Datasets:
configs:
- config_name: default
data_files: omega_tensor.parquet
- config_name: init_state
data_files: init_state_tensor.parquet
dataset_info:
- config_name: default
features:
- name: op1
dtype: int64
- name: op2
dtype: int64
- name: op3
dtype: int64
- name: op4
dtype: int64
- name: op1_name
dtype: string
- name: op2_name
dtype: string
- name: op3_name
dtype: string
- name: op4_name
dtype: string
- name: halting_count
dtype: int64
- name: omega
dtype: float64
- name: has_swp
dtype: bool
- name: has_mov
dtype: bool
- name: has_and
dtype: bool
- name: has_shr
dtype: bool
- name: involution_count
dtype: int64
- name: coupling
dtype: string
- name: avg_cycle
dtype: float64
- name: max_cycle
dtype: int64
- name: entropy_bits
dtype: float64
- name: reachable_outputs
dtype: int64
splits:
- name: train
num_examples: 38416
- config_name: init_state
features:
- name: op1
dtype: int8
- name: op2
dtype: int8
- name: op3
dtype: int8
- name: op4
dtype: int8
splits:
- name: train
num_examples: 38416
license: mit
task_categories:
- other
tags:
- halting-problem
- computability
- algorithmic-information-theory
- exhaustive-enumeration
- FPGA
size_categories:
- 1M<n<10M
Omega Tensor: Exact Halting Fractions for 38,416 Instruction Sets
Exhaustive enumeration of all 1,679,616 programs of length 8 for every possible instruction set on a two-register, 8-bit, bounded machine with JNZ-to-zero control flow. Each row is one ISA. Each halting count is exact.
The repository contains two tables:
omega_tensor.parquet(38,416 rows) — canonical single-initial-state table used in the paper, with the halting fraction at A=0, B=0 and cross-observable columns (cycle length, output entropy, reachability, structural flags).init_state_tensor.parquet(38,416 rows, 260 columns) — extends the above to every initial A value in 0..255 at B=0. Halting counts only.
Contents — omega_tensor.parquet
38,416 rows. Each row represents one ordered ISA defined by four opcode assignments (opcodes 1-4) drawn from 14 primitive operations. Opcode 0 is always INC. Opcode 5 is always JNZ.
| Column | Type | Description |
|---|---|---|
| op1-op4 | int | Operation IDs for opcodes 1-4 |
| op1_name-op4_name | str | Human-readable operation names |
| halting_count | int | Exact count of halting programs (out of 1,679,616) |
| omega | float | halting_count / 1,679,616 |
| has_swp, has_mov, has_and, has_shr | bool | Structural flags |
| involution_count | int | Number of involution operations in the ISA |
| coupling | str | Register coupling type: bidirectional, one-way, or none |
| avg_cycle | float | Mean cycle length among non-halting programs |
| max_cycle | int | Maximum cycle length among non-halting programs |
| entropy_bits | float | Shannon entropy of the output distribution (bits) |
| reachable_outputs | int | Number of distinct outputs produced by halting programs |
Statistics — omega_tensor.parquet
- Range: [0.290710, 0.816916]
- Mean: 0.612418
- Distinct values (6 decimal places): 1,148
- Total programs executed: 64,527,984,256 (38,416 x 1,679,616)
- Avg cycle length range: [23.4, 216.1] (mean 116.2)
- Max cycle length: 255 for all ISAs
- Output entropy range: [0.80, 5.44] bits (mean 2.50)
- Reachable outputs: [9, 211] / 256 (mean 53.7)
- Outputs reachable by ALL ISAs: 9
Extremes
- Minimum (0.2907): INC INC INC INC INC JNZ. Five copies of INC. 71% of programs loop forever.
- Maximum (0.8169): INC AND AND AND AND JNZ. Four copies of AND. Only 18% loop.
Method
Each ISA was evaluated by exhaustive enumeration of all 6^8 = 1,679,616 programs of length 8. Each program was executed from initial state (A=0, B=0, PC=0) for up to 256 steps. Programs that reached PC >= 8 were counted as halting. The computation used a Numba-JIT-compiled Python interpreter at 0.26 seconds per ISA (approximately 2.8 hours total for all 38,416 ISAs).
Every halting count is exact. No sampling, no approximation, no estimation.
Verification
Known ISA values verified against independent FPGA silicon measurements and CPU cross-checks:
- ISA-A (DEC SWP ADD XOR): 978,929 halting (dataset) = 978,929 (FPGA) = 978,929 (CPU)
- ISA-B (NEG MOV SUB AND): 1,235,107 = 1,235,107 = 1,235,107
- ISA-G (SHR MOV SUB AND): 1,248,490 = 1,248,490 = 1,248,490
Contents — init_state_tensor.parquet
38,416 rows, one per ISA (same ISA space as omega_tensor.parquet). 260 columns: four opcode slots plus 256 halting counts, one per initial A value.
| Column | Type | Description |
|---|---|---|
| op1-op4 | int8 | Operation IDs for opcodes 1-4 |
| h_000 through h_255 | int32 | Halting counts at initial state (A=N, B=0) for N=0..255 |
Join on (op1, op2, op3, op4) to recover names, cycle length, entropy, and structural columns from omega_tensor.parquet. The canonical value omega from that table equals h_000 / 1,679,616.
Statistics — init_state_tensor.parquet
- Total cells: 38,416 x 256 = 9,834,496 exact halting counts
- Total program evaluations: ~16.5 billion
- Omega(init=0) mean: 0.612418 (matches
omega_tensor.parquet) - Omega(init=128) mean: 0.366431, range [0.236044, 0.681003]
- Ratio Omega(0)/Omega(128): mean 1.7429, median 1.6430, range [0.8336, 3.1466]
- 99.92% of ISAs have Omega(0) > Omega(128); 0.08% have the inverse (INC-dominant ISAs)
- 7.91% of ISAs have a perfectly reflection-symmetric bowl (Omega(a) = Omega(256-a))
- No ISA has its Omega minimum at init_a = 128 exactly; 3.23% have argmin in [126, 130]
- Bowl depth Omega(0) - Omega(min) ranges [0.058, 0.513], mean 0.260
Method — init_state_tensor.parquet
Same interpreter and enumeration protocol as the canonical tensor, extended over all 256 values of the initial accumulator A (B held at 0). 9,834,496 (ISA, init_a) cells, each the exact halting count over the full 1,679,616-program space. Computed with the same Numba-JIT interpreter under an 18-worker multiprocessing pool, 110.3 wall hours, zero errors. Every count is exact.
Cross-validation: for every ISA, the h_000 column equals the canonical halting_count from omega_tensor.parquet.