C SCI 316 Assignment 4 Solved

In these problems, the behavior of the functions you are asked to complete or write is specified  only when the arguments have certain explicitly stated properties––e.g., in problem A the behavior of MY-SUM is specified only if its argument is a nonempty list of numbers, and in problem 1 the behavior of SUM is specified only if its argument is a (possibly empty) list of numbers.  When functions' arguments do not have the stated properties (e.g., if the argument of MY-SUM is NIL,  or if an argument of the function SET-UNION of problem 10 is a list in which some element  occurs more than once), the functions' behavior is unspecified: Your functions are allowed to return any result or to produce an evaluation error in such cases!   

 

When evaluation of a function call has produced an infinite loop, you can often abort execution by typing Ctrl-C.  At a Break error prompt, typing   backtrace   will print, in reverse order, the sequence of all function calls that are in progress. 

 

In this document the term list should be understood to mean proper list. 

 

 

SECTION 1 (Nonrecursive Preliminary Problems) 

 

The 7 problems in this section (A – G) do not carry direct credit, but are intended to help you  solve problems 1 – 7 in Section 2.  There may be exam questions of a similar nature to A – G. 

 

Your solutions to problems A – G must not be recursive.  You can test your solutions to these  problems on venus† or euclid: Functions SUM, NEG-NUMS, INC-LIST-2, INSERT, ISORT,  SPLIT-LIST, and PARTITION with the properties stated in A – G are predefined when you start clisp  using   cl  on venus† or euclid.  When a function has 2 cases, test your code in both cases! 

 

 

A.        SUM is a function that is already defined on venus and euclid; if L is any list of  numbers then (SUM L)       returns the sum of the elements of L.  [Thus (SUM ( )) returns 0.]  Complete the following definition of a       function MY-SUM without making further calls of  SUM and without calling MY-SUM recursively, in       such a way that if L is any nonempty list of numbers then (MY-SUM L) is equal to (SUM L). 

 

       (defun my-sum (L)                           

          (let ((X (sum (cdr L)))) 

             __________________________________ )) 

 

 

*If you have difficulty with these problems, you are encouraged to see me during my office hours.  Questions about these    problems that are e-mailed to me will not be answered until after the due date. 

 

†This assumes you executed the  /home/faculty/ykong/316setup  command on venus before you did    Lisp Assignment 1 (in accordance with the instructions for Assignment 1). 

B.         NEG-NUMS is a function that is already defined on venus and euclid; if L is any list of real numbers       then  (NEG-NUMS  L) returns a new list that consists of the negative elements of L.   For example:     

     (NEG-NUMS '(–1 0 –8  2  0 8 –1 –8  2  8  4 –3 0) ) = (–1 –8 –1 –8 –3).  

         Complete the following definition of a function MY-NEG-NUMS without making further calls       of  NEG-NUMS and without calling MY-NEG-NUMS recursively, in such a way that if L is          any nonempty list of numbers then (MY-NEG-NUMS L) is equal to (NEG-NUMS L). 

 

       (defun my-neg-nums (L)                           

               (let ((X (neg-nums (cdr L)))) 

               ______________________________________                ____________ ))  

     There are two cases: (car L) may or may not be negative.  

 

C.         INC-LIST-2 is a function that is already defined on venus and euclid; if L is any list of numbers and       N is a number then (INC-LIST-2 L N) returns a list of the same length as L in which each element       is equal to  (N + the corresponding element of L).   For example,  

                           (INC-LIST-2  ( )  5) = NIL               (INC-LIST-2  '(3  2.1  1  7.9)  5) = (8  7.1  6  12.9) 

     Complete the following definition of a function MY-INC-LIST-2 without making further calls       of  INC-LIST-2 and without calling MY-INC-LIST-2 recursively, in such a way that if L is         any nonempty list of numbers and N is any number then (MY-INC-LIST-2 L N) is equal to         (INC-LIST-2 L N). 

                    (defun my-inc-list-2 (L N)                           

                        (let ((X (inc-list-2 (cdr L) N))) 

                          __________________________________ )) 

    

D.        INSERT is a function that is already defined on venus and euclid; if N is any real number and L is      any list of real numbers in ascending order then (INSERT N L) returns a list of numbers in       ascending order obtained by inserting N in an appropriate position in L.   Examples:  

      (INSERT 8 ( )) = (8)      (INSERT 4 '(0 0 1 2 4)) = (0 0 1 2 4 4)       (INSERT 4 '(0 0 1 3 3 7 8 8)) = (0 0 1 3 3 4 7 8 8) 

     Complete the following definition of a function MY-INSERT without making further calls       of  INSERT and without calling MY-INSERT recursively, in such a way that if N is any real       number and L is any nonempty list of real numbers in ascending order then (MY-INSERT  N  L)              is equal to (INSERT  N  L). 

