General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Termination
Definition 172 Let R ⊆ A 2 be a binary relation, an infinite chain in R is a sequence
a1, a2, . . . in A, such that ∀n ∈ N1.〈an, an+1〉 ∈ R.
We say that R terminates (on a ∈ A), iff there is no infinite chain in R (that begins with a).
We say that P diverges (on a ∈ A), iff it does not terminate on a.
Theorem 173 Let P = 〈f::A → R ; R〉 be an abstract procedure and a ∈ TA(A, Σ; V),
then P terminates on a, iff the recursion relation of P does.
Definition 174 Let P = 〈f::A → R ; R〉 be an abstract procedure, then we call the function
{〈a, b〉 | a ∈ TA(A, Σ; V) and P terminates for a with b} in A ⇀ B the result function of P.
Theorem 175 Let P = 〈f::A → B ; D〉 be a terminating abstract procedure, then its result
function satisfies the equations in D.
c○: Michael Kohlhase 104
We should read Theorem 175 as the final clue that abstract procedures really do encode functions
(under reasonable conditions like termination). This legitimizes the whole theory we have
developed in this section.
Abstract vs. Concrete Procedures vs. Functions
An abstract procedure P can be realized as concrete procedure P ′ in a programming language
Correctness assumptions (this is the best we can hope for)
If the P ′ terminates on a, then the P terminates and yields the same result on a.
If the P diverges, then the P ′ diverges or is aborted (e.g. memory exhaustion or buffer
overflow)
Procedures are not mathematical functions (differing identity conditions)
compare σ : N1 → N1 with σ(o) o, σ(s(n)) n + σ(n)
with σ ′ : N1 → N1 with σ ′ (o) 0, σ ′ (s(n)) ns(n)/2
these have the same result function, but σ is recursive while σ ′ is not!
Two functions are equal, iff they are equal as sets, iff they give the same results on all
arguments
c○: Michael Kohlhase 105
2.3.5 More SML: Recursion in the Real World
We will now look at some concrete SML functions in more detail. The problem we will consider is
that of computing the n th Fibonacci number. In the famous Fibonacci sequence, the n th element
is obtained by adding the two immediately preceding ones.
This makes the function extremely simple and straightforward to write down in SML. If we look
at the recursion relation of this procedure, then we see that it can be visualized a tree, as each
natural number has two successors (as the the function fib has two recursive calls in the step
case).
Consider the Fibonacci numbers
Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
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