ECM2418 Computer Languages and Representations Continuous Assessment
Functional and Logic Programming
Dr David Wakeling
Handed out Handed in
Thursday 26th October 2023 (T1:05) Thursday 14th December 2023 (T1:12)
This Continuous Assessment is worth 40% of the module mark.
Question 1: Light Show
Every week, The Sunday Times newspaper publishes a Teaser. Teaser 3172, of Sunday 9th July 2023, was as follows.
My bedside clock displays the time and date using eight digits; for example, at 9.43am on 28th June, the display would be
Each digit in the electronic display lights up some (or all) of seven light seg- ments, the above display lighting up a total of 45 segments.
On one occasion recently, the displayed digits were all different and the total number of lit segments was prime. The same was true exactly one day later. Then, just one minute after the second occasion, the number of lit segments was the average of the numbers of lit segments on those two previous occasions.
What was that third display?
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Question 1.1
Show a Haskell function generator1 that returns a list of tuples (HR,MN,DY,MT) that may be solutions to the Teaser. That is, for which, HR, MN, DY and MT are valid hours, minutes, days (assume a non-leap year) and months.
This function will be assessed by the number of tests that it passes, as counted by the function x_generator1 below. The expected answer is 10.
x_generator1 :: Int x_generator1 =
length [t | t <- ts, t ‘elem‘ g] where
g = ts =
generator1
[ ( 2,15,14,11)
, ( 4,31,27, 9)
, ( 6,47,10, 8)
, ( 9, 3,23, 6) , (11,19, 6, 5) , (13,35,19, 3) , (15,51, 2, 2) , (18, 6,16,12)
, (20 ,22 ,29 ,10) , (22,38,11, 9) ]
Question 1.2
Show a Haskell function tester1 that returns true for tuples (HR,MN,DY,MT) that are solutions to the Teaser. That is, for which the tuple is “magic”, a second tuple exactly one day later is also “magic”, and just one minute on from this second tuple the number of lit segments on the display is the average of the number of lit segments of these two tuples. A tuple (HR,MN,DY,MT) is “magic” if the displayed digits of HR, MN, DY and MT are all be different, and the total number of lit segments in the display is prime.
This function will be assessed by the number of tests that it passes, as counted by the func- tion x_tester1 below. Note that these test cases were NOT produced by generator1. The expected answer is 10.
x_tester1 :: Int x_tester1 =
length [t | t <- ts, tester1 t] where
ts =
[ ( 6,59,17,24)
2
, ( 6,59,17,34)
, ( 6,59,27,14) , ( 6,59,27,41) , ( 8,59,12,46) , (16,59, 7,24) , (16,59, 7,42)
, (16,59, 7,43) , (16 ,59 ,27 ,40) , (18,59, 2,46) ]
Question 1.3
On blue18.ex.ac.uk, my program
computes [(16,59,27,4)], from which one can deduce the answer to Teaser 3172 is
in 0.004 seconds. Tune your program so that on the same machine, it computes this answer within 1.000 seconds.
(5 marks)
Question 2: Digital Trio
Teaser 3158, of Sunday 5th May 2023, was as follows.
“I have a couple of subtraction problems for you”, George told Martha. Look: N1-N2=N3andN3-N4=N5. CanyousolvethemifItellyouthatN1, N3 and N5 are all three-digit whole numbers whose sum is less than 2000, the same three non-zero digits appearing in all three numbers but no digit being repeated within any of those numbers? N2 and N4 are both two-digit whole numbers using two of the three digits mentioned above, and the first digit of N1 is not equal to the first digit of N2.
What is N1?
main :: IO () main =
print (filter tester1 generator1)
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Question 2.1
Show a Haskell function generator2 that returns a list of tuples (N1,N2,N3,N4,N5) that may be solutions to the Teaser. That is, for which, N1, N3 and N5 are three-digit numbers, and N2 and N4 are two-digit numbers. The same three digits appear in N1, N3 and N5, two of these digits appear in N2, and two of them appear in N4. In each number, no digit is zero and none is repeated. The first digit of N1 is not equal to the first digit on N2.
This function will be assessed by the number of tests that it passes, as counted by the function x_generator2 below. The expected answer is 10.
x_generator2 :: Int x_generator2 =
length [t | t <- ts, t ‘elem‘ g] where
g = generator2
ts =
[ ("123","21","123","12","123")
, ("162","26","261","12","621") , ("219","19","912","21","291") , ("329","92","932","32","239") , ("439","94","394","43","394") , ("549","95","945","95","945")
, ("568","68","586","56","586")
, ("769","67","679","97","796")
, ("879","79","897","98","789")
, ("987","79","789","79","789") ]
(10 marks)
Question 2.2
Show a Haskell function tester2 that returns true for tuples (N1 , N2 , N3 , N4 , N5 ) that are solutions to the Teaser. That is, for which N1 − N2 = N3, N3 − N4 = N5 and N1 + N3 + N5 < 2000.
