Procedural and Functional Abstraction

Overview

Recall questions on what procedural abstraction is, worked examples of values being abstracted away, and why the result is a procedure.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is procedural abstraction?

   Model answer

   Procedural abstraction strips out the particular values used in a computation. What is left is the computational pattern, or computational method — a procedure. Procedural abstraction therefore represents a computational method.

2. 2

   Work through an example that shows the actual values being abstracted away.

   Model answer

   Start with three particular computations: 3 × 3, 7 × 7 and 12 × 12. Each uses its own actual value. Abstract that value away by replacing it with a name, and the shared pattern appears: PROCEDURE square(x) RETURN x * x ENDPROCEDURE The procedure captures the method — multiply the number by itself — without committing to any particular number. That step, from the specific computations to the general pattern, is procedural abstraction.

3. 3

   Why is the result of procedural abstraction described as a procedure rather than a function?

   Model answer

   Because abstracting away the values leaves the method of computation still fully specified. square(x) says exactly how the answer is produced: multiply x by itself. A procedure is a stated way of computing something, so that is what you have. Getting a function, which says only what the relationship between input and output is and nothing about how it is worked out, requires a further abstraction on top of this one.

4. 4

   What is functional abstraction?

   Model answer

   Functional abstraction is a further abstraction applied to a procedure, which disregards the particular computation method. The result is a function: something defined only by the relationship between what goes in and what comes out, with how the result is produced hidden.

5. 5

   Explain the difference between procedural abstraction and functional abstraction, and why the second is needed.

   Model answer

   Treat it as two abstractions applied one after the other. The first strips the particular values out of a computation and leaves behind the method of computing — and a method is a procedure, not a function, because how the answer is worked out is still written into it. Only when that method is thrown away as well, by a second and further abstraction, is what remains a function: a bare mapping from inputs to outputs. That second step, functional abstraction, is needed because two procedures that sort a list by completely different methods describe one and the same function; without it you could neither say they were equivalent nor reason about what a program computes independently of how it computes it.

6. 6

   Use square root to illustrate the step from a procedure to a function.

   Model answer

   A procedure for square root might state a method: start with a guess g for √x, repeatedly replace g by (g + x / g) / 2 until it stops changing, then return g. That is a computational method with the values abstracted away. Apply functional abstraction and the method is disregarded, leaving only sqrt: a mapping that takes a non-negative number and yields the number whose square it is. Written as a function, sqrt(9) = 3 tells you nothing about how the 3 was arrived at, and the underlying method could be replaced by a lookup table or by hardware without changing the function at all.

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