed element independence
%0 = inet.parallel_map @normalize(%data) {
dimensions = [1024]
interaction_rule = @element_wise
}
%1 = inet.parallel_map @validate(%0) {
dimensions = [1024]
interaction_rule = @element_wise
}
%2 = inet.parallel_map @transform(%1) {
dimensions = [1024]
interaction_rule = @element_wise
}
inet.return %2
}
Coeffects Guide Coloring Strategy
The PSG’s coeffect analysis directly informs our coloring strategy:
// Coeffects determine parallelization potential
let colorByCoeffects (node: PSGNode) =
match node.Coeffects with
| Pure ->
// Pure computations can be any color
Color.Flexible
| MemoryAccess Sequential ->
// Sequential access might pipeline
Color.Pipeline
| MemoryAccess Random ->
// Random access needs careful coloring
Color.Partitioned
| AsyncBoundary ->
// Natural parallelization boundary
Color.Boundary
Advanced Pattern Recognition
The bidirectional zipper enables sophisticated pattern recognition:
Sliding Window Parallelism
// Recognize sliding window operations
let recognizeSlidingWindow (zipper: PSGZipper) =
match zipper |> lookAhead 3 with
| [Window n; Map f; Reduce g] ->
// Each window can be processed in parallel
Some (SlidingWindowNet(n, f, g))
| _ -> None
Tree Reduction Patterns
// Identify reduction trees
let recognizeReductionTree (zipper: PSGZipper) =
let rec checkTree depth =
match zipper.Focus with
| Reduce(op, [left; right]) ->
// Balanced tree reduction - perfect for parallel!
Some (ReductionTree(op, depth))
| _ -> None
Real-World Transformations
Financial Monte Carlo
// Original sequential simulation
let runSimulation() = async {
let mutable results = []
for i in 1 .. 10000 do
let! result = simulateScenario i
results <- result :: results
return results
}
// Graph coloring reveals: each scenario is independent!
// Transforms to interaction net with 10,000 parallel nodes
Image Processing Pipeline
// Sequential-looking pipeline
let pipeline image = async {
let! blurred = gaussianBlur image
let! edges = detectEdges blurred
let! enhanced = enhanceContrast edges
return enhanced
}
// Zipper analysis finds: operations are pixel-independent
// Each stage becomes a parallel interaction net
Scientific Stencil Computation
// Time-stepped simulation
let simulate grid steps = async {
let mutable current = grid
for t in 1 .. steps do
let! next = updateGrid current
current <- next
return current
}
// Graph coloring of updateGrid reveals stencil pattern
// Transforms to geometric interaction net
The Transformation Pipeline
Why This Matters: Hidden Parallelism Everywhere
The graph coloring approach reveals that much seemingly sequential code is actually parallel:
- Array operations: Often element-independent
- Validation logic: Rules rarely interact
- Data transformations: Usually pure functions
- Aggregations: Tree-reducible
By using graph coloring on our PSG with bidirectional zipper analysis, we automatically discover these opportunities.
Platform-Specific Benefits
The discovered parallelism maps efficiently to diverse hardware:
GPU: Massive Thread Arrays
// Colored regions become GPU kernels
let regions = colorGraph psg
regions |> List.filter (fun r -> r.Size > 1000)
|> List.map compileToGPU
CPU: SIMD Vectorization
// Small colored regions become SIMD
let regions = colorGraph psg
regions |> List.filter (fun r -> r.Size <= 8)
|> List.map compileToSIMD
Distributed: Actor Networks
// Large independent regions become actors
let regions = colorGraph psg
regions |> List.filter (fun r -> r.IsIndependent)
|> List.map distributeToActors
Future Directions
This graph coloring approach opens exciting possibilities:
- Automatic GPU Offloading: Identify GPU-suitable patterns automatically
- Dynamic Recoloring: Adapt parallelization based on runtime data
- Cross-Function Analysis: Find parallelism across function boundaries
- Speculative Parallelization: Color optimistically, verify at runtime
Conclusion
By applying graph coloring to our Program Semantic Graph and using bidirectional zippers for sophisticated traversal, we transform the compiler into a parallelism discovery engine. The connection to Ramsey theory isn’t about verification - it’s about understanding how simple patterns compound into complex parallel structures.
This approach turns the traditional compilation model on its head. Instead of developers manually identifying parallel regions, the compiler discovers them automatically through mathematical analysis. Sequential-looking async code transforms into massively parallel interaction nets, delivering dramatic performance improvements while maintaining F#’s elegant programming model.
The beauty is that developers write natural F# code, and the compiler does the hard work of finding and exploiting parallelism. Graph coloring keeps our parallel execution “on the beam” - not through formal verification, but through intelligent transformation that unlocks the full potential of modern hardware.