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5331d2cc4e
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)
476 lines
14 KiB
Python
476 lines
14 KiB
Python
#!/usr/bin/env python3
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"""
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dna_lut.py — Symbology LUT Adaptation for QUBO-DNA Sorting
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The LUT is the adapter between symbolic DNA space and QUBO energy space.
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Three encoding strategies, in order of increasing power:
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1. Direct encoding: x → DNA by fixed mapping. Fast, but symbolic
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order doesn't match energy order. LUT does the actual sorting.
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2. Position-dependent encoding: x → DNA with per-position base
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selection based on Q_ii sign. Fixes sign inversion for Ising-type
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QUBOs but still not monotone.
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3. Monotone encoding (the fix): sort all solutions by energy FIRST,
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then assign DNA sequences in energy order. Lexicographic DNA
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sort = energy sort BY CONSTRUCTION.
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The monotone encoding makes the DNA sequence a rank key:
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sequence "AAAA..." = lowest energy
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sequence "AAAT..." = second lowest
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...
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sequence "ZZZZ..." = highest energy
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Sorting the DNA IS sorting the energy. No thermodynamic model needed.
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The LUT maps sequence ↔ energy, and the sequence itself carries the
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rank information.
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Connection to SilverSight:
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HachimojiLUT.lean defines codon → state mapping via lookup.
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This extends that to: sequence → energy via monotone LUT.
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The PhaseCircle (ℤ/360ℤ) provides the symbolic address space.
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"""
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from __future__ import annotations
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import json
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import random
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from dataclasses import dataclass
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from typing import Dict, List, Optional, Tuple
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# ============================================================
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# §1 HACHIMOJI ALPHABET
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# ============================================================
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BASES = list("ABCGPSTZ") # 8 bases, ordered by ASCII for lexicographic monotonicity
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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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N_BASES = len(BASES)
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def int_to_dna(value: int, length: int) -> str:
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"""Convert integer to DNA sequence (base-8, fixed length)."""
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seq = []
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for _ in range(length):
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seq.append(INDEX_TO_BASE[value % N_BASES])
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value //= N_BASES
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return "".join(reversed(seq))
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def dna_to_int(sequence: str) -> int:
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"""Convert DNA sequence to integer."""
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value = 0
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for b in sequence:
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value = value * N_BASES + BASE_TO_INDEX[b]
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return value
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def sequence_length_for_solutions(n_solutions: int) -> int:
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"""Minimum sequence length to represent n_solutions unique keys."""
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if n_solutions <= 0:
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return 0
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length = 1
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while N_BASES ** length < n_solutions:
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length += 1
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return length
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# ============================================================
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# §2 QUBO ENERGY
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# ============================================================
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def qubo_energy(x: List[int], Q: List[List[float]]) -> float:
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"""Compute QUBO energy E(x) = x^T Q x."""
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n = len(x)
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return sum(Q[i][j] * x[i] * x[j] for i in range(n) for j in range(n))
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# ============================================================
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# §3 ENCODING STRATEGIES
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# ============================================================
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@dataclass
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class LUT:
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"""Lookup table: DNA sequence ↔ QUBO solution ↔ energy."""
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entries: Dict[str, Tuple[List[int], float]] # sequence → (x, energy)
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qubo_matrix: List[List[float]]
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n_vars: int
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encoding: str # "direct", "positional", "monotone"
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def lookup(self, sequence: str) -> Tuple[List[int], float]:
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"""Look up solution and energy for a sequence."""
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return self.entries.get(sequence, ([], float("inf")))
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def energy(self, sequence: str) -> float:
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return self.lookup(sequence)[1]
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def solution(self, sequence: str) -> List[int]:
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return self.lookup(sequence)[0]
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def size(self) -> int:
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return len(self.entries)
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def sort_by_sequence(self) -> List[Tuple[str, List[int], float]]:
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"""Sort entries by DNA sequence (lexicographic)."""
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return sorted(
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[(s, x, e) for s, (x, e) in self.entries.items()],
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key=lambda item: item[0],
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)
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def sort_by_energy(self) -> List[Tuple[str, List[int], float]]:
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"""Sort entries by energy (ascending). Ground truth."""
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return sorted(
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[(s, x, e) for s, (x, e) in self.entries.items()],
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key=lambda item: item[2],
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)
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def verify_monotone(self) -> Tuple[bool, float]:
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"""Check if lexicographic sort matches energy sort.
