A ternary cell is the fundamental computational unit in balanced ternary logic — a cell that holds one of three states: Negative (−1), Neutral (0), or Positive (+1). This Python package implements ternary cell grids with formula evaluation, providing the building blocks for ternary spreadsheet computation and ternary strategy simulation.
Binary logic (0 and 1) is the foundation of digital computing, but balanced ternary (−1, 0, +1) offers several theoretical advantages: ternary numbers are more compact (log₂(3) ≈ 1.585 bits per trit), balanced ternary arithmetic simplifies sign handling (no separate sign bit), and three-valued logic naturally represents unknown/neutral states that binary requires hacky workarounds for. The ternary cell is the atomic unit — analogous to a bit in binary systems — and composing grids of ternary cells enables ternary spreadsheet computation, strategy simulation, and the study of emergent ternary dynamics that are central to the SuperInstance framework.
Each cell holds a value from {−1, 0, +1}. The elementary operations differ from binary:
| Operation | Truth Table | Notation |
|---|---|---|
| Negation | −(−1)=+1, −0=0, −(+1)=−1 | ā |
| Min (AND-like) | min(a, b) | a∧b |
| Max (OR-like) | max(a, b) | a∨b |
| Consensus | −1 if both = −1, +1 if both = +1, else 0 | a⊙b |
| Sum (truncated) | a+b clamped to [−1, +1] | a⊕b |
A grid of N×M ternary cells supports:
- Direct values: set cell(r,c) = −1, 0, or +1
- Formulas: SUM, PRODUCT, THRESHOLD over rectangular ranges
- Evaluation: topological sort for dependency resolution
Formula evaluation uses clamped ternary arithmetic:
SUM(a, b, c) = clamp(a + b + c, -1, +1)
THRESHOLD(range, t) = +1 if Σ > t, −1 if Σ < −t, else 0
| Operation | Time | Space |
|---|---|---|
| Cell access | O(1) | O(1) |
| Formula eval (k×k range) | O(k²) | O(k²) |
| Grid evaluate (all formulas) | O(N·M + E) | O(N·M) |
where E = total formula dependencies across the grid.
fromternary_cellimportTernaryCell, TernaryValuecell=TernaryCell(TernaryValue.POSITIVE)
print(cell.value) # +1# Negatecell_neg=-cellprint(cell_neg.value) # -1# Consensusneutral=TernaryCell(TernaryValue.NEUTRAL)
result=cell.consensus(cell_neg)
print(result.value) # 0 (positive and negative cancel)| Class | Methods | Description |
|---|---|---|
TernaryCell | value, negate(), consensus() | Single ternary value holder |
TernaryValue | NEGATIVE, NEUTRAL, POSITIVE | Enum of ternary states |
TernaryGrid | get(), set(), evaluate() | N×M grid with formula support |
The ternary cell is the atomic primitive of the γ + η = C framework. The three values map directly: +1 = γ (constructive, chosen, present), −1 = η (avoidant, rejected, absent), 0 = neutral (undecided, unknown, transitional). A grid of ternary cells computes the dynamics of γ and η interaction over space and time, producing emergent competence C. This is the discrete computational substrate on which all SuperInstance ternary experiments run. See ARCHITECTURE.md.
- Knuth, D. E. (1981). The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Addison-Wesley. — Section 4.1 on balanced ternary number systems.
- Frieder, G., & Luk, C. (1975). "Ternary Computers." In Proceedings of the IEEE. — Survey of ternary computing hardware.
- Hayes, B. (2001). "Third Base." American Scientist, 89(6), 490–494. — Popular introduction to balanced ternary.
| Property | Binary (0, 1) | Balanced Ternary (−1, 0, +1) |
|---|---|---|
| States per digit | 2 | 3 |
| Information per digit | 1 bit | log₂(3) ≈ 1.585 bits |
| Sign representation | Separate sign bit | Native (−1 is a value) |
| Negation | Flip all bits + add 1 | Swap −1 ↔ +1, keep 0 |
| Unknown state | Requires hack (nullable) | Native (0 = neutral/unknown) |
The ternary cell's native support for an unknown/neutral state eliminates the need for nullable types or sentinel values — a significant advantage in decision-making contexts where "undecided" is a genuine third state.
MIT