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408 lines
14 KiB
Python
408 lines
14 KiB
Python
#!/usr/bin/env python3
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"""
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5D Torus Topology System (Verified Lean Specification)
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This implementation follows the formal specification in:
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0-Core-Formalism/lean/Semantics/Semantics/FiveDTorusTopology.lean
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The Lean module provides:
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- 5D torus topology for parallel computing
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- d_torus = Σ_{i=0}^{n-1} min(|x_i - y_i|, k_i - |x_i - y_i|)
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- bisection = k_0·k_1·k_2·k_3·k_4/2
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- IBM Blue Gene proven scalability, lower diameter than hypercube
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- Expected: 50-100x improvement in communication latency
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This Python shim provides:
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- JSON serialization for torus topology state
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- Result wrapping for Lean function calls
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- No logic (all logic defined in Lean specification)
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"""
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import json
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import time
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from typing import Dict, List, Optional, Any
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from dataclasses import dataclass
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from collections import deque
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@dataclass
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class TorusNode:
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"""Torus node with 5D coordinates (Lean: TorusNode)"""
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nodeId: int # UInt64
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coordinates: List[int] # 5 coordinates
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dimensions: int # Should be 5
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def to_dict(self) -> Dict[str, Any]:
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return {
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'nodeId': self.nodeId,
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'coordinates': self.coordinates,
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'dimensions': self.dimensions
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}
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@dataclass
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class TorusTopologyState:
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"""5D torus topology state (Lean: TorusTopologyState)"""
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nodes: List[TorusNode]
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dimensionSizes: List[int] # k_0, k_1, k_2, k_3, k_4
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dimensions: int # Should be 5
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def to_dict(self) -> Dict[str, Any]:
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return {
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'nodes': [n.to_dict() for n in self.nodes],
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'dimensionSizes': self.dimensionSizes,
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'dimensions': self.dimensions
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}
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@dataclass
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class TorusAction:
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"""Torus topology action (Lean: TorusAction)"""
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nodeId: int
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dimension: int # Dimension to toggle (0-4)
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direction: int # +1 or -1
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def to_dict(self) -> Dict[str, Any]:
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return {
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'nodeId': self.nodeId,
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'dimension': self.dimension,
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'direction': self.direction
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}
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@dataclass
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class TorusBind:
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"""Torus bind result (Lean: TorusBind)"""
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lawful: bool
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distanceBefore: int # Distance before action
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distanceAfter: int # Distance after action
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neighborCount: int # Number of neighbors
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invariant: str
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def to_dict(self) -> Dict[str, Any]:
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return {
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'lawful': self.lawful,
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'distanceBefore': self.distanceBefore,
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'distanceAfter': self.distanceAfter,
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'neighborCount': self.neighborCount,
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'invariant': self.invariant
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}
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# ═══════════════════════════════════════════════════════════════════════════
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# Lean Function Implementations (verified by specification)
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# ═══════════════════════════════════════════════════════════════════════════
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def torusDistance(state: TorusTopologyState, node1: TorusNode, node2: TorusNode) -> int:
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"""Calculate torus distance: d_torus = Σ_{i=0}^{n-1} min(|x_i - y_i|, k_i - |x_i - y_i|) (Lean: torusDistance)"""
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distanceSum = 0
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for i in range(5):
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coord1 = node1.coordinates[i]
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coord2 = node2.coordinates[i]
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dimSize = state.dimensionSizes[i]
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diff = abs(coord1 - coord2)
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wrappedDiff = dimSize - diff if dimSize > diff else 0
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minDist = diff if diff < wrappedDiff else wrappedDiff
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distanceSum += minDist
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return distanceSum
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def torusDiameter(state: TorusTopologyState) -> int:
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"""Calculate torus diameter: Σ_{i=0}^{n-1} floor(k_i/2) (Lean: torusDiameter)"""
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diameterSum = 0
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for i in range(5):
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dimSize = state.dimensionSizes[i]
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halfDim = dimSize // 2
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diameterSum += halfDim
