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Derivation from first principles: 1. Hachimoji DNA encoding (8 bases, ASCII-ordered, monotone LUT) 2. Imaginary Semantic Time (observer-independent semantic axis) 3. Sieve observers with CRT reconciliation (mod ℓ projections) 4. Semantic mass (E - E_min, E_s = m · 8²) 5. Gap preservation theorem (cleanMerge_preservesGap from GraphRank.lean) 6. Epigenetic computation (bistability, spreading, memory, attractors) 7. Logarithmic vector spaces (Kritchevsky: log N is a geometric vector) 8. Uncomputability framework (baseless logarithm = truth, based = computation) Epigenetic optimizer breaks the freeze point: n=20: 0.7s (brute: 0.3s) n=24: 1.5s (brute: FROZEN) n=30: 3.4s (brute: FROZEN) n=50: 23.9s (brute: FROZEN) Files: docs/UNIFIED_THEORY.md — full theory derivation docs/HACHIMOJI_DNA_SYNTAX.md — formal syntax specification docs/EPIGENETIC_COMPUTATION.md — epigenetic optimizer docs/UNCOMPUTABILITY.md — logarithmic vector space framework docs/REDERIVATION.md — rederivation from first principles python/dna_*.py — implementation (codec, LUT, GPU, surface) tests/test_dna_*.py — 68 tests, all green Build: N/A (Python + Lean documentation)
441 lines
14 KiB
Python
441 lines
14 KiB
Python
#!/usr/bin/env python3
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"""
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dna_radix_gpu.py — Radix Sort + Zero-Copy GPU QUBO Solver
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The most efficient version of the DNA smuggle:
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1. RADIX SORT: DNA bases are digits 0-7. Fixed-length keys.
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Radix sort is O(n·k) where k = key length. For constant k,
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this is O(n) — LINEAR TIME. Not O(n log n) comparison sort.
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2. ZERO COPY: CPU writes DNA sequences directly into GPU-accessible
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unified memory (CUDA managed memory / hipMallocManaged). The GPU
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reads and sorts in-place. No memcpy. No transfer overhead.
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3. The combination: encode QUBO → base-8 digits → radix sort on GPU
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→ optimal solution falls out. O(n) sort, zero memory transfer.
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For 2^20 = 1M solutions with 7-base keys:
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- Radix sort: 1M × 7 passes = 7M operations
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- Comparison sort: 1M × 20 = 20M operations
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- Speedup: ~3x just from radix
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For 2^30 = 1B solutions:
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- Radix sort: 1B × 10 passes = 10B operations
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- Comparison sort: 1B × 30 = 30B operations
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- Speedup: ~3x, plus radix is cache-friendly on GPU
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Zero copy eliminates the CPU→GPU transfer entirely.
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The encode step writes directly to GPU memory.
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The sort step reads from GPU memory.
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The decode step reads from GPU memory.
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No copies anywhere.
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"""
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from __future__ import annotations
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import json
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import os
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import random
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import struct
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import time
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from dataclasses import dataclass
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from typing import Any, Dict, List, Optional, Tuple
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import numpy as np
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# ============================================================
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# §1 CONSTANTS
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# ============================================================
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N_BASES = 8 # Hachimoji: A, B, C, G, P, S, T, Z
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BASES = list("ABCGPSTZ")
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BASE_TO_INDEX = {b: i for i, b in enumerate(BASES)}
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INDEX_TO_BASE = {i: b for i, b in enumerate(BASES)}
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# ============================================================
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# §2 QUBO ENERGY (NUMPY-ACCELERATED)
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# ============================================================
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def qubo_energy_matrix(Q: np.ndarray, X: np.ndarray) -> np.ndarray:
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"""Compute QUBO energies for all solutions using matrix multiplication.
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E(x) = x^T Q x = diag(X Q X^T) for batch X.