This function will be assessed by the number of tests that it passes, as counted by the function x_tester2 below. The expected answer is 10.
x_tester2 :: Int x_tester2 =
length [t | t <- ts, tester2 t] where
ts =
[ ("138","01","137","50","87")
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, ("143","01","142","52","90")
, ("171","02","169","79","90") , ("152","03","149","54","95") , ("159","04","155","61","94") , ("161","05","156","63","93") , ("182","06","176","80","96")
, ("151","07","144","57","87") , ("165","08","157","64","93") , ("174","09","165","71","94") ]
(10 marks)
Question 2.3
On blue18.ex.ac.uk, my program
computes [("594","45","549","54","495")] in 0.003 seconds. Tune your program so that on the same machine, it computes this answer within 1.000 seconds.
(5 marks)
Question 3: Easier to Ask the Audience
Teaser 3145, of Sunday 1st January 2023, was as follows.
“I have forgotten the phone number!” complained Martha, about to phone a friend. “So have I!” replied George, “but I have some vague memories of it. It is a perfect square with all the digits different, and the last digit is equal to the number of digits to be dialled. The last-but-one digit is odd and one of the digits is zero. Also the second and third and last-but-one digits are all exact multiples of the first digit. Maybe you can work it out.”
Martha proceeded to dial the number correctly.
What number did she dial?
main :: IO () main =
print (filter tester2 generator2)
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Question 3.1
Show a Prolog predicate generator3 that yields successive numbers N between 1,000 to 1,000,000 (inclusive) that may be solutions to the Teaser. That is, integers N that are perfect squares.
This predicate will be assessed by the number of tests that it passes, as counted by the predicate x_generator3 below. The expected answer is 10.
x_generator3( N ) :- x_generator3_loop(
[ 1024 , 9409 , 23716 , 51529 , 123904 , 185761 , 868624 , 962361
, 982081, 1000000 ], 0, N ).
x_generator3_loop( [], C, C ). x_generator3_loop( [T|TS], C, N ) :-
generator3( T ),
C1 is C + 1,
x_generator3_loop( TS, C1, N ). x_generator3_loop( [_|TS], C, N ) :-
x_generator3_loop( TS, C, N ).
(10 marks)
Question 3.2
Show a Prolog predicate tester3 that is true for phone numbers N that are solutions to the Teaser. That is, for integers N where all of the digits are different, the last digit is equal to the number of digits, the last-but-one digit is odd and one of the digits is zero. In addition, the second and third and last-but-one digits are all exact multiples of the first digit.
This predicate will be assessed by the number of tests that it passes, as counted by a predicate x_tester3 below. The expected answer is 10.
x_tester3( N ) :- x_tester3_loop(
[ 123056 , 128036 , 139076 , 142076 , 148056 , 159076 , 173096 , 189036
, 193056, 198076 ], 0, N ).
x_tester3_loop( [], C, C ). x_tester3_loop( [T|TS], C, N ) :-
tester3( T ),
C1 is C + 1,
x_tester3_loop( TS, C1, N ).
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x_tester3_loop( [_|TS], C, N ) :-
x_tester3_loop( TS, C, N ).
(10 marks)
Question 3.3
On Swish Prolog, my program
computes 173056 in 0.40 seconds. Tune your program so that on the same system, it computes this answer within 2.00 seconds.
(5 marks)
Question 4: Cube Route
Teaser 3149, of Sunday 29th January 2023, was as follows.
I have a set of ten cards, each of which has a different digit written on it. All the cards have been used to make a set of prime numbers. After discarding the smallest prime, and without changing the order of any cards, I have placed the remaining primes in order of decreasing size to give a large number. It is possible, without changing the order of any cards, to break this number into a set composed entirely of cubes. Neither set contains a number with more than four digits.
List, in order of decreasing size, my set of prime numbers.
Question 4.1
Show a Prolog predicate generator4 that yields arrangements of the digits 0 to 9 divided into runs of one, two, three or four digits that form prime numbers. Importantly (and somewhat surprisingly) leading zero digits do not count, so “251” is considered to be prime, but “0251” is not.
This predicate will be assessed by the number of tests that it passes, as counted by a predicate x_generator4 below. The expected answer is 10.
main :-
generator3( N ), tester3( N ), write( N ).
