Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/HypercubeTopology.lean

import Semantics.FixedPoint

namespace Semantics.HypercubeTopology

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Hypercube Topology for Unified Topology
-- 
-- This module implements hypercube topology based on Connection Machine architecture.
-- 
-- Key equations:
-- d_hc = Σ_{i=0}^{n-1} |x_i - y_i|
-- neighbor_count = 2n
-- connectivity = 2^n nodes
-- 
-- where:
-- - d_hc = Hypercube distance between nodes
-- - n = Number of dimensions (12 for Connection Machine)
-- - x_i, y_i = Node coordinates in dimension i
-- - neighbor_count = Number of neighbors per node
-- - connectivity = Total number of nodes in hypercube
-- 
-- Concept:
-- - 12-dimensional hypercube topology for unified topology
-- - 4,096 nodes with direct neighbor communication
-- - Avoids Von Neumann memory bottleneck
-- - Each node has local memory and communicates with neighbors
-- ═══════════════════════════════════════════════════════════════════════════

/-- Hypercube node position -/
structure HypercubeNode where
  nodeId : UInt64
  coordinates : Array UInt64  -- n-dimensional coordinates (n = 12 for CM)
  dimensions : UInt32  -- Number of dimensions (typically 12)
  deriving Repr, Inhabited

/-- Hypercube topology state -/
structure HypercubeTopologyState where
  nodes : Array HypercubeNode
  dimensions : UInt32  -- Number of dimensions
  maxNodeId : UInt64  -- Maximum node ID
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Hypercube Distance Calculation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate hypercube distance: d_hc = Σ_{i=0}^{n-1} |x_i - y_i| -/
def hypercubeDistance (node1 : HypercubeNode) (node2 : HypercubeNode) : UInt64 :=
  let minDim := min node1.dimensions node2.dimensions
  let dist := node1.coordinates.zipWith node2.coordinates (fun x y => if x > y then x - y else y - x)
  let sumDist := dist.take (minDim.toNat) |> List.foldl (fun acc d => acc + d) 0
  sumDist

/-- Check if two nodes are neighbors (distance = 1) -/
def areNeighbors (node1 : HypercubeNode) (node2 : HypercubeNode) : Bool :=
  hypercubeDistance node1 node2 == 1

/-- Get neighbors of a node -/
def getNeighbors (state : HypercubeTopologyState) (node : HypercubeNode) : Array HypercubeNode :=
  let dim := node.dimensions
  let neighborCoords := (List.range dim.toNat).map (fun i =>
    let newCoords := node.coordinates.mapIdx (fun idx coord =>
      if idx == i then (coord + 1) % (2 ^ dim.toNat) else coord
    )
    newCoords
  )
  
  neighborCoords.map (fun coords =>
    state.nodes.find? (fun n => n.coordinates == coords)
  ) |> Array.filterMap (fun x => x)

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Hypercube Topology Properties
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate neighbor count: neighbor_count = 2n -/
def neighborCount (dimensions : UInt32) : UInt32 :=
  2 * dimensions

/-- Calculate connectivity: connectivity = 2^n nodes -/
def connectivity (dimensions : UInt32) : UInt64 :=
  2 ^ dimensions.toNat

/-- Calculate hypercube diameter: max distance between any two nodes = n -/
def hypercubeDiameter (dimensions : UInt32) : UInt32 :=
  dimensions

/-- Calculate bisection bandwidth: 2^(n-1) edges cut by splitting hypercube in half -/
def bisectionBandwidth (dimensions : UInt32) : UInt64 :=
  2 ^ (dimensions.toNat - 1)

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Bind Primitive for Hypercube Topology
-- ═══════════════════════════════════════════════════════════════════════════

/-- Hypercube topology action -/
structure HypercubeAction where
  nodeId : UInt64
  dimension : UInt32  -- Dimension to toggle (0 to n-1)
  deriving Repr, Inhabited

/-- Hypercube bind result -/
structure HypercubeBind where
  lawful : Bool  -- Whether action is lawful
  distanceBefore : UInt64  -- Distance before action
  distanceAfter : UInt64  -- Distance after action
  neighborCount : UInt32  -- Number of neighbors
  invariant : String  -- Invariant description
  deriving Repr, Inhabited

/-- Check if hypercube action is lawful -/
def isHypercubeActionLawful (state : HypercubeTopologyState) (action : HypercubeAction) : Bool :=
  action.dimension < state.dimensions ∧
  action.nodeId < state.maxNodeId

/-- Toggle coordinate in specified dimension -/
def toggleCoordinate (node : HypercubeNode) (dimension : UInt32) : HypercubeNode :=
  let newCoords := node.coordinates.mapIdx (fun idx coord =>
    if idx == dimension.toNat then (coord + 1) % (2 ^ node.dimensions.toNat) else coord
  )
  {
    nodeId := node.nodeId,
    coordinates := newCoords,
    dimensions := node.dimensions
  }

/-- Bind primitive for hypercube topology -/
def hypercubeBind (state : HypercubeTopologyState) (action : HypercubeAction) : Q16_16 → HypercubeBind
  | currentTime =>
    let lawful := isHypercubeActionLawful state action
    
    let oldNode := state.nodes.find? (fun n => n.nodeId == action.nodeId)
    let referenceNode := state.nodes.get! 0  -- Use first node as reference
    let distanceBefore := match oldNode with
      | some n => hypercubeDistance n referenceNode
      | none => 0
    
    let newNode := if lawful then
      match oldNode with
      | some n => toggleCoordinate n action.dimension
      | none => oldNode.get!
    else
      match oldNode with
      | some n => n
      | none => {
        nodeId := action.nodeId,
        coordinates := Array.mk (List.replicate state.dimensions.toNat 0),
        dimensions := state.dimensions
      }
    
    let distanceAfter := if lawful then hypercubeDistance newNode referenceNode else distanceBefore
    let nCount := neighborCount state.dimensions
    
    {
      lawful := lawful,
      distanceBefore := distanceBefore,
      distanceAfter := distanceAfter,
      neighborCount := nCount,
      invariant := if lawful then "hypercube_topology_satisfied" else "hypercube_constraint_violated"
    }

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Invariant Preservation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Lawful hypercube actions preserve neighbor count -/
theorem lawfulActionPreservesNeighborCount (state : HypercubeTopologyState) (action : HypercubeAction) :
    (hypercubeBind state action (ofNat 0)).lawful →
    (hypercubeBind state action (ofNat 0)).neighborCount = neighborCount state.dimensions := by
  intro h
  cases h

/-- Hypercube distance is symmetric -/
theorem hypercubeDistanceSymmetric (node1 node2 : HypercubeNode) :
    hypercubeDistance node1 node2 = hypercubeDistance node2 node1 := by

/-- Hypercube diameter equals number of dimensions -/
theorem hypercubeDiameterEqualsDimensions (state : HypercubeTopologyState) :
    hypercubeDiameter state.dimensions = state.dimensions := by

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  #eval Examples
-- ═══════════════════════════════════════════════════════════════════════════

#let node1 := {
  nodeId := 1,
  coordinates := #[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
  dimensions := 12
}

#let node2 := {
  nodeId := 2,
  coordinates := #[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
  dimensions := 12
}

#let node3 := {
  nodeId := 3,
  coordinates := #[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
  dimensions := 12
}

#eval hypercubeDistance node1 node2

#eval hypercubeDistance node1 node3

#eval areNeighbors node1 node2

#eval areNeighbors node1 node3

#eval neighborCount 12

#eval connectivity 12

#eval hypercubeDiameter 12

#eval bisectionBandwidth 12

end Semantics.HypercubeTopology