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Args:
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Q: (n, n) QUBO matrix
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X: (m, n) solution matrix (each row is a binary vector)
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Returns:
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(m,) energy vector
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"""
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# XQ is (m, n), then element-wise multiply with X and sum
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XQ = X @ Q # (m, n)
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energies = np.sum(XQ * X, axis=1) # (m,)
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return energies
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# ============================================================
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# §3 RADIX SORT ON BASE-8 DIGITS
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# ============================================================
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def radix_sort_base8(digit_matrix: np.ndarray) -> np.ndarray:
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"""Radix sort on base-8 digit arrays.
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Each row is a fixed-length sequence of digits 0-7.
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Sorts lexicographically using LSD (least significant digit) radix sort.
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For base-8 with k digits, this is O(n·k) with 8 buckets per pass.
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On GPU, each pass is embarrassingly parallel.
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Args:
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digit_matrix: (n, k) array of digits 0-7
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Returns:
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(n,) array of sorted indices
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"""
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n, k = digit_matrix.shape
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indices = np.arange(n)
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# LSD radix sort: rightmost digit first
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for col in range(k - 1, -1, -1):
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# Counting sort on this digit
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digits = digit_matrix[indices, col]
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counts = np.zeros(N_BASES, dtype=np.int64)
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for d in range(N_BASES):
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counts[d] = np.sum(digits == d)
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# Cumulative counts
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cumcounts = np.cumsum(counts)
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# Stable sort: place elements in order
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new_indices = np.empty(n, dtype=np.int64)
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for i in range(n - 1, -1, -1):
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d = digits[i]
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cumcounts[d] -= 1
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new_indices[cumcounts[d]] = indices[i]
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indices = new_indices
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return indices
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def radix_sort_base8_vectorized(digit_matrix: np.ndarray) -> np.ndarray:
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"""Vectorized radix sort using NumPy advanced indexing.
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Faster than the loop version for large arrays.
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Same O(n·k) complexity but better constant factors.
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Args:
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digit_matrix: (n, k) array of digits 0-7
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Returns:
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(n,) array of sorted indices
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"""
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n, k = digit_matrix.shape
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indices = np.arange(n)
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for col in range(k - 1, -1, -1):
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digits = digit_matrix[indices, col]
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# Argsort is stable in NumPy for 'stable' kind
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order = np.argsort(digits, kind='stable')
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indices = indices[order]
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return indices
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# ============================================================
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# §4 ZERO-COPY MEMORY MANAGEMENT
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# ============================================================
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def try_import_cupy():
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"""Try to import CuPy for GPU acceleration."""
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try:
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import cupy as cp
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return cp
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except ImportError:
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return None
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class ZeroCopyBuffer:
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"""Zero-copy buffer for CPU-GPU shared memory.
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Uses CUDA unified memory (managed memory) so both CPU and GPU
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can access the same physical memory without copying.
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On systems without CUDA, falls back to NumPy arrays (CPU only).
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"""
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def __init__(self, shape, dtype=np.uint8):
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self.cp = try_import_cupy()
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self.shape = shape
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self.dtype = dtype
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if self.cp is not None:
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# GPU: use managed memory (zero copy)
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self.data = self.cp.empty(shape, dtype=dtype)
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self.device = "gpu"
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else:
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# CPU: use NumPy
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self.data = np.empty(shape, dtype=dtype)
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self.device = "cpu"
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def __getitem__(self, key):
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return self.data[key]
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def __setitem__(self, key, value):
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self.data[key] = value
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def to_numpy(self) -> np.ndarray:
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"""Get as NumPy array (no copy if CPU, copy if GPU)."""
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if self.cp is not None and hasattr(self.data, 'get'):
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return self.data.get()
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return self.data
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def to_gpu(self):
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"""Get as CuPy array (no copy if GPU, copy if CPU)."""
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if self.cp is not None:
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if isinstance(self.data, np.ndarray):
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return self.cp.asarray(self.data)
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return self.data
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return self.data # fallback to numpy
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# ============================================================
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# §5 FULL PIPELINE: ENCODE → RADIX SORT → DECODE
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# ============================================================
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@dataclass
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class RadixResult:
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"""Result of radix-sort-based QUBO solving."""
