Comp Sci 538 
Lecture Notes for April 21

Merge Sort in Prolog
	Merge([], L, L).
	Merge(L, [], L).
	Merge([A | T1], [B | T2], [A | L2]) :- A =< B, Merge(T1, [B | T2] , L2).
	Merge([A | T1], [B | T2], [B | L2]) :- A > B, Merge([A | T1, T2, L2).

	Split([], [], []).
	Split([A], [A], []).
	Split([A, B | T], [A | P1], [B | P2]) :- Split(T, P1, P2).

	MergeSort([], []).
	MergeSort([A], [A]).
	MergeSort(L1, L2) :- Split(L1, P1, P2), MergeSort(P1, S1), MergeSort(P2, S2), Merge(S1, S2, L2).

	Keeps dividing the list until there are lists of size 1 or 0 (depends if you start with an even or odd number of elements).
	Takes log2 n to divide
	Total time is n log2 n

Arithmetic in Prolog
	2 = 1 +1 returns no because 2 (an integer) is not the same as 1+1 (a function)

	?- 1 = 2.	comparison
	no
	?- x = 2.	assignment
	x = 2
	?- x = y.	assigning unknowns to the same unknown
	x = _1
	y = _1	_1 is an internal variable
	?- x = y, y = joe.
	x = _1
	y = _1	does left side first
	y = _1 = joe
	x = _1 = joe

	x = y is same as y = x, Prolog is symmetric

	from above 1 + 1 is a standard form of +(1, 1) which is more evident that 2 = +(1, 1). returns no

	is	form of = used only in arithmetic expression, both assignment and comparison
	2 is 1 + 1.
	yes

	x is 1 * 2.
	x = 2

	y is x + 1	no good if x is unbound

	(0 is (I**N) + (J**N) - (K**N)), N > 2.	unfortunately won't work, very difficult to implement

	len(L, N)	L will be a list N will be its length
	len([], 0).
	len([H | T], N) :- len(T, M), N is M+1.

	Prolog can count the number of terms and functions with different number of terms are unique.

	adj([A, B | T]) :- A = B.
	adj([_, B | T]) :- adk([B | T]).	adj is mistyped adk
	will always return no to adk([B | T]) since no rules exist for it, run-time error

	member(A, [A | _]).
	member(A, [_ | T]) :- member(A, [T]).	T is already a list, now it is a nested list

	member(2, [1, 2]).
	infinite loop
		A = 2 T = [2]
		member(2, [[2]]).	[[2]] is equivalent to [[2] | []]
			A = 2 T = []
			member(2, [[]]).

	odd(1).
	odd(n) :- odd(M), N is M+2.

	odd(x), x = 2.
	infinite search pattern
	generates M = 1 does 1 = 2 no
		M = 3 does 3 = 2 no
		M = 5 does 5 = 2 no ...