Lists, Lists, Lists
 Due: 11:00pm Thursday, 17 September
 Starter files:
 fork wellesleycs251 / cs251lists, add bpw as admin
 Programming answers in
functions.rkt
 Text answers in
answers.txt
 Submission: commit and push your completed work to your Bitbucket fork as in assignment zero (but with the repository for this assignment).
 Relevant reading:
 Computability and the Halting Problem
 Racket
 McCarthy, Recursive Functions of Symbolic Expressions and Their Computation by Machine, 1960. Sections 12 (pages 18). Other sections optional.
 Tools:
Notes
 This assignment will take longer than the last. Start early!
 The problems are not sorted by difficulty. Feel free to jump around.
 Inspiration: ‘(1 2 3 4
1. Racket Programming^{1} (70 points)
Please do not use DrRacket’s “box comments” in your code. They interact poorly with our grading infrastructure.
Complete the following programming problems. If at all possible, avoid the use of the builtin list append
function (and the equivalent listappend
we wrote in lecture today). One or two functions below may seem to require append
or something equivalent—feel free to use append
there, but note its running time is O(n) given an nelement first argument. Soon we will see a new style of recursion for an efficient, elegant alternative.

Write a function
merge
that merges two lists into one as in mergesort. Assuming the two argument lists are each sorted lists of numbers, this function will return a sorted list containing all elements from the two lists. Your function does not need to work for inputs that are not sorted.> (merge (list 1 3 6 7 9 10) (list 2 4 5 8)) '(1 2 3 4 5 6 7 8 9 10) > (merge (list 4 5 6) (list 1 2 3)) '(1 2 3 4 5 6) > (merge null null) '() ; this means null

Write a function
rev
that takes a listxs
and reverses its order. You may not use the builtinreverse
function.> (rev (list 1 (list 2 3) (list 4 5 (list 6 7 8)))) '((4 5 (6 7 8)) (2 3) 1) > (rev (list 1 2 3 4 5)) '(5 4 3 2 1) > (rev (list 1)) '(1) > (rev null) '()

Write a function
deeprev
that takes any argumentx
and deeply reverses it. Ifx
is an atom,deeprev
returns it as is. Ifx
is acons
cell,deeprev
assumes it is a list, reverses the list, and deeply reverse any lists that are elements inx
. Assume thatcons
cells are used only to represent lists. Thecons?
function returns#t
if its argument is acons
cell and#f
otherwise.^{2}> (deeprev (list 1 (list 2 3) (list 4 5 (list 6 7 8)))) '(((8 7 6) 5 4) (3 2) 1) > (deeprev (list 1 2 3 4 5)) '(5 4 3 2 1) > (deeprev (list 1)) '(1) > (deeprev null) '()

Write a function
containsmultiple
that takes an integerm
and a list of integersns
that returns#t
ifm
evenly divides at least one element of the integer listns
; otherwise it returns#f
. Usemodulo
to determine divisibility.> (containsmultiple 5 (list 8 10 14)) #t > (containsmultiple 3 (list 8 10 14)) #f > (containsmultiple 5 null) #f

Write a function
allcontainmultiple
that takes an integern
and a list of lists of integersnss
(pronounced “enziz”) and returns#t
if each list of integers innss
contains at least one integer that is a multiple ofn
; otherwise it returns#f
.> (allcontainmultiple 5 (list (list 17 10 2) (list 25) (list 3 7 5)) #t > (allcontainmultiple 3 (list (list 17 10 2) (list 25) (list 3 7 5)) #f > (allcontainmultiple 3 null) #t

Write a function
bits
that takes a natural numbern
and returns a list of the bits (0
s and1
s) in the binary representation ofn
.> (bits 5) '(1 0 1) > (bits 10) '(1 0 1 0) > (bits 11) '(1 0 1 1) > (bits 22) '(1 0 1 1 0) > (bits 23) '(1 0 1 1 1) > (bits 46) '(1 0 1 1 1 0) > (bits 1) '(1) > (bits 0) '()
Hint: The above sequence of examples has been chosen carefully to illustrate the divide/conquer/glue nature of the solution.
Optional 240tinted challenge: Write a separate function
bits2s
that takes a natural numberb
and an integern
and a returns a list of0
s and1
s representing all digits of theb
bit two’scomplement representation of integern
.> (bits2s 4 6) '(0 1 1 0) > (bits2s 4 6) '(1 0 1 0)

