(*
** Course: Concepts of Programming Languages (BU CAS CS 320)
** Semester: Summer I, 2009
** Instructor: Hongwei Xi (hwxi AT cs DOT bu DOT edu)
*)

//
// Assignment One (Due: Tuesday, May 26, 2009)
//
// Total points: 100
//

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(*

//
// Exercise 1 (10 points)
//

Exercise 2.7: Old English money had 12 pence in a shilling and 20 shillings
in a pound. Write functions to add and substract two amounts, working with
triples.

*)

typedef money = '{
  pound= int, shilling= int, pence= int
}

fun assgn1ex1_add (m1: money, m2: money): money

//
// please note that the following function needs to handle the case
// correctly where the amount of [m1] is less than the amount of [m2]:
//

fun assgn1ex1_sub (m1: money, m2: money): money


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(*

//
// Exercise 2 (20 points)
//

Exercise 2.19: Declare a function in ATS for computing the Greatest Common
Divisor based on the following equations, where m and n range ove positive
integers:

GCD(m, n) = GCD (n, m)
GCD(2m, 2n) = 2 * GCD(m, n)
GCD(2m, 2n + 1) = GCD (m, 2n + 1)
GCD(2m + 1, 2n + 1) = GCD (n-m, 2m+1)
GCD(m, m) = m 

*)

fun assgn1ex2_gcd (m: int, n: int): int

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(*

//
// Exercise 3 (15 points)
//

Exercise 2.23

P(1) = 1
P(n) = 1 + P(1) + ... + P(n-1) for n > 1

Please implement a function in ATS that computes P(n) for given integers n
that are greater than or equal to 1.

*)

fun assgn1ex3_P (n: int): int

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(*

//
// Exercise 4 (25 points)
//

For each natural number $n$, let $vFib(n)$ be the vector
$(Fib(n), Fib(n+1))^T$, that is, the transpose of the vector $(Fib(n),
Fib(n+1))$. Clearly, we have the following equation for each natural number
$n$.
\[vFib(n+1)=A\cdot vFib(n) \mbox{, where } A=\left(\begin{array}{cc}
0 & 1 \\
1 & 1 \\
\end{array}\right)\]
Hence, we have the equation $vFib(n)=A^n\cdot vFib(0)$ for each
natural number $n$.  Please use this equation to implement a
$\Theta(\log n)$-time and $\Theta(1)$-space procedure in ATS that
computes the Fibonacci numbers $Fib(n)$.

\def\buid{{\it buid}}
\item Please find the last 8 digits of $Fib(\buid)$, where $\buid$ is the
8-digit number in your BU id.

Please see the file [assgn1ex4.dats].

*)

fun fastfib (n: int): int

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(*

//
// Exercise 5 (10 points)
//

Exercise 3.2: Write a function to return the last element of a nonempty
list. You may assume that the input of this function is a non-empty list.

*)

fun{a:t@ype} list_last (xs: list0 a): a

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(*

//
// Exercise 6 (20 points)
//

Please implement a function that returns the product of two given lists.
For instance, given lists [a, b] and [1, 2, 3], the product should be the
following 6-element list:

[(a,1), (a,2), (a, 3), (b, 1), (b, 2), (b, 3)]

*)

fun{a,b:t@ype} listprod (xs: list0 a, ys: list0 b): list0 @(a, b)

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(* end of [assignment1.sats] *)