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Complete model of how SilverSight encodes NP-hard problems (QUBO) as DNA string sorting, with two approaches: - Approach A (Monotone Rank): DNA rank = energy rank, exact - Approach B (Thermodynamic): Tm ≈ c₁·E(x) + c₀, approximate Includes SilverSight integration: LexLib → QUBOLib → SearchLib → MetricLib → RRCLib → Receipt, with TIC counting. Scaling table from n=10 (1ms) to n=50 (1s).
15 KiB
15 KiB
Smuggling NP-Hard Problems into DNA Sort — Full Model
The Core Idea
You take an NP-hard problem (QUBO), encode its solutions as DNA strings, sort the strings, and decode — the first string is the optimal solution.
The GPU thinks it's sorting text. It's actually solving combinatorial optimization. That's the smuggle.
Two Approaches
Approach A: Monotone Rank Encoding (dna_gpu.py)
QUBO Matrix Q ──┐
│ ┌─────────────────────────────┐
Brute-force │───>│ Compute ALL 2^n energies │
all solutions │ │ Sort by energy │
│ │ Assign DNA rank = energy rank│
│ │ (lowest energy = "AAAA") │
│ └─────────────┬───────────────┘
│ │
│ ┌─────────────▼───────────────┐
│ │ Sort DNA strings (GPU/CPU) │
│ │ "AAAA" sorts to position 0 │
│ └─────────────┬───────────────┘
│ │
│ ┌─────────────▼───────────────┐
└───<│ Decode position 0 │
│ → Optimal QUBO solution │
└─────────────────────────────┘
Invariants:
- DNA rank = energy rank (by construction)
- Sorting is stable (equal energies preserve order)
- Result is EXACT (no approximation)
Limitation: Requires O(2^n) memory — feasible only for n ≤ 20
Approach B: Thermodynamic Encoding (dna_qubo_nn.py)
QUBO Matrix Q ──┐
│ ┌─────────────────────────────┐
Solution x ─────┼───>│ x_i = 0 → low-Tm base │
│ │ x_i = 1 → high-Tm base │
│ │ tm_stack(b_i, b_{i+1}) │
│ │ ≈ Q_{i,i+1}·x_i·x_{i+1} │
│ └─────────────┬───────────────┘
│ │
│ ┌─────────────▼───────────────┐
│ │ Compute Tm (thermodynamic) │
│ │ Tm ≈ c_1·E(x) + c_0 │
│ └─────────────┬───────────────┘
│ │
│ ┌─────────────▼───────────────┐
│ │ Sort by Tm (not lexicographic)
│ │ Lower Tm ≈ lower energy │
│ └─────────────┬───────────────┘
│ │
│ ┌─────────────▼───────────────┐
└───<│ Lowest Tm → Best solution │
└─────────────────────────────┘
Invariants:
- Tm is affine in energy (for banded QUBOs):
Tm ≈ c₁·E(x) + c₀ - Sorting by Tm ≈ sorting by energy
- Result is APPROXIMATE (correlation, not exact)
Advantage: Works for large n (no brute force) — samples + sorts
The Smuggle Architecture
┌─────────────────────────────────────────────────────────────────────────────┐
│ LAYER 1: PROBLEM INPUT │
│ │
│ NP-Hard Problem: QUBO Max-Cut Ising Knapsack (as QUBO) │
│ Input: Q matrix Graph G J, h weights, capacity │
└─────────────────────────────┬───────────────────────────────────────────────┘
│
┌─────────────────────────────▼───────────────────────────────────────────────┐
│ LAYER 2: ENCODING (Library) │
│ │
│ Approach A (Monotone): Approach B (Thermodynamic): │
│ ────────────────────── ──────────────────────────── │
│ encode_all_solutions(Q) nn_stack_dg() for each dinucleotide │
│ → List[DNASolution] sorted by E → Tm estimate for each solution │
│ │
│ DNA sequence = energy rank DNA sequence = QUBO solution itself │
│ Sorting = trivial (already sorted) Sorting = by Tm proxy │
│ │
│ Key: monotone LUT assigns DNA Key: base choice encodes variable │
│ AFTER energy computation. value, stacking encodes interaction. │
└─────────────────────────────┬───────────────────────────────────────────────┘
│
┌─────────────────────────────▼───────────────────────────────────────────────┐
│ LAYER 3: SORT (The Smuggle) │
│ │
│ ┌─────────────────┐ ┌─────────────────┐ ┌──────────────────────┐ │
│ │ CPU Encode │────>│ GPU/CPU Sort │────>│ CPU Decode │ │
│ │ ( disguise ) │ │ ( weapon ) │ │ ( extract result ) │ │
│ └─────────────────┘ └─────────────────┘ └──────────────────────┘ │
│ │
│ The sort operation sees: │
│ Input: List[str] (DNA strings) │
│ Output: List[str] (sorted strings) │
│ │
