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

IntInf (infinite precision integers) from the SML standard library (a library of modules that
comes with the SML distributions). The IntInf module provides a type IntINF.int and a set of
infinite precision integer functions.
A better, larger Fibonacci Function
Idea: Use a type with more Integers (Fortunately, there is IntInf)
use "/usr/share/smlnj/src/smlnj-lib/Util/int-inf.sml";
val zero = IntInf.fromInt 0;
val one = IntInf.fromInt 1;
fun bigfob (0) = (zero,one)
| bigfob (1) = (one,one)
| bigfob (n:int) = let val (a, b) = bigfob(n-1) in (b,IntInf.+(a,b)) end;
fun bigfib (n) = let val (a, _) = bigfob(n) in IntInf.toString(a) end;
c○: Michael Kohlhase 108
We have seen that functions are just objects as any others in SML, only that they have functional
type. If we add the ability to have more than one declaration at at time, we can combine function
declarations for mutually recursive function definitions. In a mutually recursive definition we
define n functions at the same time; as an effect we can use all of these functions in recursive calls.
In our example below, we will define the predicates even and odd in a mutual recursion.
Mutual Recursion
generally, we can make more than one declaration at one time, e.g.
- val pi = 3.14 and e = 2.71;
val pi = 3.14
val e = 2.71
this is useful mainly for function declarations, consider for instance:
fun even (zero) = true
| even (suc(n)) = odd (n)
and odd (zero) = false
| odd(suc(n)) = even (n)
trace: even(4), odd(3), even(2), odd(1), even(0), true.
c○: Michael Kohlhase 109
This mutually recursive definition is somewhat like the children’s riddle, where we define the “left
hand” as that hand where the thumb is on the right side and the “right hand” as that where the
thumb is on the right hand. This is also a perfectly good mutual recursion, only — in contrast to
the even/odd example above — the base cases are missing.
2.3.6 Even more SML: Exceptions and State in SML
Programming with Effects
Until now, our procedures have been characterized entirely by their values on their arguments
(as a mathematical function behaves)
This is not enough, therefore SML also considers effects, e.g. for
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