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Returns:
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(is_monotone, rank_correlation)
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"""
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by_seq = self.sort_by_sequence()
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by_energy = self.sort_by_energy()
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# Check if the orderings are identical
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seq_order = [s for s, _, _ in by_seq]
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energy_order = [s for s, _, _ in by_energy]
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is_monotone = seq_order == energy_order
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# Rank correlation
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n = len(by_seq)
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if n < 2:
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return True, 1.0
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seq_rank = {s: i for i, (s, _, _) in enumerate(by_seq)}
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energy_rank = {s: i for i, (s, _, _) in enumerate(by_energy)}
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d_sq = sum((seq_rank[s] - energy_rank[s]) ** 2 for s in seq_rank)
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corr = 1.0 - (6.0 * d_sq) / (n * (n * n - 1))
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return is_monotone, corr
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def to_json(self) -> str:
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return json.dumps({
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"encoding": self.encoding,
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"n_vars": self.n_vars,
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"qubo_matrix": self.qubo_matrix,
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"entries": {
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s: {"x": x, "energy": round(e, 6)}
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for s, (x, e) in self.entries.items()
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},
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}, indent=2)
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@classmethod
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def from_json(cls, data: str) -> "LUT":
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d = json.loads(data)
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entries = {
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s: (v["x"], v["energy"]) for s, v in d["entries"].items()
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}
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return cls(
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entries=entries,
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qubo_matrix=d["qubo_matrix"],
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n_vars=d["n_vars"],
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encoding=d["encoding"],
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)
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def build_direct_lut(
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Q: List[List[float]],
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n_vars: int,
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base_map: Optional[Dict[int, str]] = None,
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) -> LUT:
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"""Build LUT with direct encoding: x_i → base by fixed mapping.
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Default: 0 → A, 1 → G.
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Symbolic order does NOT match energy order.
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"""
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if base_map is None:
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base_map = {0: "A", 1: "G"}
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entries = {}
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for i in range(2**n_vars):
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x = [(i >> j) & 1 for j in range(n_vars)]
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seq = "".join(base_map[v] for v in x)
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energy = qubo_energy(x, Q)
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entries[seq] = (x, energy)
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return LUT(
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entries=entries,
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qubo_matrix=Q,
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n_vars=n_vars,
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encoding="direct",
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)
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def build_monotone_lut(
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Q: List[List[float]],
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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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) -> LUT:
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"""Build LUT with monotone encoding: DNA sequence = energy rank.
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1. Enumerate all solutions (or sample)
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2. Sort by energy
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3. Assign DNA sequences in energy order
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Result: lexicographic DNA sort = energy sort BY CONSTRUCTION.
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Args:
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Q: QUBO matrix
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n_vars: number of variables
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n_samples: 0 for brute force (n_vars ≤ 12), else random sampling
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seed: RNG seed
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Returns:
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LUT with monotone encoding
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"""
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# Step 1: enumerate solutions
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solutions = []
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if n_samples == 0 and n_vars <= 12:
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for i in range(2**n_vars):
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x = [(i >> j) & 1 for j in range(n_vars)]
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energy = qubo_energy(x, Q)
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solutions.append((x, energy))
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else:
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n_samples = n_samples or 10000
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rng = random.Random(seed)
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seen = set()
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for _ in range(n_samples):
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x = tuple(rng.randint(0, 1) for _ in range(n_vars))
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if x in seen:
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continue
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seen.add(x)
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energy = qubo_energy(list(x), Q)
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solutions.append((list(x), energy))
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# Step 2: sort by energy
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solutions.sort(key=lambda item: item[1])
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# Step 3: assign DNA sequences in energy order
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seq_len = sequence_length_for_solutions(len(solutions))
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entries = {}
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for rank, (x, energy) in enumerate(solutions):
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seq = int_to_dna(rank, seq_len)
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entries[seq] = (x, energy)
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return LUT(
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entries=entries,
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qubo_matrix=Q,
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n_vars=n_vars,
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encoding="monotone",
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)
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def build_positional_lut(
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Q: List[List[float]],
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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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) -> LUT:
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"""Build LUT with position-dependent encoding.
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Each variable gets a base pair chosen by Q_ii sign.
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Not monotone, but fixes sign inversion for Ising-type QUBOs.
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"""
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# Design position-dependent base pairs
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pairs = [("A", "G"), ("T", "C"), ("B", "S"), ("P", "Z")]
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pos_map = []
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for i in range(n_vars):
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pair = pairs[i % len(pairs)]
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if Q[i][i] < 0:
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pos_map.append((pair[1], pair[0])) # swap for negative diagonal
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else:
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pos_map.append(pair)
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entries = {}
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if n_samples == 0 and n_vars <= 12:
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for i in range(2**n_vars):
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x = [(i >> j) & 1 for j in range(n_vars)]
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seq = "".join(pos_map[j][x[j]] for j in range(n_vars))
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energy = qubo_energy(x, Q)
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entries[seq] = (x, energy)
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else:
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n_samples = n_samples or 10000
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rng = random.Random(seed)
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seen = set()
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for _ in range(n_samples):
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x = tuple(rng.randint(0, 1) for _ in range(n_vars))
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if x in seen:
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continue
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seen.add(x)
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seq = "".join(pos_map[j][x[j]] for j in range(n_vars))
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energy = qubo_energy(list(x), Q)
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entries[seq] = (list(x), energy)
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return LUT(
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entries=entries,
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qubo_matrix=Q,
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n_vars=n_vars,
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encoding="positional",
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)
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# ============================================================
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# §4 VERIFICATION
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# ============================================================
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def verify_all_encodings(
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Q: List[List[float]],
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n_vars: int,
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n_samples: int = 0,
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) -> Dict[str, dict]:
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"""Compare all three encoding strategies.
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Returns:
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Dict with results for each encoding method.
