Phase 0 Cheat Sheet: Dtypes and Primitives

A quick reference for the two most important practical questions:

1. What data types exist in Phase 0?
2. What primitive operations are available in core IR?

This is a companion document, not a normative spec. If this file and phase-0/ disagree, phase-0/ is authoritative.

1) Type building blocks

Phase 0 type grammar (simplified) includes:

• Tensor[Dtype, Shape, Refinements]
• Scalar[Dtype]
• Type -> Type
• Unit

Dtypes in Phase 0

Shape and refinement reminders

• Shape is ordered ([B, S, D] != [S, B, D]).
• Shape expressions are integer expressions in the Presburger-style subset.
• Refinements are logical predicates checked by SMT (Z3).
• Default symbolic dimension rule is α >= 1 unless explicitly overridden.

2) Ten core primitives

Phase 0 commits to ten core operations. reduce is one IR operation with two typing rules (reduce and reduce-keep).

Grouped by shape behavior

• Shape-preserving: unary, cast
• Shape-combining: binary, matmul
• Shape-reducing: reduce
• Shape-rearranging: transpose, reshape, slice
• Shape-extending: concat, broadcast

3) Three tiny typed examples

A) cast: dtype change, shape unchanged

let to_bf16 [B, S, D]
    (x : Tensor[F32, [B, S, D]])
    : Tensor[BF16, [B, S, D]] =
  cast[BF16] x

B) matmul: shape-combining primitive

let proj [B, M, K, N]
    (a : Tensor[F32, [B, M, K]])
    (b : Tensor[F32, [B, K, N]])
    : Tensor[F32, [B, M, N]] =
  matmul a b

C) reduce_keep: same rank, collapsed axis size 1

let row_sum_keep [B, S, D]
    (x : Tensor[F32, [B, S, D]])
    : Tensor[F32, [B, S, 1]] =
  reduce_keep[sum, axis=2] x

4) Beginner pitfalls (and fixes)

Pitfall 1: assuming implicit broadcast

Wrong mental model: x + bias should just work if bias is [D].

Phase 0 fix: broadcast explicitly first.

let bias_b = broadcast bias [B, S, D] in
x + bias_b

Pitfall 2: assuming implicit dtype promotion

Wrong mental model: I32 + F32 auto-promotes.

Phase 0 fix: cast explicitly, then apply binary op.

let xi = cast[F32] x_i32 in
xi + y_f32

Pitfall 3: confusing value-level mul with shape arithmetic

• mul primitive multiplies tensor values at runtime.
• Shape arithmetic lives in type/refinement expressions.

Keep these domains separate when reading typing rules.

5) Quick decision guide

When writing a new line of code, ask in order:

1. Did dtype change? Use cast.
2. Did axis order change? Use transpose.
3. Did total element count stay same but layout changed? Use reshape.
4. Need dimension expansion? Use explicit broadcast.
5. Need to aggregate an axis? Use reduce / reduce_keep.
6. Need to combine two tensors by contraction? Use matmul.

6) References

• phase-0/01-type-system-foundation.md
• phase-0/02-core-operations-and-typing.md
• phase-0/03-subtyping-and-coercion.md
• companions/phase-0-learning-path.md
• companions/real-world-starter-pack.md