Affine accesses and legal execution order

For fixed integer coefficients, an expression such as 32 i+j+4 maps loop coordinates to a linearized element index. Multiplying two varying induction variables, such as i×j, is not affine. MLIR's Affine dialect also distinguishes dimensions and symbols under defined validity rules; do not assume every loop-invariant expression is automatically a legal symbol. Dynamic strides can require a different representation or stronger assumptions than a constant-coefficient example. The lesson uses mathematical index expressions rather than version-sensitive parser syntax.

The iteration domain matters as much as the subscript. Bounds, guards, and shape assumptions determine which operations actually execute and whether addresses are in range. Two distinct logical accesses may alias through views or shared buffers, so different variable names are not proof of independence. Analysis must include relevant reads, writes, and effects. Affine structure makes some questions tractable; it does not erase unknown calls or permit unsupported assumptions about external memory.

A flow dependence connects a write that produces a value to a later read that must observe it. Anti-dependences and output dependences also constrain transformations when buffers are reused. For lexicographic loop execution, a legal schedule keeps every required source before its destination. An interchange permutes schedule coordinates and can turn a previously positive dependence into a negative one. Inspect the first nonzero component after permutation rather than merely observing that the loop bounds remain unchanged.

For a recurrence reading A[i−1, j+1], the dependence from its producer to consumer has displacement(1, −1). With i outermost, the positive first component preserves row-to-row production. Swapping to an increasing j-outer schedule makes the first component negative and can read a value before its producer runs. Some transformations can repair a schedule through skewing or other changes, but an unmodified interchange is not justified by a locality benefit alone.

Integer overflow, floating-point reductions, exceptions, and side effects can impose constraints beyond ordinary array dependences. Mathematical equality over real numbers does not establish identical IEEE floating-point behavior. LLVM 20.1.0 documents explicit fast-math permissions; changing reduction order needs the relevant source and IR contract rather than an assumption that neural workloads tolerate anything. Save the original domain, alias assumptions, and numeric mode with a transformation test so a passing easy input is not mistaken for a proof.

The matrix lab illustrates an affine-looking geometric map only. Reset and set both diagonal scale a=1 and shear b=1, predict determinant 1 and the unit-square shear, then Step. Its selected matrix is [[a, b], [0,1]], and readouts describe that final matrix while playback interpolates geometry. It does not analyze loop domains, integer lattices, aliases, or dependences. Explain why a visually invertible map cannot certify that a reordered recurrence reads the same versions of memory.