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"""
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results = {}
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# Direct encoding
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lut_direct = build_direct_lut(Q, n_vars)
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mono, corr = lut_direct.verify_monotone()
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best_seq = lut_direct.sort_by_energy()[0]
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worst_seq = lut_direct.sort_by_sequence()[0]
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results["direct"] = {
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"encoding": "direct",
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"n_entries": lut_direct.size(),
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"is_monotone": mono,
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"rank_correlation": round(corr, 4),
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"min_energy": round(best_seq[2], 4),
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"min_energy_seq": best_seq[0],
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"first_lex_seq": worst_seq[0],
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"first_lex_energy": round(worst_seq[2], 4),
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}
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# Positional encoding
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lut_pos = build_positional_lut(Q, n_vars, n_samples)
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mono, corr = lut_pos.verify_monotone()
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best_seq = lut_pos.sort_by_energy()[0]
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worst_seq = lut_pos.sort_by_sequence()[0]
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results["positional"] = {
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"encoding": "positional",
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"n_entries": lut_pos.size(),
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"is_monotone": mono,
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"rank_correlation": round(corr, 4),
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"min_energy": round(best_seq[2], 4),
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"min_energy_seq": best_seq[0],
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"first_lex_seq": worst_seq[0],
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"first_lex_energy": round(worst_seq[2], 4),
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}
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# Monotone encoding
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lut_mono = build_monotone_lut(Q, n_vars, n_samples)
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mono, corr = lut_mono.verify_monotone()
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best_seq = lut_mono.sort_by_energy()[0]
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worst_seq = lut_mono.sort_by_sequence()[0]
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results["monotone"] = {
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"encoding": "monotone",
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"n_entries": lut_mono.size(),
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"is_monotone": mono,
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"rank_correlation": round(corr, 4),
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"min_energy": round(best_seq[2], 4),
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"min_energy_seq": best_seq[0],
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"first_lex_seq": worst_seq[0],
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"first_lex_energy": round(worst_seq[2], 4),
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}
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return results
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# ============================================================
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# §5 DEMO QUBO GENERATORS
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# ============================================================
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def demo_qubo(n: int = 6, seed: int = 42, style: str = "random") -> List[List[float]]:
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"""Generate a demo QUBO."""
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rng = random.Random(seed)
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Q = [[0.0] * n for _ in range(n)]
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if style == "diagonal":
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for i in range(n):
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Q[i][i] = rng.uniform(1, 5)
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elif style == "banded":
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for i in range(n):
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Q[i][i] = rng.uniform(1, 5)
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if i + 1 < n:
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val = rng.uniform(-2, 2)
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Q[i][i + 1] = val
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Q[i + 1][i] = val
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elif style == "ising":
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for i in range(n):
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Q[i][i] = -1.0
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if i + 1 < n:
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Q[i][i + 1] = -0.5
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Q[i + 1][i] = -0.5
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else: # random
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for i in range(n):
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Q[i][i] = rng.uniform(-2, 5)
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for j in range(i + 1, n):
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val = rng.uniform(-2, 2)
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Q[i][j] = val
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Q[j][i] = val
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return Q
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# ============================================================
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# §6 CLI
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# ============================================================
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def run_demo(name: str, Q: List[List[float]], n_vars: int):
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"""Run a single demo comparing all encodings."""
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print(f"\n{'='*60}")
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print(f" {name}")
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print(f"{'='*60}")
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results = verify_all_encodings(Q, n_vars)
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for method in ["direct", "positional", "monotone"]:
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r = results[method]
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print(f"\n {method.upper()} encoding:")
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print(f" Entries: {r['n_entries']}")
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print(f" Monotone: {r['is_monotone']}")
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print(f" Rank corr: {r['rank_correlation']}")
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print(f" Min energy: {r['min_energy']} (seq: {r['min_energy_seq']})")
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print(f" First in lex: {r['first_lex_energy']} (seq: {r['first_lex_seq']})")
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if r["is_monotone"]:
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print(f" ✓ Lexicographic sort = Energy sort")
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else:
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print(f" ✗ Lexicographic sort ≠ Energy sort")
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if __name__ == "__main__":
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print("=" * 60)
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print("Symbology LUT Adaptation — Fixed")
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print("=" * 60)
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# Diagonal
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Q1 = demo_qubo(6, seed=42, style="diagonal")
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run_demo("Diagonal QUBO (6 vars)", Q1, 6)
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# Banded
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Q2 = demo_qubo(6, seed=42, style="banded")
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run_demo("Banded QUBO (6 vars)", Q2, 6)
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# Ising
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Q3 = demo_qubo(6, seed=42, style="ising")
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run_demo("1D Ising Chain (6 vars)", Q3, 6)
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# Random
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Q4 = demo_qubo(6, seed=42, style="random")
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run_demo("Random QUBO (6 vars)", Q4, 6)
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# Larger
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Q5 = demo_qubo(8, seed=42, style="banded")
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run_demo("Banded QUBO (8 vars)", Q5, 8)
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print(f"\n{'='*60}")
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print("DONE")
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print(f"{'='*60}")
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