Write a function
censorword
that takes a stringw
and returnsw
ifw
is not is a word in the bad words list'(algorithms midterm extension databases systems grade)
and returns'XXXX
ifw
is a bad word. The builtinmember
function takes a valuex
and a listxs
and returns#f
ifx
is not an element inxs
or non#f
otherwise. We have been avoiding writing the quote notation in Racket programs so far. Make an exception here to use symbols. We will discuss these more later.> (censorword 'apple) 'apple > (censorword 'midterm) 'XXXX
Write a second function
censor
that usescensorword
and the builtinmap
function to censor sentences by replacing all bad words:censor
takes a list of strings and returns a list of strings with bad words replaced by'XXXX
. Do not use recursion incensor
.^{2}> (censor '(I need an extension because I have an algorithms midterm)) '(I need an XXXX because I have an XXXX XXXX) > (censor '(Programming languages is more fun than systems)) '(Programming languages is more fun than XXXX)
2. Error Detection Constructs^{2} (10 points)
Evaluation of a Racket expression can either terminate normally (and return a value), terminate abnormally with an error, or run forever. Some examples of expressions that terminate with an error are (/ 3 0)
, division by 0; (car 2)
, taking the car
of an atom; and (+ 3 #f)
, adding a boolean to a number. As we have seen in evaluation rules for Racket expressions, dynamic typechecking detects these errors. In DrRacket, this terminates evaluation and prints a message to the screen. Suppose that you work at a software company that builds text editors in RacketwithSideEffects. (It’s been done in a very similar language: Emacs is built in a dialect of Lisp!) Side effects are observable changes that are not just values returned by evaluation. They include: changing the value stored in a mutable variable (like variables work in Java), sending data across the network, displaying text or images on a screen, and countless other operations. The subset of Racket we have studied so far lacks side effects, but the full language includes operations with side effects.
Your boss, who is notoriously skittish about side effects, wants to handle errors in Racket programs without terminating the computation, but doesn’t know how. Thus, the job has fallen to you (the programming language expert).

Your boss asks you to implement a Racket construct
(error? e)
that detects whether an expressione
will cause an error when evaluated. More specifically, your boss wants evaluation of(error? e)
to return the value#t
if evaluatinge
terminates with an error and return the value#f
if evaluatinge
behave in any way except terminating in error. Explain why it is not possible to add theerror?
construct to Racket. 
Your boss asks you to implement a Racket construct
(guarded e)
that either evaluatese
and returns its value, or ife
would halt with an error, returns0
without performing any side effects. This could be used to try to evaluatee
and if an error would occur, just use0
instead. For example,(+ (guarded e1) e2) ; just (+ 0 e2) if e1 halts with an error ; (+ e1 e2) otherwise
will have the value of
e2
if evaluation ofe1
would halt in error, and the value of(+ e1 e2)
otherwise. Describe how you might implement theguarded
construct. What difficulties might you encounter? Notice that unlike(error? e)
, evaluation of(guarded e)
does not need to finish and produce a result in all cases.
3. Conditional Expressions (20 points)
We introduced the if
expression in Racket. The original definition of Lisp focused instead on a related cond
expression, presented by McCarthy in his 1960 paper, Recursive Functions of Symbolic Expressions and Their Computation by Machine, which discussed foundations and implementation details of Lisp. The name cond
stands for conditional. Read sections 12 (pages 18) of McCarthy’s paper to answer the questions below.
Reading Notes
McCarthy uses mathematical notation that differs somewhat from Racket syntax. Here is a conversion from McCarthy’s notation to a syntax that would fit Racket.
 (p_{1} → e_{1}, …, p_{n} → e_{n}) would be written
(cond [p1 e1] ... [pn en])
, where allpi
andei
are expressions.  T is
#t
and F is#f
.  ∧ is boolean and (logical conjunction), ∨ is boolean or (logical disjunction), and ¬ is boolean not (logical negation). (The text defines these standard notations.)
 λ((x_{1},…,x_{n}), e) would be
(lambda (x1 ... xn) e)
, wherex1
throughxn
are variable names ande
is an expression.
Section 2 gets denser on pages 68. The most important material is earlier, but try to pick up what you can from the later parts as well.
Optional Reading
The remainder of McCarthy’s paper is an interesting read as well if you are curious. It describes more features of the Lisp language (still present in Racket) that we will consider briefly later. We will soon return to at least the section describing implementation.
Questions
Where asked for prose answers, please write no more than a few sentences.

McCarthy describes undefined results for some conditional expressions.
What does it mean for an expression’s result to be undefined? (McCarthy gives an indirect definition of this notion early in Section 2.)

Write two Racket expressions (using only the Racket features we have examined in this course) whose results are undefined, one using recursion and one without.

Write an evaluation rule for
cond
(using the Racketstyle syntax listed above and the style of evaluation rules we have defined in class) to describe the semantics of McCarthy’s conditional expression. 
Given the
cond
syntax shown above and your evaluation rule, we can implementcond
as syntactic sugar. Describe in English how to construct an equivalent Racket expression to replace acond
expression. Feel free to show concrete examples of replacement by an equivalent expression, but please describe the general transformation or “desugaring” you would perform given any validcond
expression.For inspiration, recall how we desugared the shortcircuit and operation,
(and e1 e2)
, to the semantically equivalent expression(if e1 e2 #f))
or how we desugared theor
expresion in class. If you need to produce an undefined result explicitly, call the zero argument functionvoid
; it produces “no value.”