# CSCI3180 Assignment 4 — Declarative Programming solved

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## Description

1 Introduction
Declarative programming is a programming paradigm that emphasises the expression of “what” the
problem is over “how” to solve the problem. You will gain experience of declarative programming
with Prolog (Logic Programming) and ML (Functional Programming) in this assignment.
You are supposed to test your work on the machine solar1.cse.cuhk.edu.hk using
• SICStus 3.12.7
• Standard ML of New Jersey, Version 110.0.7
They can be invoked using the commands sics and sml respectively.
2 Logic Programming
Implement the predicates or queries of the following problems in a Prolog program file named
“asg4.pl”, and recall the successor notation s() to represent natural numbers in this task. You
should clearly indicate using comments the corresponding question number of each problem.
1. Recall the list operations taught in lectures and tutorials.
(a) Define element last(X, L) which is true if the last element in list L is X. Example:
?- element last(e, [a,b,c,d,e]).
yes
(b) Define element n(X, L, N) which is true if the N-th element in list L is X. Example:
?- element n(c, [a,b,c,d,e], s(s(s(0)))).
yes
(c) Define remove n(X, L1, N, L2) which is true if the resulting list L2 is obtained from
L1 by removing the N-th element, and X is the removed element. Example:
?- remove n(X, [a,b,c,d,e], s(s(s(0))), L2).
X = c,
L2 = [a, b, d, e]
(d) Based on (c), give a query to find which list will become [c,b,d,e] after removing its
second element “a”.
(e) Define insert n(X, L1, N, L2) which is true if the resulting list L2 is obtained by
inserting X to the position before the N-th element of list L1. Example:
?- insert n(h, [a,b,c], s(s(0)), L2).
L2 = [a,h,b,c]
(f) Define repeat three(L1, L2) which is true if the resulting list L2 has each element in
list L1 repeated three times. Example:
?- repeat three([a,b,c,d,e],X).
X = [a,a,a,b,b,b,c,c,c,d,d,d,e,e,e]
(g) Based on (f), give a query to find which list will become [i,i,i,m,m,m,n,n,n] after
repeating each element of it for three times.
2. A multi-way tree is composed of a root and a sequence of sub-trees (children), which are
multi-way tree themselves. The list of sub-trees of a certain node is ordered from left to
right and also called a forest. The root of a multi-way tree is a node containing a value.
Diagrammatically, a multi-way tree consists of a set of nodes and lines connecting parents
1
a
b c d
e f g
Figure 1: An example of multi-way tree
and children. The nodes are depicted by circles with values written inside. A multi-way tree
is never empty.
In Prolog, we can represent a multi-way tree by the term “mt(X, F)”, where X denotes the
root and F denotes the forest of sub-trees. Please finish the following questions using the
successor notation s() to represent natural numbers in this task. You can use the sum/3
predicate from the lecture.
(a) Represent the multi-way tree in Figure 1 as a Prolog term, with order of the sub-trees
from left to right. (Hint: represent forest as a list of multi-way tree(s)).
(b) Define the predicate is tree(Term) which is true if Term represents a multi-way tree.
(c) Define the predicate num node(Tree, N) which is true if N is the number of nodes of
the given multi-way tree Tree.
(d) Define sum length(Tree, L) which is true if L is the sum of lengths of all internal
paths in Tree. The length of an internal path from the root node r to an internal node
n is the distance from r to n. For example, the sum of lengths of all internal paths in
the multi-way tree in Figure 1 is: 1 (b) + 1 (c) + 1 (d) + 2 (e) + 2 (f) + 2 (g) = 9
(i.e., s(s(s(s(s(s(s(s(s(0))))))))).
3 Functional Programming
Implement the required functions of the following problems in an ML program file named “asg4.ml”.
You should clearly indicate using comments the corresponding question number of each problem.
This task involves a card game simplified from Texas Hold’em. In each round, two players will
get five cards respectively. The player who has the better hand will win the game. You have to
implement several ML helper functions to determine the better hand.
Please consider the following eight different hands. (We refer to the rules defined on https:
//www.pagat.com/poker/rules/ranking.html.)
• Four of a kind
Four cards of the same rank – such as four queens. The fifth card, known as the kicker, can
be anything. Between two fours of a kind, the one with the higher set of four cards is higher,
so 3-3-3-3-A is beaten by 4-4-4-4-2. If two players have four of a kind of the same rank, the
rank of the kicker decides.
• Full House
2
This combination consists of three cards of one rank and two cards of another rank – for
example three sevens and two tens. When comparing full houses, the rank of the three cards
determines which is higher. For example 9-9-9-4-4 beats 8-8-8-A-A. If the threes of a kind
are equal, the rank of the pairs decides.
• Flush
Five cards of the same suit. When comparing two flushes, the highest card determines which
is higher. If the highest cards are equal then the second highest card is compared; if those
are equal too, then the third highest card, and so on. For example ♠K-♠J-♠9-♠3-♠2 beats
♦K-♦J-♦7-♦6-♦5 because the nine beats the seven. If all five cards are equal, the flushes
are equal.
