Computer Science 538
Lecture Notes 4-12-00

Type Inference in SML (cont)

How does it work with patterns?
	
	Fun Leng [] = 0
		|  Leng (a::b) = 1 + Leng b;

	Need to determine the type of Leng, a, and b.
	   First each item is given a separate type identifier 
	   T(Leng) =>  ‘x -> ‘y  refined to ‘z List -> int because Leng[] and returns zero
	   T(a) => ‘v => ‘z
	   T(b) => ‘w	=> ‘z list 	because the cons operator both are the same type	

Finding errors with type inference.
	
	f() + f(1)
	Type of F: ‘a -> ‘b
	‘a = unit from f() and ‘a = int from f(1) so this produces an error

Where type inference fails

	Fun F x = x x;		This can not be type infered.
		x = ‘a = ‘b ->c  which changes to ‘a = ‘a -> ‘c which can not be solved

	Fun f(g) = (g(3), g(true))
		Type of g = ? 
		You have a function that takes some types but not all types.  
		This example will take int or bool
		You can get a type error if you give something other than a int or bool
	Solution:
	
	Fun f(g) = (g(I (3)), g(b(bool)))
		T(g) = T -> ‘b



Prolog - Programming in Logic, 1972 by Kowalski and Comeraver

	Conventinal programming model "Algorithms + Data structures = Programs".
	Prolog: "Algorithms = Logic + Control" , Logic is esscential and control is
	  an efficiency issue.

Example prolog program.  Is a list in order?
	
	iNORDER ([]).		is the empty list in order: yes
	iNORDER ([_]).		is a single element in order: yes
	iNORDER ([a, b | c] :- (a =< b, iNORDER([b|c]))).

	:- equals if,  | = the reminder of the list

Terminology in prolog
	
	Elementary data objects = Integers and Atoms
		Integer is a numeric literal: 1 2 3 ~17
		Atoms are identifiers that begin with lowercase

	Prolog data objects = Terms
		to program, we define relations among Terms.  Ex: iNORDER([])
	
	A Predicate names a relation.  Ex: iNORDER

	To define a relation using a predicate we write clauses made up of either
	a fact or rule.