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# cps721: Assignment 2

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cps721: Assignment 2 (100 points).

YOU SHOULD NOT USE ”;” , ”!” AND ”–” IN YOUR PROLOG RULES.
You cannot use any library predicates not mentioned in class or below.
You MUST work in groups of TWO, or THREE, you cannot work alone. You can discuss this assignment only with your CPS721 group partners or with the CPS721 instructor. By submitting this assignment you acknowledge that you read and understood the course Policy on Collaboration in homework
assignments stated in the CPS721 course management form.
1 (24 points). For each of the following pairs of Prolog lists, state which pairs can be made identical, and which
cannot. Write brief explanations (name your file lists.txt ): it is not acceptable to give an answer without explanations. For those pairs that mention variables, and that can be made identical, give the values of their variables that
make the two lists the same. As a proof for your answer provide transformation from one representation to another
(e.g., from “,”-based notation to “j”-based notation, or vice versa, when possible). Make sure that you apply only
equivalent transformations to a list when you rewrite a list into a different representation. You lose marks if you give
only short answers, but do not explain.
[ [] ] and [F | F]
[F, [a,G|V]] and [V, [F,d,a]]
[F|G] and [[q,p,r,s]]
[F, [] | [c | G]] and [a | [G, c]]
[P,a | [d,P|R]] and [[a | [b,c]]|[F,G | [I]]]
[F, [G], J | W] and [mth110, mth210 | [cps305, cps721]]
[F, G | [k, [G,l]]] and [k | [l | [F,J]]]
[a,b | F] and [a, G, d | [e, [G | [c]]]]
Handing in solutions: An electronic copy of your file lists.txt must be included in your zip archive.
2 (25 points) Write the following Prolog programs. In this part of the assignment you are asked to implement in
Prolog a few programs with recursion over lists. If you wish, you may use (you do not have to) any of the programs
we wrote in class, but if you do, be sure to include them in your program file. (If one of the predicates that you
would like to use is a part of the ECLiPSe Prolog’s standard library of predicates, then rename it and provide rules
for the renamed predicate. See the handout “How to use Eclipse Prolog in labs” for details). Whenever possible,
try to write a recursive program directly. You cannot use programs that we did not discuss in class: you may lose
all marks if you use external programs that are not allowed. Test each of your programs on some examples of your
choice (including queries with variables when possible). Keep all these programs in one file named recursion.pl
1. add(N,InputList,OutputList): OutputList is an initial sequence of N elements from InputList. You can consider
this as follows: we read InputList element by element, count its elements and add elements one by one to
OutputList until we added exactly N elements or exhausted all elements from InputList, whatever happens
first. Examples of queries:
The following all succeed The following fail
2. convert(List1,List2): List2 is a list of elements of List1 converted element-by-element from one symbolic
representation to another. You can assume in your program that List1 is given as an input, that List1 does
not have lists nested inside (i.e., List1 is a “flat” list), and the 2nd argument is either a given list, or a variable.
1
Use in your implementation the predicate di tionary(E lem1; E lem2). Assume that you are given a finite
number of atomic statements with this predicate, e.g., di tionary(one; 1) and di tionary(two; 2) and so on,
but for simplicity, there are no rules defining these predicate. You can write your own atomic statements, e.g.,
if you would like to translate from one language to another word by word. Examples of queries:
The following all succeed: The following fail:
convert([],[]). convert([two,four],[]).
convert([five], ). convert([two,six],[two,six]).
convert([one,two | [three,four]], X). convert([four], 4).
convert([one,two | [three,four]], [1,2,3,4]).
3. grep(E,L,Occurrences): Occurrences is the list of all occurrences of the element E in a given input list L. The
list L may have nested lists inside. Also, the element E can be a list, and in this case the program has to find all
occurrences of E as a sub-list inside L. The query grep([],[a,b,c],X) should fail, i.e., Prolog should
say ”No”. Make sure your program returns only one answer to any query. Hint: you might wish to introduce
a helping predicate grep(E ; L; O urren es; Counter), where the last argument keeps track of the position
of an element in L that the program ic currently processing. Examples of queries that succeed:
grep([a], [a,b,[c,a],[a],d,[a],[[a]],e],X). /* query returns X=[4,6] */
grep(live,[learn,as,if,you,were,to,live,forever],A). /* returns A= */
grep([1,2],[a,b,[1,2,3],c,[1|],d],).
