Paper 97: Biological Computing under the Rei D-FUMT₈ Lens — Physarum TSP and DNA 3-SAT
Author: Fujimoto Nobuki (藤本伸樹) / fc0web
Date: 2026-04-16 | License: CC-BY-4.0
Keywords: Physarum polycephalum, DNA computing, Adleman, Lipton, TSP, 3-SAT, D-FUMT₈
Abstract
STEP 816 implements two biological analog-computing approaches to NP-hard problems:
-
Physarum slime mold solving TSP (Tero-Kobayashi-Nakagaki 2010 tube adaptation model) — finds optimal tour on an 8-city instance (ratio 1.000).
-
DNA computing for 3-SAT (Adleman 1994 / Lipton 1995 style) — on 10 random 5-variable 10-clause instances, 10/10 satisfiable with 79 total satisfying assignments.
Both are integrated into the D-FUMT₈ ontology: Physarum tube pruning = FLOWING, DNA superposition pool = BOTH.
1. Physarum TSP
Nakagaki 2000 demonstrated that Physarum polycephalum can find the shortest path through a maze. The Tero et al. 2010 model formalizes this as:
dD_ij / dt = |Q_ij|^μ − D_ij (tube conductance adaptive)
Q_ij = D_ij (p_i − p_j) / L_ij (Hagen-Poiseuille-like flux)
Tubes with strong flux strengthen, weak ones decay. At equilibrium, surviving tubes form a near-minimum transport network.
Result (8-city random instance):
| method | length | ratio |
|---|---|---|
| brute force optimum | 2.5328 | 1.000 |
| nearest neighbour | 2.5328 | 1.000 |
| Physarum (simulated) | 2.5328 | 1.000 |
2. DNA computing 3-SAT
Adleman's 1994 method:
- encode each clause as a DNA strand pair,
- mix all 2^n possible assignments into a pool,
- hybridize + ligate + PCR-amplify: only satisfying assignments survive.
Result (10 trials, 5 variables, 10 clauses each):
| metric | value |
|---|---|
| satisfiable trials | 10/10 |
| total satisfying assignments | 79 |
| mean per trial | 7.9 |
3. D-FUMT₈ interpretations
| element | D-FUMT₈ | rationale |
|---|---|---|
| Physarum tube network (dynamical) | FLOWING | continuous adaptation, analog optimization |
| nutrient source / city | TRUE | decidable goal state |
| Physarum near-optimal tour | TRUE | solution, decidable |
| Failed Physarum (trapped in local min) | NEITHER | no valid tour emerges |
| DNA satisfying assignment pool | BOTH | paraconsistent: many valid assignments coexist |
| DNA empty pool | FALSE | unsatisfiable |
| DNA unique survivor after amplification | SELF | self-verifying assignment |
4. Why biological computing is a separate category
Unlike electrical circuits (Paper 91) which compute in discrete-event cycles, biological computing is:
- Massively parallel (10^14 DNA strands per µL)
- Stochastic (not deterministic)
- Energy-efficient (chemical potential, no electricity)
This adds a fourth physical realization to the triple of Paper 95 (electrical + photonic + particle), making the peak-9232 type invariants potentially testable in four independent media.
5. Open questions
- Can Physarum detect the peak-9232 invariant? Requires mapping Collatz orbits to spatial maze structure.
- Can DNA encode Collatz termination as a SAT instance? Yes in principle (Cook-Levin), at astronomical strand count.
- Rei's 25 atomic cores could become DNA strand sets — SAT solver could verify Paper 60 classification.
6. Scope and honest limits
- Physarum simulation is a numerical model of the 2010 paper, not real slime mold.
- DNA SAT is brute-force enumeration, not actual wet-lab DNA.
- Results agree with established literature; we add only D-FUMT₈ mapping.
7. Reproducibility
python scripts/step816-physarum-dna-computing.py
# → data/step816-physarum-dna.json
CC-BY-4.0