12.10.4 Tensor Substitution Partial Evaluation Relation

Tensor Substitution Partial Evaluation Relation is the identification of slot substitution as the elementary building block underlying tensor evaluation on fewer than all of its arguments, showing that the tensor result case of partial evaluation is nothing other than a single application, or a sequence of applications, of slot substitution.

Connecting the Two Concepts

Partial Evaluation as Iterated Substitution

Partial evaluation of a tensor on some, but not all, of its required arguments can be understood entirely in terms of slot substitution: each argument supplied during partial evaluation corresponds to a substitution performed into one specific target slot, with the remaining slots left open exactly as slot substitution describes.

A Single Substitution as the Simplest Case

The simplest instance of this relation occurs when only one argument is supplied during partial evaluation, in which case partial evaluation and a single slot substitution are literally the same operation, both producing a tensor of reduced rank with exactly one fewer open slot than the original.

Building Up Multiple Substitutions

Sequential Substitution Reproducing Full Partial Evaluation

When partial evaluation involves supplying several arguments at once, this can be equivalently reproduced by performing slot substitution repeatedly, one argument at a time, into successive target slots, with the remaining slot structure updating after each individual substitution as described previously:

showing that filling two slots simultaneously produces the same result as filling one slot first, then filling the remaining open slot of the resulting reduced tensor.

Order Independence Inherited from Substitution

Because individual slot substitutions into different slots commute with one another, as established for slot substitution generally, the order in which the arguments of a partial evaluation are conceptually inserted does not affect the final result, even though partial evaluation is often written as though all arguments are supplied at once.

Why This Relation Is Useful

Reducing a Complex Operation to a Simpler One

Understanding partial evaluation as repeated substitution allows a seemingly more complicated operation, involving several simultaneous arguments, to be analyzed using the simpler, single-slot mechanics of substitution, including its componentwise contraction formula and its effect on the remaining slot structure.

Justifying the Multilinearity of Partial Evaluation

Since each individual substitution preserves multilinearity in the remaining open slots, and partial evaluation is built from a sequence of such substitutions, this relation directly explains why the tensor result case of partial evaluation remains multilinear in whatever arguments have not yet been supplied.

Full Evaluation as the Endpoint of This Relation

Completing All Substitutions

Carrying the relation to its natural conclusion, if slot substitution is performed for every one of a tensor's slots in turn, no open slots remain, and the process terminates in the scalar result case of tensor evaluation, showing that full evaluation is simply the special case of this iterated substitution process in which every slot has been addressed.

Illustration