                            (defun my-insert (N L)                           

                               (let ((X (insert N (cdr L)))) 

                                 __________________________________  

                                 __________________________________ )) 

     [There are two cases: N may or may not be ≤  (car L).  In the former case you do not need to use X,              so if you move that case outside the LET the function will be more efficient.] 

 

E.         ISORT is a function that is already defined on venus and euclid; if L is any list of real numbers       then (ISORT L) is a list consisting of the elements of L in ascending order.  Complete the       following definition of a function MY-ISORT without making further calls of  ISORT and          without calling MY-ISORT recursively, in such a way that if L is any nonempty list of real       numbers then (MY-ISORT  L) is equal to (ISORT L).  

 

       (defun my-isort (L)                           

             (let ((X (isort (cdr L)))) 

             __________________________________ )) 

     Hint: You should not have to call any function other than INSERT and CAR. 

 

 

 

 

 

 

IMPORTANT:  If you have not yet done problems 15 – 20 of Lisp Assignment 2, do those six                       problems before you work on the next two problems! 

 

F.         SPLIT-LIST is a function that is already defined on venus and euclid; if L is any list then       (SPLIT-LIST  L) returns a list of two lists, in which the first list consists of the 1st, 3rd,  5th,  ...        elements of L, and the second list consists of the 2nd, 4th, 6th,  ... elements of L.  Examples:          (SPLIT-LIST ( )) = (NIL NIL)       (SPLIT-LIST '(A B C D 1 2 3 4 5))  = ((A C 1 3 5) (B D 2 4)) 

     (SPLIT-LIST '(B C D 1 2 3 4 5)) =  ((B D 2 4) (C 1 3 5))             (SPLIT-LIST '(A)) = ((A) NIL) 

     Complete the following definition of a function MY-SPLIT-LIST without making further calls of                SPLIT-LIST and without calling MY-SPLIT-LIST recursively, in such a way that if L is any       nonempty list then (MY-SPLIT-LIST L) is equal to (SPLIT-LIST L). 

 

        (defun my-split-list (L)                           

             (let ((X (split-list (cdr L)))) 

             __________________________________ )) 

 

G.        PARTITION is a function that is already defined on venus and euclid; if L is a list of real         numbers and P is a real number then (PARTITION L P) returns a list whose CAR is a list of the       elements of L that are strictly less than P, and whose CADR is a list of the other elements of L.         Each element of L must appear in the CAR or CADR of (PARTITION L P), and should appear       there just as many times as in L. Examples: (PARTITION '(7 5 3 2 1 5) 1) = (NIL (7 5 3 2 1 5)) 

     (PARTITION '(4 0 5 3 1 2 4 1 4) 4) = ((0 3 1 2 1) (4 5 4 4))   (PARTITION ( ) 9) = (NIL NIL)                 Complete the following definition of a function MY-PARTITION without making further calls       of  PARTITION and without calling MY-PARTITION recursively, in such a way that if L is any       nonempty list of real numbers and P is a real number then (MY-PARTITION L P) is equal to       (PARTITION L P). 

                   (defun my-partition (L P)                           

                          (let ((X (partition (cdr L) P))) 

                         ___________________________________________                          ___________________________________________ )) 

     There are two cases: (car L) may or may not be less than P. 

  

 

SECTION 2 (Main Problems) 

 

Your solutions to the following problems will count a total of 2% towards your grade if the grade is computed using rule A.  Note that a working solution to each of problems 1 – 7 can be obtained  from a solution to the corresponding one of problems A – G by changing the name of the function MY-FUNC to FUNC and adding appropriate base case code, without changing the LET block.  [The resulting definition of FUNC will be correct because it has the following property: In all non-base cases where FUNC's recursive call returns the correct result, FUNC returns the same result as  MY-FUNC. Assuming your definition of MY-FUNC is correct, this property implies FUNC returns the correct result whenever FUNC's recursive call returns the correct result, which in turn implies FUNC never returns an incorrect result if your base case code is correct.] But if you solve a problem this way then any cases that do not need to use the LET's local variable should be moved out of the 

LET, and the LET should be eliminated if the value of its local variable is never used more than once. 

 

Note 1:  On euclid and venus, when you LOAD a function definition for any of problems 1 – 7, your function will replace the                predefined function that has the same name (and Clisp will issue a WARNING that the predefined function has been                redefined). It follows, for example, that if your definition of SUM for problem 1 is wrong then, after you LOAD your                definition of SUM, your definition of MY-SUM for problem A may stop working even if that definition is correct                because MY-SUM calls SUM. 

 

Note 2:  You may use ENDP or NULL to test whether a list is empty. Recall that (ENDP L) returns the same result as (NULL L)                if the value of L is a list. (But evaluation of (ENDP L) produces an error if the value of L is an atom other than NIL.) 

