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

val it = [2,3,4,5] : int list
Question: What is happening here, we define a function by itself? (circular?)
c○: Michael Kohlhase 64
A constructor is an operator that “constructs” members of an SML data type.
The type of lists has two constructors: nil that “constructs” a representation of the empty list,
and the “list constructor” :: (we pronounce this as “cons”), which constructs a new list h::l
from a list l by pre-pending an element h (which becomes the new head of the list).
Note that the type of lists already displays the circular behavior we also observe in the function
definition above: A list is either empty or the cons of a list. We say that the type of lists is
inductive or inductively defined.
In fact, the phenomena of recursion and inductive types are inextricably linked, we will explore
this in more detail below.
Defining Functions by Recursion
SML allows to call a function already in the function definition.
fun upto (m,n) = if m>n then nil else m::upto(m+1,n);
Evaluation in SML is “call-by-value” i.e. to whenever we encounter a function applied to
arguments, we compute the value of the arguments first.
So we have the following evaluation sequence:
upto(2,4) 2::upto(3,4) 2::(3::upto(4,4)) 2::(3::(4::nil)) = [2,3,4]
Definition 92 We call an SML function recursive, iff the function is called in the function
definition.
Note that recursive functions need not terminate, consider the function
fun diverges (n) = n + diverges(n+1);
which has the evaluation sequence
diverges(1) 1 + diverges(2) 1 + (2 + diverges(3)) . . .
Defining Functions by cases
c○: Michael Kohlhase 65
Idea: Use the fact that lists are either nil or of the form X::Xs, where X is an element and
Xs is a list of elements.
The body of an SML function can be made of several cases separated by the operator |.
Example 93 Flattening lists of lists (using the infix append operator @)
fun flat [] = [] (* base case *)
| flat (l::ls) = l @ flat ls; (* step case *)
val flat = fn : ’a list list -> ’a list
- flat [["When","shall"],["we","three"],["meet","again"]]
["When","shall","we","three","meet","again"]
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