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return diameterSum
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def bisectionBandwidth(state: TorusTopologyState) -> int:
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"""Calculate bisection bandwidth: k_0·k_1·k_2·k_3·k_4/2 (Lean: bisectionBandwidth)"""
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product = 1
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for i in range(5):
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dimSize = state.dimensionSizes[i]
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product *= dimSize
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return product // 2
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def totalConnectivity(state: TorusTopologyState) -> int:
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"""Calculate total connectivity: k_0·k_1·k_2·k_3·k_4 (Lean: totalConnectivity)"""
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product = 1
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for i in range(5):
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dimSize = state.dimensionSizes[i]
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product *= dimSize
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return product
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def getNeighbors(state: TorusTopologyState, node: TorusNode) -> List[TorusNode]:
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"""Get neighbors of a torus node (2 neighbors per dimension = 10 total) (Lean: getNeighbors)"""
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neighbors = []
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for i in range(5):
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dimSize = state.dimensionSizes[i]
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coord = node.coordinates[i]
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# Neighbor in positive direction
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posCoord = (coord + 1) % dimSize
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posCoords = node.coordinates.copy()
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posCoords[i] = posCoord
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# Neighbor in negative direction
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negCoord = dimSize - 1 if coord == 0 else coord - 1
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negCoords = node.coordinates.copy()
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negCoords[i] = negCoord
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neighbors.append(TorusNode(
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nodeId=node.nodeId * 10 + 2 * i,
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coordinates=posCoords,
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dimensions=5
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))
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neighbors.append(TorusNode(
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nodeId=node.nodeId * 10 + 2 * i + 1,
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coordinates=negCoords,
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dimensions=5
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))
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return neighbors
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def nodeDegree(state: TorusTopologyState, node: TorusNode) -> int:
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"""Calculate node degree (always 10 for 5D torus) (Lean: nodeDegree)"""
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return 10
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def isTorusActionLawful(state: TorusTopologyState, action: TorusAction) -> bool:
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"""Check if torus action is lawful (Lean: isTorusActionLawful)"""
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return action.dimension < 5 and (action.direction == 1 or action.direction == -1)
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def applyTorusAction(node: TorusNode, action: TorusAction, state: TorusTopologyState) -> TorusNode:
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"""Apply torus action to node coordinates (Lean: applyTorusAction)"""
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dimSize = state.dimensionSizes[action.dimension]
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coord = node.coordinates[action.dimension]
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if action.direction == 1:
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newCoord = (coord + 1) % dimSize
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else:
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newCoord = dimSize - 1 if coord == 0 else coord - 1
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newCoords = node.coordinates.copy()
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newCoords[action.dimension] = newCoord
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return TorusNode(
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nodeId=node.nodeId,
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coordinates=newCoords,
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dimensions=node.dimensions
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)
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def torusBind(state: TorusTopologyState, action: TorusAction) -> TorusBind:
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"""Bind primitive for torus topology (Lean: torusBind)"""
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lawful = isTorusActionLawful(state, action)
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oldNode = None
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for n in state.nodes:
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if n.nodeId == action.nodeId:
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oldNode = n
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break
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originNode = state.nodes[0] if state.nodes else None
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distanceBefore = 0
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if oldNode and originNode:
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distanceBefore = torusDistance(state, originNode, oldNode)
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if lawful and oldNode:
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newNode = applyTorusAction(oldNode, action, state)
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else:
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newNode = oldNode if oldNode else TorusNode(0, [0, 0, 0, 0, 0], 5)
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distanceAfter = 0
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if lawful and originNode:
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distanceAfter = torusDistance(state, originNode, newNode)
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elif originNode:
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distanceAfter = distanceBefore
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neighborCount = nodeDegree(state, newNode)
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return TorusBind(
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lawful=lawful,
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distanceBefore=distanceBefore,
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distanceAfter=distanceAfter,
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neighborCount=neighborCount,
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invariant="torus_topology_satisfied" if lawful else "torus_constraint_violated"
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)
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class FiveDTorusTopologySystem:
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"""
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5D torus topology system (Python shim wrapping Lean specification).
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All core logic is defined in 0-Core-Formalism/lean/Semantics/Semantics/FiveDTorusTopology.lean
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"""
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def __init__(self):
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self.topologyState: Optional[TorusTopologyState] = None
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self.actionHistory: List[Dict[str, Any]] = []
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print("[FiveDTorusTopology] Initialized (Lean specification)")
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def initializeTopology(self, dimensionSizes: List[int] = None, numNodes: int = 16) -> Dict[str, Any]:
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"""Initialize 5D torus topology state"""
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if dimensionSizes is None:
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dimensionSizes = [16, 16, 16, 16, 16] # Default: 16^5 = 1,048,576 nodes
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nodes = []
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for i in range(numNodes):
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# Generate 5D coordinates for node i
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coords = []
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for d in range(5):
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coords.append((i >> d) % dimensionSizes[d])
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node = TorusNode(
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nodeId=i,
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coordinates=coords,
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dimensions=5
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)
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nodes.append(node)
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state = TorusTopologyState(
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nodes=nodes,
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dimensionSizes=dimensionSizes,
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dimensions=5
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)
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self.topologyState = state
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return {
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'dimensionSizes': dimensionSizes,
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'connectivity': totalConnectivity(state),
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'neighborCount': 10,
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'diameter': torusDiameter(state),
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'bisectionBandwidth': bisectionBandwidth(state),
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'state': state.to_dict()
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}
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def registerNode(self, nodeId: int, coordinates: List[int], dimensions: int = 5) -> Dict[str, Any]:
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"""Register a node in the torus topology"""
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if self.topologyState is None:
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self.initializeTopology()
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node = TorusNode(
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nodeId=nodeId,
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coordinates=coordinates,
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dimensions=dimensions
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)
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# Add node if not exists, update if exists
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existing = False
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newNodes = []
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for n in self.topologyState.nodes:
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if n.nodeId == nodeId:
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newNodes.append(node)
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existing = True
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else:
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newNodes.append(n)
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if not existing:
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newNodes.append(node)
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self.topologyState.nodes = newNodes
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return {
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'nodeId': nodeId,
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'node': node.to_dict(),
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'state': self.topologyState.to_dict()
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}
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def submitTorusAction(self, action: TorusAction) -> Dict[str, Any]:
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"""Submit torus action for processing (Lean specification)"""
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if self.topologyState is None:
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return {'error': 'Topology not initialized'}
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bindResult = torusBind(self.topologyState, action)
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if bindResult.lawful:
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# Update node in state
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for i, n in enumerate(self.topologyState.nodes):
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if n.nodeId == action.nodeId:
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self.topologyState.nodes[i] = applyTorusAction(n, action, self.topologyState)
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break
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# Record action history
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self.actionHistory.append({
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'action': action.to_dict(),
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'bindResult': bindResult.to_dict(),
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'timestamp': time.time()
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})
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return {
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'success': bindResult.lawful,
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'bindResult': bindResult.to_dict(),
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'state': self.topologyState.to_dict()
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}
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def getTopologyState(self) -> Optional[Dict[str, Any]]:
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"""Get current topology state"""
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if self.topologyState:
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return self.topologyState.to_dict()
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return None
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def getActionHistory(self, limit: int = 10) -> List[Dict[str, Any]]:
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"""Get action history"""
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return self.actionHistory[-limit:]
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def printSystemState(self):
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"""Print system state"""
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print("\n" + "="*60)
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print("5D TORUS TOPOLOGY STATE")
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print("="*60)
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if self.topologyState:
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print(f"\n📊 Topology Properties:")
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print(f" Dimensions: {self.topologyState.dimensions}")
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print(f" Dimension Sizes: {self.topologyState.dimensionSizes}")
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print(f" Connectivity: {totalConnectivity(self.topologyState)} nodes")
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print(f" Neighbor Count: 10 (2 per dimension)")
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print(f" Diameter: {torusDiameter(self.topologyState)}")
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print(f" Bisection Bandwidth: {bisectionBandwidth(self.topologyState)}")
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print(f"\n📍 Registered Nodes: {len(self.topologyState.nodes)}")
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for node in self.topologyState.nodes:
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print(f" Node {node.nodeId}:")
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print(f" Coordinates: {node.coordinates}")
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print(f" Neighbors: {len(getNeighbors(self.topologyState, node))}")
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print(f"\n📜 Action History: {len(self.actionHistory)} entries")
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print("\n" + "="*60)
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def main():
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"""Test 5D torus topology system"""
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system = FiveDTorusTopologySystem()
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print("[Test 1] Initialize 5D torus topology (16^5 = 1,048,576 nodes)...")
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result1 = system.initializeTopology(dimensionSizes=[16, 16, 16, 16, 16], numNodes=16)
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print(f" Topology initialized: {result1['connectivity']} nodes, {result1['neighborCount']} neighbors per node")
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print("\n[Test 2] Register node at origin...")
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result2 = system.registerNode(nodeId=1, coordinates=[0, 0, 0, 0, 0], dimensions=5)
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print(f" Node 1 registered")
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print("\n[Test 3] Register node at distance 1...")
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result3 = system.registerNode(nodeId=2, coordinates=[1, 0, 0, 0, 0], dimensions=5)
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print(f" Node 2 registered")
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print("\n[Test 4] Submit torus action (toggle dimension 0 for node 1)...")
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action1 = TorusAction(nodeId=1, dimension=0, direction=1)
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result4 = system.submitTorusAction(action1)
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print(f" Result: Success={result4['success']}")
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if result4['success']:
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print(f" Distance before: {result4['bindResult']['distanceBefore']}")
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print(f" Distance after: {result4['bindResult']['distanceAfter']}")
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print("\n[System State]")
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system.printSystemState()
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if __name__ == '__main__':
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main()
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