Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 213 — #221
7.4. Arithmetic Expressions 213
The constant function equal to 1 will satisfy (7.5), but it’s not known if another
function does as well. The problem is that the third case specifies f 4 .n/ in terms
of f 4 at arguments larger than n, and so cannot be justified by induction on N. It’s
known that any f 4 satisfying (7.5) equals 1 for all n up to over 10 18 .
A final example is the Ackermann function, which is an extremely fast-growing
function of two nonnegative arguments. Its inverse is correspondingly slow-growing—
it grows slower than log n, log log n, log log log n, . . . , but it does grow unboundly.
This inverse actually comes up analyzing a useful, highly efficient procedure known
as the Union-Find algorithm. This algorithm was conjectured to run in a number
of steps that grew linearly in the size of its input, but turned out to be “linear”
but with a slow growing coefficient nearly equal to the inverse Ackermann function.
This means that pragmatically, Union-Find is linear, since the theoretically
growing coefficient is less than 5 for any input that could conceivably come up.
The Ackermann function can be defined recursively as the function A given by
the following rules:
A.m; n/ D 2n if m D 0 or n 1; (7.6)
A.m; n/ D A.m 1; A.m; n 1// otherwise: (7.7)
Now these rules are unusual because the definition of A.m; n/ involves an evaluation
of A at arguments that may be a lot bigger than m and n. The definitions
of f 2 above showed how definitions of function values at small argument values in
terms of larger one can easily lead to nonterminating evaluations. The definition of
the Ackermann function is actually ok, but proving this takes some ingenuity (see
Problem 7.25).
7.4 Arithmetic Expressions
Expression evaluation is a key feature of programming languages, and recognition
of expressions as a recursive data type is a key to understanding how they can be
processed.
To illustrate this approach we’ll work with a toy example: arithmetic expressions
like 3x 2 C 2x C 1 involving only one variable, “x.” We’ll refer to the data type of
such expressions as Aexp. Here is its definition:
Definition 7.4.1.
Base cases: