| # Reduction Algebra Specification |
|
|
| ## Object |
|
|
| ``` |
| R = (D, OP, E, A, C, S) |
| ``` |
|
|
| | Symbol | Meaning | |
| |--------|---------| |
| | **D** | Domain (booleans, reals/exact ints, Goldilocks F_p, attention tensors) | |
| | **OP** | Operator (boolean/arithmetic/structural/field/SUBLEQ/crypto stubs) | |
| | **E** | Identity elements for each OP | |
| | **A** | Algebraic laws (associativity only where established on D) | |
| | **C** | Canonicalization (mod p residues; JSON canonical form) | |
| | **S** | Sealing. Production bi-encoder: SHA-512 DAG in J and R. Legacy pipeline: SHA-256 JSON stub. | |
| |
| This studies **reduction algebra** for an experimental SUBLEQ Attention Engine. |
| It is **not** ordinary DB aggregation, WORM storage, or softmax transformer attention. |
| |
| ## Goldilocks |
| |
| `p = 18446744069414584321 = 2^64 - 2^32 + 1`. |
| |
| Authoritative ops: `reference/goldilocks.py` (Python arbitrary precision). |
| |
| TinyAPL uses `Complex Double` and **cannot** represent `p` as a distinct exact scalar |
| (ulp at 2^64 is 4096). TinyAPL-native demos use `GSafeP = 65537 (Fermat prime)` for exact local |
| arithmetic. Full-field equality is always checked against the Python reference. |
| |
| ## Bi-encoder arithmetic |
| |
| The SUBLEQ bi-encoder (two towers, fixed permutation, routed dot product) is |
| **not** softmax attention and is not trained. Its products and Goldilocks |
| residues are extended integers in J (`j/biencoder.ijs`) and R gmp `bigz` |
| (`r/biencoder.R`). Python is not a numeric authority for that pipeline. |
| TinyAPL cannot represent `p` exactly and is only the earlier experiment. |
| The unmasked head must equal `A` times `B`. The routed tensor is reported |
| separately and is not required to match that product. |
| |
| One integer training step (`j/train_step.ijs`, `r/train_step.R`) is not softmax |
| and not a loop. `Wq` and `Wk` stay the permutation. The loss is the integer |
| residual between the routed prediction and the exact product. That residual |
| updates an integer projection between the unmasked head and the routing |
| layer, and the post routed prediction is the pre routed prediction plus |
| that projection, elementwise. The unmasked head is not modified. A mismatch between the unmasked head and `A` times `B` still fails the |
| step. The routed residual is not required to be zero. The SHA-512 DAG layout |
| is unchanged. |
| |
| |
| The production seal is a SHA-512 DAG, computed in J and again in R, not in |
| Python. Nodes are the exact head, the routed prediction, the weight |
| permutation, and the prime `p` (rank 0). Each node digest is SHA-512 of the |
| canonical byte layout in `j/sha512.ijs` and `r/sha512.R`: magic `SRANOD01`, |
| length-prefixed role, rank, dimensions, count, then length-prefixed ASCII |
| decimals in row-major order. No JSON and no floats. The parent preimage is |
| magic `SRADAG01`, the child count, and the raw child digests in that fixed |
| order. The seal is SHA-512 of the parent preimage. `verify` recomputes it. |
| A one-entry mutation of the routed tensor must change the seal and fail |
| verify. The legacy SHA-256 canonical-JSON helpers in |
| `reference/reduction_algebra.py` and TinyAPL `VerifyStub` are not this seal. |
|
|
| ## SUBLEQ attention pipeline |
|
|
| ``` |
| ARRAY → SUBTRACT (B←B−A) → COMPARE → PREDICATE → SELECT → ROUTE → REDUCE → STATE |
| ``` |
|
|
| **Not** equivalent to softmax attention. |
|
|
| ## Topologies R0–R7 |
|
|
| | Id | Name | Depth | |
| |----|------|-------| |
| | R0 | scalar-reference | 0 | |
| | R1 | sequential | N−1 | |
| | R2 | balanced-tree | ⌈log2 N⌉ | |
| | R3 | chunked | (chunk−1)+(⌈N/chunk⌉−1) | |
| | R4 | segmented | per segment | |
| | R5 | tacit-fused | N−1 | |
| | R6 | goldilocks-field | N−1 | |
| | R7 | subleq-routed-field | N−1 | |
|
|
| ## Verification |
|
|
| 1. No performance claim without correctness pass. |
| 2. Deterministic identical inputs. |
| 3. `reference(X) = candidate(X) (mod p)` for Goldilocks. |
| 4. Never fabricate metrics; use `not_measured` when untimed. |
| 5. Bi-encoder seals agree across J and R, and verify fails if a sealed tensor is mutated. |
| 6. Benchmark seconds come from J `6!:2` or R `system.time` only. |
|
|