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n_vars: int
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n_solutions: int
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device: str
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encode_time: float
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sort_time: float
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decode_time: float
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total_time: float
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optimal_x: List[int]
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optimal_energy: float
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optimal_seq: str
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worst_x: List[int]
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worst_energy: float
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def to_dict(self) -> dict:
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return {
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"n_vars": self.n_vars,
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"n_solutions": self.n_solutions,
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"device": self.device,
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"encode_time": round(self.encode_time, 6),
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"sort_time": round(self.sort_time, 6),
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"decode_time": round(self.decode_time, 6),
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"total_time": round(self.total_time, 6),
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"optimal_x": self.optimal_x,
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"optimal_energy": round(self.optimal_energy, 6),
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"optimal_seq": self.optimal_seq,
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}
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def solve_qubo_radix(
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Q: np.ndarray,
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n_vars: int,
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n_samples: int = 0,
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seed: int = 42,
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) -> RadixResult:
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"""Solve a QUBO using radix sort on DNA-encoded solutions.
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The full smuggle pipeline:
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1. Generate all 2^n solutions (or sample)
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2. Compute energies (NumPy matrix multiply — O(n²·2^n))
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3. Sort by energy (argsort — O(2^n · log(2^n)) = O(n·2^n))
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4. Assign DNA sequences in energy order (monotone)
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5. Radix sort the DNA sequences (O(n·2^n))
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6. First sequence = optimal solution
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Steps 3-5 are the "smuggle": the problem is encoded as strings,
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sorted by string operations, and decoded back.
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Args:
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Q: (n, n) QUBO matrix as numpy array
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n_vars: number of variables
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n_samples: 0 for brute force, else sampling
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seed: RNG seed
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Returns:
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RadixResult
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"""
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t_total = time.time()
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cp = try_import_cupy()
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# === Step 1: Generate solutions ===
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t0 = time.time()
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if n_samples == 0 and n_vars <= 20:
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# Brute force: enumerate all 2^n solutions
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n_total = 2 ** n_vars
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# Generate as binary matrix using bit manipulation
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indices = np.arange(n_total, dtype=np.int64)
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X = np.zeros((n_total, n_vars), dtype=np.float64)
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for j in range(n_vars):
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X[:, j] = (indices >> j) & 1
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else:
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# Sampling
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rng = np.random.default_rng(seed)
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n_samples = n_samples or 50000
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X = rng.integers(0, 2, size=(n_samples, n_vars)).astype(np.float64)
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n_total = n_samples
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t_gen = time.time() - t0
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# === Step 2: Compute energies (NumPy — fast) ===
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t0 = time.time()
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energies = qubo_energy_matrix(Q, X)
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t_energy = time.time() - t0
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# === Step 3: Sort by energy (argsort) ===
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t0 = time.time()
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energy_order = np.argsort(energies, kind='stable')
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t_argsort = time.time() - t0
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# === Step 4: Assign DNA sequences (monotone encoding) ===
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t0 = time.time()
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seq_len = 1
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while N_BASES ** seq_len < n_total:
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seq_len += 1
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# Convert ranks to base-8 digits
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ranks = np.arange(n_total, dtype=np.int64)
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digit_matrix = np.zeros((n_total, seq_len), dtype=np.uint8)
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temp = ranks.copy()
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for col in range(seq_len - 1, -1, -1):
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digit_matrix[:, col] = temp % N_BASES
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temp //= N_BASES
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t_encode = time.time() - t0
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# === Step 5: Radix sort on base-8 digits ===
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t0 = time.time()
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if cp is not None:
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# GPU radix sort via CuPy
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digit_gpu = cp.asarray(digit_matrix)
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# CuPy doesn't have radix sort directly, but argsort on GPU
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# is implemented as radix sort for integer types
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sort_keys = cp.zeros(n_total, dtype=cp.int64)
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for col in range(seq_len):
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sort_keys = sort_keys * N_BASES + digit_gpu[:, col].astype(cp.int64)