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x_generator4( N ) :-
x_generator4_loop(
[ [[9 ,6 ,7] ,[4 ,0 ,1] ,[2 ,8 ,3] ,[5]]
, [[9 ,8 ,3] ,[6 ,0 ,1] ,[5] ,[4 ,7] ,[2]]
, [[9 ,8 ,3] ,[6 ,7] ,[4 ,2 ,0 ,1] ,[5]]
, [[9 ,8 ,5 ,1] ,[2] ,[4 ,3] ,[6 ,0 ,7]]
, [[9 ,8 ,5 ,1] ,[2] ,[3] ,[6 ,0 ,4 ,7]]
, [[9 ,8 ,5 ,1] ,[2] ,[7] ,[4 ,6 ,0 ,3]]
, [[8 ,9] ,[7] ,[6 ,0 ,1] ,[2 ,5 ,4 ,3]]
, [[8 ,9] ,[7] ,[5 ,6 ,3] ,[4 ,0 ,2 ,1]]
, [[8 ,9] ,[5] ,[4 ,7] ,[6 ,0 ,1] ,[3] ,[2]]
, [[3],[5],[6,0,7],[2],[4,1],[8,9]] ], 0, N ).
x_generator4_loop( [], C, C ). x_generator4_loop( [T|TS], C, N ) :-
generator4( T ),
C1 is C + 1,
x_generator4_loop( TS, C1, N ). x_generator4_loop( [_|TS], C, N ) :-
x_generator4_loop( TS, C, N ).
(10 marks)
Question 4.2
Show a Prolog predicate tester4 that is true for lists of lists of digits that form prime numbers and may be solutions to the Teaser. That is, for collections of prime numbers that after discarding the smallest prime, may be arranged in order of decreasing size to give a large number that may be divided into runs of one, two three or four digits that form cubes.
This predicate will be assessed by the number of tests that it passes, as counted by the predicate x_tester4 below. The expected answer is 10.
x_tester4( N ) :- x_tester4_loop(
[ [[8 ,2 ,7] ,[6 ,1] ,[5 ,3] ,[4 ,0 ,9]] , [[8 ,2 ,7] ,[6 ,1] ,[4 ,0 ,9] ,[5 ,3]]
, [[8 ,2 ,7] ,[5 ,3] ,[6 ,1] ,[4 ,0 ,9]]
, [[8 ,2 ,7] ,[4 ,0 ,9] ,[6 ,1] ,[5 ,3]]
, [[6 ,1] ,[8 ,2 ,7] ,[4 ,0 ,9] ,[5 ,3]]
, [[6 ,1] ,[4 ,0 ,9] ,[5 ,3] ,[8 ,2 ,7]]
, [[5 ,3] ,[6 ,1] ,[4 ,0 ,9] ,[8 ,2 ,7]]
, [[5 ,3] ,[4 ,0 ,9] ,[6 ,1] ,[8 ,2 ,7]]
, [[4 ,0 ,9] ,[5 ,3] ,[8 ,2 ,7] ,[6 ,1]]
, [[4,0,9],[8,2,7],[6,1],[5,3]] ], 0, N ).
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x_tester4_loop( [], C, C ). x_tester4_loop( [T|TS], C, N ) :-
tester4( T ),
C1 is C + 1,
x_tester4_loop( TS, C1, N ).
x_tester4_loop( [_|TS], C, N ) :- x_tester4_loop( TS, C, N ).
(10 marks)
Question 4.3
On Swish Prolog, my program
computes a first result in in 33.0 seconds, from which one can deduce the answer to the Teaser is 827, 409, 61, 53. Tune your program so that on the same system, it computes this answer within 120.0 seconds.
(5 marks)
Submission
You should submit a single “.zip” file to the ELE system. Other compression formats, such as “.rar”, “.7z”, “.gz” and “.bz2” are unacceptable, and will receive a mark of zero. The “.zip” file should contain four completed text files “Light.hs” (containing the answer to Question 1), “Trio.hs” (containing the answer to Question 2), “Audience.pl” (con- taining the answer to Question 3) and “Cube.pl” (containing the answer to Question 4).
If there is any question as to whether your functional programs compute the correct result, these questions will be answered on the implementation at
https://www.tutorialspoint.com/compile_haskell_online.php
If there is any question as to whether your logic programs compute the correct result, these questions will be answered on the implementation at
https://swish.swi-prolog
main :-
generator4( XS ), tester4( XS ), write( XS ).
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All students are reminded of the University regulations on academic honesty and plagiarism.
In particular, functions an predicates clearly intended ONLY to pass the given tests will be treated as malpractice (“an attempt to deceive the examiners”).
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