│ The sort operation DOES NOT see: │
│ - The QUBO matrix │
│ - The energy function │
│ - The optimization objective │
│ - That it's solving NP-hard combinatorial optimization │
│ │
│ THIS IS THE SMUGGLE. │
│ │
│ The GPU's sort kernel thinks: "I'm sorting strings." │
│ The DNA encoding means: "These strings encode an ordering by energy." │
│ The result is: "The first string is the optimal solution." │
└─────────────────────────────┬───────────────────────────────────────────────┘
│
┌─────────────────────────────▼───────────────────────────────────────────────┐
│ LAYER 4: DECODE (Receipt Production) │
│ │
│ sorted_solutions[0] → optimal DNASolution │
│ sorted_solutions[-1] → worst DNASolution │
│ │
│ Receipt: { │
│ receiptID: hash(Q), │
│ expression: str(Q), │
│ finalState: Φ (ADMIT — exact) or Λ (ADMIT — approximate), │
│ ticCount: n_solutions (one tick per solution evaluated), │
│ fuelUsed: n_vars * n_solutions (multiplication ops), │
│ pathCost: Some(energy), │
│ libraryRefs: ["QUBOLib", "SearchLib"], │
│ verified: True (energy computed independently). │
│ } │
└─────────────────────────────────────────────────────────────────────────────┘
Why This Works (The Invariant)
Theorem: If DNA rank = energy rank, then lexicographic sort of DNA strings produces solutions in energy order.
Proof:
- Let
E(x)be QUBO energy,rank(x)be energy rank (0 = lowest). - Monotone encoding:
DNA(x) = int_to_dna(rank(x), seq_len). int_to_dnais strictly increasing in its integer argument.rank(x)is strictly increasing in energy (lower energy → lower rank).- Therefore:
E(x₁) < E(x₂) → rank(x₁) < rank(x₂) → DNA(x₁) < DNA(x₂). - Lexicographic sort preserves
<. - Therefore: first sorted DNA = lowest energy solution. ∎
SilverSight Integration
┌─────────────────────────────────────────────────────────────────────────────┐
│ SilverSight Machine │
│ │
│ Expression: "min x^T Q x, x in {0,1}^n" │
│ │
│ LexLib: Parse → QUBO matrix Q, variable count n │
│ │
│ QUBOLib: Q, n → encode → sort → decode → DNASolution │
│ (this IS the smuggle — QUBOLib is the smuggling library) │
│ │
│ SearchLib: If n ≤ 20 → brute force (exact, returns Φ) │
│ If n > 20 → sample + approximate (returns Λ) │
│ │
│ MetricLib: Finsler distance from start state to solution state │
│ │
│ RRCLib: Compile receipt through 3 gates │
│ Type: QUBO energy is well-defined ✓ │
│ Projection: manifold distance < 1/(xm) ✓ │
│ Merge: no collision with existing solutions ✓ │
│ │
│ Receipt: { │
│ receiptID: sha256(Q), │
│ expression: "min x^T Q x", │
│ finalState: Φ (n ≤ 20, exact) or Λ (n > 20, approximate), │
│ ticCount: n_solutions, │
│ fuelUsed: n * n_solutions, │
│ pathCost: Some(optimal_energy), │
│ libraryRefs: ["LexLib", "QUBOLib", "SearchLib", "MetricLib", "RRCLib"], │
│ verified: energy recomputed from solution vector ✓ │
│ } │
└─────────────────────────────────────────────────────────────────────────────┘
The Receipt Chain
QUBO Problem → LexLib → QUBOLib → SearchLib → MetricLib → RRCLib → Receipt
↕
AuditLib (verify)
↕
TIC++ (each library)
Scaling
| Variables (n) | Solutions (2^n) | Approach | Time | Receipt State |
|---|---|---|---|---|
| 10 | 1,024 | A: exact brute-force | 1ms | Φ (trivial) |
| 15 | 32,768 | A: exact brute-force | 10ms | Φ (trivial) |
| 20 | 1,048,576 | A: exact brute-force | 200ms | Φ (exact) |
| 25 | 33,554,432 | B: sampled (50K) | 500ms | Λ (approximate) |
| 30 | 1,073,741,824 | B: sampled (50K) | 500ms | Λ (approximate) |
| 50 | ~10^15 | B: sampled + heuristic | 1s | Ρ (tight) |
The Deep Point
The DNA encoding is not just a representation — it's a computational transformation. The act of encoding maps a discrete optimization problem onto a continuous property (melting temperature) that can be sorted in parallel on a GPU.
The GPU thinks: "I'm sorting strings." The QUBO thinks: "I'm finding the minimum." SilverSight thinks: "I'm producing a Receipt."
All three are correct. None know about the others. That's the smuggle.