The following to test: ?- grep(1,[1,2,3,4,5,6,3,2,1], X]).
Handing in solutions: (a) An electronic copy of your file recursion.pl with all your Prolog rules must be included
in your zip archive; (b) your session with Prolog, showing the queries you submitted and the answers returned (the
name of the file must be recursion.txt). It is up to you to formulate a range of queries that demonstrates that your
programs are working properly.
3. (31 points) This part will exercise what you have learned about recursion over terms and over lists in Prolog.
More specifically, this part is about doing transformations between lists and terms in Prolog. In this part of the
2nd assignment, you can use the library predicate atomi (X) that is true if and only if X is either a constant, or a
number, or the empty list. In all other cases, it is false.
? Implement the predicate ons(List; T erm) that constructs a term representing a given input List. Let the
constant null represent the empty list [ ℄, and the term next(H ead; T ail) represent the list [H eadjT ail℄. Your
program takes as an input a given list, and constructs a term representation of this list (a kind of symbolic
analog of a linked list). First, for simplicity, assume that List has no nested lists inside, i.e., that List is a
“flat” list. The second argument in queries can be either a term, or a variable. Examples of queries:
The following all succeed: The following fail:
cons([],null). cons([a,b],next(a,next(b,next(null,null)))).
cons([a], next(a,null)). cons([a,b],null).
cons([a,b],next(a,next(b,null))). cons([a],[]).
?- cons([a,b,c],X). /* query returns X=next(a,next(b,next(c,null)))
? Once you have completed the previous program, re-write it to make sure it works also for a more complex
case, when an input list has other nested lists inside. Examples of queries:
The following all succeed: The following fail:
cons([[a]],next(next(a,null),null)). cons([a,[b]],next(a,next(b,null))).
cons([[[a]]], next(next(next(a,null),null),null). cons([[a]],next(a,null)).
?- cons([a,[b,[c]]],X).
/*query returns X=next(a, next(next(b, next(next(c, null), null)), null)) */
2
? Implement the predicate linked2List(T ; L) that takes as an input a term T representing a list and converts it
back to the usual Prolog list L corresponding to T . Of course, for every correctly constructed term T there is
only one corresponding list. Note that if you run the predicates ons(L1; T ) and linked2list(T ; L2) consecutively, then you get the list L2 identical to the initial list L1. In other words, the predicate linked2List(T ; L)
is reciprocal to ons(L; T ), and they do mutually opposite transformations. Examples:
?- cons([a, [b, [c], d], e], X), linked2List(X, List).
X= next(a, next(next(b, next(next(c, null), next(d,null)) ), next(e, null)) )
List = [a, [b, [c], d], e]
?- cons([a, [b, [c], d]], X), linked2List(X, List).
X= next(a, next(next(b, next(next(c, null), next(d, null))), null))
List = [a, [b, [c], d]]
?- cons([a, [b, [c]]], Term), linked2List(Term, L), cons(L, T).
Term = next(a, next(next(b, next(next(c, null), null)), null))
L = [a, [b, [c]]]
T = next(a, next(next(b, next(next(c, null), null)), null))
Handing in solutions. An electronic copy of: (a) your program (terms.pl) that includes all your Prolog rules;
(b) your session with Prolog, showing the queries you submitted and the answers returned (the name of the file
must be terms.txt) . It is up to you to formulate a range of queries that demonstrates that your program is working
properly. Your queries should include not only the examples given above, but a few other tests as well (e.g., test all
bases cases). Request all answers (using either “;” or the button more).
