What earns an instruction

An instruction sits in the vocabulary iff something real speaks it and no composition of the rest reproduces its bits.

Why#

A compiler that will record reverse mode cannot treat the op set as a convenience menu. Every derivative rule expands into instructions that themselves have rules. If that expansion leaves the vocabulary, the transform has nowhere to write. If it stays inside by duplicating an instruction that composition already expresses in bits, the spec grows a synonym, dumps get noisier, and two spellings of the same math start to drift.

The other failure is the kernel zoo: a faster or stabler form lands as a new instruction because a facade wanted a name. Speed and accuracy then hide in the IR instead of living where they belong — accuracy in the vocabulary, speed in the catalog.

The idea#

   0  Input                             [4, 8]
   1  Input                             [4, 8]
   2  Input                             [4, 8]
   3  Leaf                              [4, 4]
   4  Leaf                              []
   5  Permute        1  order=[1, 0]    [8, 4]
   6  MatMul         0, 5               [4, 4]
   7  Broadcast      4  shape=[4, 4]    [4, 4]
   8  Mul            6, 7               [4, 4]
   9  Add            8, 3               [4, 4]
  10  LogSoftmax     9  axis=1          [4, 4]
  11  Exp            10                 [4, 4]
  12  MatMul         11, 2              [4, 8]
tape: 13 nodes, 0 parameters, 3 inputs

One membership test, two clauses, both required:

• A real consumer records it, or a derivative rule speaks it.
• No composition of the remaining instructions reproduces its bits.

Bit-identity makes the second clause decidable. The first keeps the set from growing "in case." Together they are the closed core: small because redundant spellings are refused, complete because adjoints and float-breaking formulas are not.

Adjoint pairing is the organizing structure. Reductions pair with broadcasts, windows with their folds, gathers with scatters; some instructions are self-adjoint; a one-way mask can terminate the closure. Recorded reverse mode is possible because that pairing is a theorem of the set, not a hope about the engine.

Composition is the default spelling. A formula moves down a tier and earns an instruction only when floating point breaks the composed form. A fused or native form that preserves bits is not a new instruction — it is an offer the catalog may elect. Accuracy lives in the vocabulary. Speed lives in the catalog.

A documented exception is allowed when the bit-test would make the spec unreadable and no later tier recovers the cost. The exception is the judgment, written down; it is not a second test.

Consequences#

• Adding an instruction is a breaking change of the IR, judged by the test, never by convenience.
• Changing a derivative rule that speaks a new instruction means the instruction earns its seat the same day, or the rule is restated in the existing set.
• Silently replacing a stable primitive with a composition that does not preserve bits is a different spec, not a rewrite.
• Unary elementwise transcendentals are one instruction kind parameterized by the map, not a new instruction per function.
• Facades compose through the public surface. Their names are not a reason to grow the vocabulary.

Not this#

• A kernel zoo. Faster spellings elect; they do not mint opcodes.
• Retiring an instruction that still has a consumer or a rule.
• Treating "we might need it" as clause (a).
• Opening the set so a dependent can register an instruction. New primitives land in the crate, under this test.

See Recorded reverse mode for why the set must close under adjoints, and Vision for composition as the default.

Spelled today#

Opcode / Op is the closed set. Map carries MapOperation. log_softmax and log_sum_exp earned their seats on bits; relu is maximum against a counted zero; Sub is the documented exception on the bit clause (Add of Neg is bit-exact). Powf is the exception on the consumer clause: kept ahead of its consumers by decision — its composition is neither bit-faithful nor defined on negative bases, and the expected consumers (learned pooling exponents, fractional robust-loss powers) are on file in the op-set audit. This section may rot; the rest must not.

See Fusion raises for the catalog's second life as an idiom raiser.