We know that in the echo program the mapping do-echo-computation is built up by combining the operations attenuate, delay, and mix, as shown here:
Figure 1a shows the individual character of the computational operations shown in combination in Figure 1:
We can see that attenuate and delay each take a sequence of samples as input and produce one as output. The double tailed arrow is used to show that mix takes two sequences of samples as input and produces one as output. Figure 1 shows just one particular way of combining these operations so as to produce a new operation, one of many. Clearly we could have a program that just does attenuate, as shown in a representational system in Figure 2:
Mathematically, the process of combining operations is composition. Given mappings whose outputs fit the inputs of other mappings, we can create a composite mapping that combines the effects of these mappings.
Against this background, a program can be seen as specifying two separable things: a collection of operations, and a way of composing those operations (to create a mapping that corresponds to some desired operation in the target domain.) Because these two things are separable, we do not have to start from scratch when we want a new combination of operations, just as we don't have to start from scratch when we want to add an echo to a new sound.
Composition and Notation
As we've seen, a program specifies a collection of operations, and a way of composing the operations to create a desired mapping. How do programming languages support this activity? This is itself a problem in representation: our target domain has mappings, with operations that include the various ways we might wish to compose them. A programming language represents these things in a way that satisfies two requirements:
(1) There has to be an operation on programs that allows us to get the effect of composing mappings.
(2) Working with programs has to allow people to obtain these effects easily and accurately, in a sufficient range of situations.
Note that these conditions correspond to Mackinlay's expressiveness and effectiveness conditions, as discussed in the post for February 17, 2008
Representation and Cost.
What does Condition (1) entail about programs? One might think that a picture like Figure 3a would capture the essentials of the situation:
Figure 3aHere there are two mappings in a target domain, m1 and m2, and their composition, m3. There is a mapping notate that takes mappings to programs, and another mapping, interpret, that takes a program to a mapping. It would seem that programs in this representational system, and an operation on them, accurately represent the mappings and the composition operation.
But how can interpret actually work? Since the idea of a representational system is that one obtains the actual results of work in the target domain, by passing through the representational domain, interpret would have to actually produce the mapping m3. But how can that be done? What does it mean to actually produce a mapping?
An auxiliary picture, Figure 3b, explains this.
Figure 3bHere a mapping m is shown in a setting that includes the things on which it operates and which it produces. The picture also includes not only the program p, but also an implementation, ip, of program p. The implementation is an actual physical apparatus that operates on and produces things in the target domain. For example, in the echo example, the combination of record, do-echo-computation, and playback, operates on an actual sound and produces an actual sound.
As Figure 3b suggests, for p to represent m requires not just that there be some correspondence between p and m, but that there be an implementation of p that can actually do the work of m. Figure 3c elaborates Figure 3a to show the implementations as well as the programs:
Figure 3cRather than including these entities whenever we show programs representing mappings, however, we can just assume that whenever we show mappings notate and interpret between mappings and programs that there are implementations available that satisfy the requirement in Figure 3b. It is these implementations that actually produce the mappings that programs represent.
Turning now to condition (2), note that this condition is about what people can do: if people can't do the required work, a programming language can't be a good representation. We'll use the term notation for representations that have to be operated on in this way by people.