4 (20 points). This part is asking you to write a recursive program over terms representing binary trees. Let the
term tree(X; Lef t; Right) represent a binary tree with the element X in the root, and two branches. The constant
void represents the empty tree. Recall that terms must be either arguments of predicates or arguments of equality.
? Implement the predicate repla e(X; Y ; T 1; T 2) that computes the tree T 2 by replacing all occurrences of X
(if any) in the tree T 1 with Y . You can assume that T 1 is an input binary tree and that the arguments X and
Y are given, but the argument T 2 is either a variable (representing a binary tree that should be computed) or
a given binary tree. Examples of queries that succeed:
?- replace(a,b, tree(a, tree(c,void,void), tree(a,void,void)), T).
T = tree(b, tree(c,void,void), tree(b,void,void))
?- replace(7,9, tree(4, tree(2,tree(1,void,void),tree(3,void,void)),
tree(6,tree(5,void,void),tree(7,void,void))), X).
X = tree(4, tree(2,tree(1,void,void),tree(3,void,void)),
tree(6,tree(5,void,void),tree(9,void,void)))
? Implement the predicate expand(X; T ree1; T ree2) that holds if T ree2 is an ordered tree resulting from
inserting number X into the ordered tree T ree1. If X already occurs in T ree1, then T ree1 and T ree2 are
identical. You can assume that the number X and T ree1 are given as an input and that T ree1 is indeed a
binary sorted tree (there is no need to check this). Recall that in a binary sorted tree tree(Root; Lef t; Right)
the element Root is greater than or equal than all elements in the Lef t branch, and also Root is less than or
equal than all elements in the Right branch. Recursively, same condition applies to all other non-leaf nodes
in the sorted binary tree. If you need more details, read http://en.wikipedia.org/wiki/Binary search tree
3
?- expand(2, tree(4,void,void),T2), expand(1,T2,T1), expand(3,T1,T3),
expand(6,T3,T6), expand(5,T6,T5), expand(7,T5,T7).
T2= tree(4, tree(2,void,void), void)
T1= tree(4, tree(2, tree(1,void,void), void), void)
T3= tree(4, tree(2, tree(1,void,void), tree(3,void,void)), void)
T6= tree(4, tree(2, tree(1,void,void), tree(3,void,void)), tree(6,void,void))
T5= tree(4, tree(2, tree(1,void,void), tree(3,void,void)),
tree(6, tree(5,void,void), void))
T7= tree(4, tree(2, tree(1,void,void), tree(3,void,void)),
tree(6, tree(5,void,void), tree(7,void,void)))
Handing in solutions: (a) An electronic copy of your file tree.pl with all your Prolog rules must be included in your
zip archive; (b) your session with Prolog, showing the queries you submitted and the answers returned (the name of
the file must be tree.txt). It is up to you to formulate a range of queries that demonstrates that your programs are
working properly.
How to submit this assignment. Read regularly Frequently Answered Questions and replies to them that are linked
from the Assignments Web page at
http://www.scs.ryerson.ca/˜mes/courses/cps721/assignments.html
If you write your code on a Windows machine, make sure you save your files as plain text that one can easily read
on Linux machines. Before you submit your Prolog code electronically make sure that your files do not contain
any extra binary symbols: it should be possible to load recursion.pl or term.pl or tree.pl into a recent
release 6 of ECLiPSe Prolog, compile your program and ask testing queries. TA will mark your assignment using
ECLiPSe Prolog. If you run any other version of Prolog on your home computer, it is your responsibility to make
sure that your program will run on ECLiPSe Prolog (release 6 or any more recent release), as required. For example,
you can run a command-line version of eclipse on moon remotely from your home computer to test your program
(read handout about running ECLiPSe Prolog). To submit files electronically do the following. First, create a zip
archive on moon:
zip yourLoginName.zip lists.txt recursion.pl recursion.txt
terms.pl terms.txt tree.pl tree.txt
where yourLoginName is the Ryerson login name of the person who submits this assignment from a group.
Remember to mention at the beginning of each file student, section numbers and last names of all people who
participated in discussions (see the course management form). You may be penalized for not doing so. Second,