• Straight
Five cards of mixed suits in sequence. For example ♠Q-♦J-♥10-♠9-♣8. When comparing
two sequences, the one with the higher ranking top card is better. Ace can count high or low
in a straight, but not both at once. So A-K-Q-J-10 and 5-4-3-2-A are valid straights, but
2-A-K-Q-J is not. 5-4-3-2-A, known as a wheel, is the lowest kind of straight, the top card
being the five.
• Three of a Kind
Three cards of the same rank plus two unequal cards. When comparing two threes of a kind
the rank of the three equal cards determines which is higher. If the sets of three are of equal
rank, then the higher of the two remaining cards in each hand are compared, and if those
are equal, the lower odd card is compared. So for example 5-5-5-3-2 beats K-4-4-4-5, which
beats Q-9-4-4-4, which beats Q-8-4-4-4.
• Two Pairs
A pair consists of two cards of equal rank. In a hand with two pairs, the two pairs are of
different ranks (otherwise you would have four of a kind), and there is an odd card to make
the hand up to five cards. When comparing hands with two pairs, the hand with the highest
pair wins, irrespective of the rank of the other cards – so J-J-2-2-A beats 10-10-9-9-8 because
the jacks beat the tens. If the higher pairs are equal, the lower pairs are compared, so that
for example 8-8-6-6-3 beats 8-8-5-5-4. Finally, if both pairs are the same, the odd cards are
compared, so Q-Q-5-5-4 beats Q-Q-5-5-3.
• Pair
A hand with two cards of equal rank and three cards which are different from these and from
one another. When comparing two such hands, the hand with the higher pair is better – so
for example 6-6-4-3-2 beats 5-5-4-3-2. If the pairs are equal, compare the highest ranking
odd cards from each hand; if these are equal compare the second highest odd card, and if
these are equal too compare the lowest odd cards. So J-J-9-3-2 beats J-J-8-7-3 because the
9 beats the 8.
• Nothing
Five cards which do not form any of the combinations listed above. The cards must all be
of different ranks, not consecutive, and contain at least two different suits. When comparing
two such hands, the one with the better highest card wins. If the highest cards are equal the
second cards are compared; if they are equal too the third cards are compared, and so on.
So A-J-9-5-3 beats A-10-9-6-4 because the jack beats the ten.
Note that cards are already sorted in descending order according to the rank (with A always
ordered last). The order of cards of the same rank is arbitrary. Each card has two attributes,
i.e., suit and rank. The type definition of a card in ML is shown below. The joker cards are not
included in this game. To further simplify the problem, we convert all ranks to integers. i.e., A =
1, J = 11, Q = 12, and K = 13.
datatype suit = Clubs | Diamonds | Hearts | Spades;
3
When you implement the following functions, please use pattern matching as much as possible.
1. Write an ML function check flush, which takes a list of five cards and returns if the hand
is a flush.
2. Write an ML function compare flush, which takes two flush card lists. The return value is
a string selected from three candidates. i.e., “Hand 1 wins”, “Hand 2 wins” and “This is a
tie”.
3. Write an ML function check straight, which takes a list of five cards and returns if the
hand is a straight.
4. Write an ML function compare straight, which takes two straight card lists. The return
value is a string selected from three candidates. i.e., “Hand 1 wins”, “Hand 2 wins” and
“This is a tie”.
5. Write an ML function count patterns, which takes a list of five cards and returns the hand
type (Nothing, Pair, Two Pairs, Three of a Kind, Full House, Four of a Kind) and a list of
rank-quantity pairs. Note that the list should be ordered according to the order of comparison
of each type of hand.
• The card list [(Clubs, 13), (Clubs, 11), (Spades, 7), (Spades, 3), (Hearts, 2)] should
return (Nothing, [(13, 1), (11, 1), (7, 1), (3, 1), (2, 1)]).
• The card list [(Spades, 11), (Spades, 9), (Hearts, 8), (Diamond, 8), (Diamonds, 3)]
should return (Pair, [(8, 2), (11, 1), (9, 1), (3, 1)]).
• The card list [(Clubs, 13), (Spades, 13), (Hearts, 6), (Spades, 1), (Diamonds, 1)] should
return (Two Pairs, [(13, 2), (1, 2), (6, 1)]).
• The card list [(Clubs, 10), (Clubs, 9), (Hearts, 9), (Spades, 9), (Spades, 3)] should return
(Three of a Kind, [(9, 3), (10, 1), (3, 1)]).
• The card list [(Diamonds, 6), (Clubs, 6), (Spades, 6), (Spades, 4), (Diamonds, 4)] should
return (Full House, [(6, 3), (4, 2)]).
• The card list [(Diamonds, 11), (Spades, 11), (Clubs, 11), (Hearts, 11), (Hearts, 10)]
should return (Four of a Kind, [(11, 4), (10, 1)]).
6. Write an ML function compare count, which takes two card lists and returns a string selected
from three candidates. i.e., “Hand 1 wins”, “Hand 2 wins” and “This is a tie”. You only
need to consider the remaining six cases except flush and straight, and assume that