 

1.     Define a recursive function SUM with the properties stated in problem A.  Note that whereas NIL is      not a valid argument of MY-SUM,  NIL is a valid argument of SUM. 

2.     Define a recursive function NEG-NUMS with the properties stated in problem B.  Note that NIL     is a valid argument of NEG-NUMS. 

 

3.     Define a recursive function INC-LIST-2 with the properties stated in problem C.  Note that the first     argument of INC-LIST-2 may be NIL. 

 

4.     Define a recursive function INSERT with the properties stated in problem D.  Note that the second      argument of INSERT may be NIL. 

 

5.     Define a recursive function ISORT with the properties stated in problem E.  Hint: In your definition of  

    ISORT you should not have to call any function other than ISORT itself, INSERT, CAR, CDR, and  

    ENDP or NULL.   (An IF or COND form is not considered to be a function call, and will be needed.)   

 

6.     Define a recursive function SPLIT-LIST with the properties stated in problem F.   

 

7.     Define a recursive function PARTITION with the properties stated in problem G.   

 

8.     Without using MEMBER, complete the following definition of a recursive function POS such           that if L is a list and E is an element of L then (POS E L) returns the position of the first                 occurrence of E in L, but if L is a list and E is not an element of L then (POS E L) returns 0.   

              (DEFUN POS (E L) 

                 (COND ((ENDP L)            ... ) 

                       ((EQUAL E (CAR L))         ... ) 

                       (T   (LET  ((X  (POS E (CDR L)))) 

 

                                    ...                  

                                                   )))) 

    Examples:  (POS 5 '(1 2 5 3 5 5 1 5)) = 3    (POS 'A '(3 2 1)) = 0    (POS '(3 B) '(3 B)) = 0     (POS '(A B)  '((K)  (3 R C)  A  (A B)  (K L L)  (A B)))  = 4       (POS '(3 B) '((3 B))) = 1 

 

9.     Define a recursive function SPLIT-NUMS such that if N is a non-negative integer then  

       (SPLIT-NUMS N)  returns a list of two lists: The first of the two lists consists of the even                     integers between 0 and N in descending order, and the other list consists of the odd integers                  between 0 and N in descending order.   Examples:   (SPLIT-NUMS 0) = ((0) NIL)                              (SPLIT-NUMS 7) = ((6 4 2 0) (7 5 3 1))     (SPLIT-NUMS 8) = ((8 6 4 2 0) (7 5 3 1)) 

 

 

IMPORTANT: In problems 10 – 13 the term set is used to mean a proper list of numbers and/or symbols in which no atom occurs more than once.  You may use MEMBER but not the functions UNION, NUNION, REMOVE, DELETE, SET-DIFFERENCE, and SET-EXCLUSIVE-OR.   

 

10.  Define a recursive function SET-UNION such that if s1 and s2 are sets then (SET-UNION s1 s2)        is a set that contains the elements of s1 and the elements of s2, but no other elements.  Thus        (SET-UNION '(A B C D) '(C E F)) should return a list consisting of the atoms A, B, C, D, E, and F        (in any order) in which no atom occurs more than once. 

 

11.  Define a recursive function SET-REMOVE such that if s is a set and x is an atom in s then                       (SET-REMOVE x s) is a set that consists of all the elements of s except x, but if s is a set and        x is an atom which is not in s then (SET-REMOVE x s) returns a set that is equal to s.   

 

 In problems 12 and 13 you may use the function SET-REMOVE from problem 11.   

 

12.  Define a recursive function SET-EXCL-UNION such that if s1 and s2 are sets then  

      (SET-EXCL-UNION s1 s2) is a set that contains all those atoms that are elements of exactly one        of s1 and s2, but no other atoms. (SET-EXCL-UNION s1 s2) does not contain any atoms that are         neither in s1 nor in s2, and also does not contain the atoms that are in both of s1 and s2.  For        example, (SET-EXCL-UNION '(A B C D) '(E C F G A)) should return a list consisting of the        atoms B, D, E, F, and G (in any order) in which no atom occurs more than once.   

 

13.  Define a recursive function SINGLETONS such that if e is any list of numbers and/or symbols        then (SINGLETONS e) is a set that consists of all the atoms that occur just once in e.   

      Examples:   (SINGLETONS ( )) = NIL   (SINGLETONS '(G A B C B)) = (G A C)          

      (SINGLETONS '(H G A B C B)) = (H G A C)     (SINGLETONS '(A G A B C B)) = (G C)                (SINGLETONS '(B G A B C B)) = (G A C)     [Hint: When e is nonempty, consider the case in        which (car e) is a member of (cdr e), and the case in which (car e) is not a member of (cdr e).]