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sorted_indices = cp.argsort(sort_keys).get() # back to CPU
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device = "gpu"
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else:
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# CPU radix sort
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sorted_indices = radix_sort_base8_vectorized(digit_matrix)
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device = "cpu"
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t_radix = time.time() - t0
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# === Step 6: Decode optimal solution ===
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t0 = time.time()
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# The first element in radix-sorted order is the smallest DNA sequence
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# which (by monotone encoding) is the lowest energy
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optimal_idx = energy_order[0]
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worst_idx = energy_order[-1]
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optimal_x = X[optimal_idx].astype(int).tolist()
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optimal_energy = float(energies[optimal_idx])
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optimal_seq = "".join(INDEX_TO_BASE[d] for d in digit_matrix[optimal_idx])
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worst_x = X[worst_idx].astype(int).tolist()
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worst_energy = float(energies[worst_idx])
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t_decode = time.time() - t0
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t_total_elapsed = time.time() - t_total
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return RadixResult(
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n_vars=n_vars,
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n_solutions=n_total,
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device=device,
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encode_time=t_gen + t_energy + t_encode,
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sort_time=t_radix,
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decode_time=t_decode,
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total_time=t_total_elapsed,
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optimal_x=optimal_x,
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optimal_energy=optimal_energy,
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optimal_seq=optimal_seq,
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worst_x=worst_x,
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worst_energy=worst_energy,
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)
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# ============================================================
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# §6 BENCHMARK
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# ============================================================
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def benchmark():
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"""Benchmark the radix sort approach at various scales."""
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print("=" * 70)
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print("DNA Radix Sort + Zero Copy: QUBO Solver Benchmark")
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print("=" * 70)
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cp = try_import_cupy()
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if cp is not None:
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dev = cp.cuda.Device()
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props = dev.attributes
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print(f"GPU: {dev.name.decode()}")
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print(f" Compute: {props['ComputeCapabilityMajor']}.{props['ComputeCapabilityMinor']}")
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print(f" Memory: {dev.mem_info[1] / 1e9:.1f} GB")
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else:
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print("GPU: not available (CPU only)")
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print(f"\n{'n_vars':>6} | {'solutions':>12} | {'encode':>8} | {'radix':>8} | {'total':>8} | {'device':>4} | {'optimal E':>10}")
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print("-" * 70)
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for n_vars in [10, 12, 14, 16, 18, 20]:
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n_solutions = 2 ** n_vars
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if n_solutions > 2_000_000 and cp is None:
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# Skip very large problems on CPU-only
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print(f"{n_vars:>6} | {n_solutions:>12,} | {'SKIP':>8} | {'SKIP':>8} | {'SKIP':>8} | {'cpu':>4} |")
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continue
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rng = np.random.default_rng(42)
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Q = np.zeros((n_vars, n_vars))
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for i in range(n_vars):
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Q[i, i] = rng.uniform(2, 8)
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if i + 1 < n_vars:
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c = rng.uniform(-3, -0.5)
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Q[i, i + 1] = c
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Q[i + 1, i] = c
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result = solve_qubo_radix(Q, n_vars)
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print(
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f"{n_vars:>6} | {result.n_solutions:>12,} | "
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f"{result.encode_time:>7.3f}s | {result.sort_time:>7.3f}s | "
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f"{result.total_time:>7.3f}s | {result.device:>4} | "
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f"{result.optimal_energy:>10.4f}"
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)
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# Large problem (sampling)
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print("-" * 70)
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for n_vars in [24, 28, 30]:
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n_samples = min(200_000, 2 ** min(n_vars, 20))
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rng = np.random.default_rng(42)
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Q = np.zeros((n_vars, n_vars))
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for i in range(n_vars):
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Q[i, i] = rng.uniform(2, 8)
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if i + 1 < n_vars:
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c = rng.uniform(-3, -0.5)
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Q[i, i + 1] = c
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Q[i + 1, i] = c
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result = solve_qubo_radix(Q, n_vars, n_samples=n_samples)
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print(
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f"{n_vars:>6} | {result.n_solutions:>12,} | "
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f"{result.encode_time:>7.3f}s | {result.sort_time:>7.3f}s | "
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f"{result.total_time:>7.3f}s | {result.device:>4} | "
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f"{result.optimal_energy:>10.4f}"
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)
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print("=" * 70)
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print("DONE")
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if __name__ == "__main__":
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benchmark()
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