Higher-Order Functions (Due October 4)

Due Tuesday, October 4, 2022 at 11:59PM

Due Tuesday, October 4, 2022 at 11:59PM

(A printable version of this homework, with some loss of formatting, is available in PDF format.)

This assignment is all individual work. There is no pair programming.


Higher-order functions are a cornerstone of functional programming. They are found in all of the web/scripting languages, including JavaScript, Python, Perl, and Lua. This assignment will help you incorporate first-class and higher-order functions into your programming practice. You will use existing higher-order functions, define higher-order functions that consume functions, and define higher-order functions that return functions. The assignment builds on what you’ve already done, and it adds new ideas and techniques that are described in sections 2.7, 2.8, and 2.9 of Build, Prove, and Compare.


The executable μScheme interpreter is in /comp/105/bin/uscheme; if you are set up with use comp105, you should be able to run uscheme as a command. The interpreter accepts a -q (“quiet”) option, which turns off prompting. Your homework will be graded using uscheme. When using the interpreter interactively, you may find it helpful to use ledit, as in the command

  ledit uscheme

We don’t give you a template—by this time, you know how to identify solutions and where to put contracts, algebraic laws, and tests.

Dire Warnings

The μScheme programs you submit must not use any imperative features. Banish set, while, print, println, printu, and begin from your vocabulary! If you break this rule for any exercise, you get No Credit for that exercise. You may find it useful to use begin and println while debugging, but they must not appear in any code you submit. As a substitute for assignment, use let or let*.

No code may compute the length of a list.

Except as noted below, do not define helper functions at top level. Instead, use let or letrec to define helper functions. When you do use let to define inner helper functions, avoid passing as parameters values that are already available in the environment. (An example of what to avoid appears under “Avoid common mistakes” below.)

Your solutions must be valid μScheme; in particular, they must pass the following test:

    /comp/105/bin/uscheme -q < myfilename > /dev/null

without any error messages or unit-test failures. If your file produces error messages, we won’t test your solution and you will earn No Credit for functional correctness. (You can still earn credit for structure and organization). If your file includes failing unit tests, you might possibly get some credit for functional correctness, but we cannot guarantee it.

Every function should be accompanied by a short contract and by unit tests. If the function does case analysis, it must also be accompanied by algebraic laws. Submissions without algebraic laws will earn No Credit.

We will evaluate functional correctness by testing your code extensively. Because this testing is automatic, each function must be named be exactly as described in each question. Misnamed functions earn No Credit.

Reading comprehension

Before starting the programming problems, answer the reading-comprehension questions for this module, which you will find online. It is also OK to alternate between reading-comprehension questions and related homework questions.

Programming and Proof


For this assignment, you will do Exercises 28 (b, e, f), 29 (a, c), 30, and 38, from pages 186 to 189 of Build, Prove, and Compare, plus the exercises F, M, O, and V below.

A summary of μScheme’s initial basis can be found on page 99. While you’re working on this homework, keep it handy.

Each top-level function you define must be accompanied by a contract and unit tests. Each named internal function written with lambda should be accompanied by a contract, but internal functions cannot be unit-tested. (Anonymous lambda functions need not have contracts.) Algebraic laws are required only where noted below; each problem is accompanied by a Laws section, which says what is needed in the way of algebraic laws.

Book problems

28. Higher-order functions and nonempty lists. Do exercise 28 on page 186 of Build, Prove, and Compare, parts (b), (e), and (f). These functions accept nonempty lists and produce scalar results. Code accordingly. You must not use recursion—solutions using recursion will receive No Credit.

Because you are not defining recursive functions, you need not write any algebraic laws.

For this problem only, you may define one helper function at top level.

Related reading: For material on higher-order functions, see section 2.8, which starts on page 129.

Laws: These functions must not be recursive, should not do any case analysis,1 and do not return functions. Therefore, no algebraic laws are needed.

29. Higher-order functions for list functions. Do exercise 29 on page 186 of Build, Prove, and Compare, parts (a) and (c): use higher-order functions to implement two standard recursive functions without recursion. You must not use recursion—solutions using recursion will receive No Credit.

Because you are not defining recursive functions, you need not write any algebraic laws.

Related reading: For material on higher-order list functions, see section 2.8, which starts on page 129. For material on curry, see section 2.7.2, which starts on page 127.

Laws: These functions must not be recursive, should not do any case analysis,2 and do not return functions. Therefore, no algebraic laws are needed.

30. Folds for higher-order functions. Do exercise 30 on page 186: implement four standard higher-order functions using only folds. You must not use recursion—solutions using recursion will receive No Credit. (It is OK if recursion happens in functions you call, like foldl and foldr. But it must not happen in functions you write.)

Because you are not defining recursive functions, you need not write any algebraic laws.

For this problem, you get full credit if your implementations return correct results. You get extra credit3 if you can duplicate the behavior of exists? and all? exactly. To earn the extra credit, it must be impossible for an adversary to write a μScheme program that produces different output with your version than with a standard version. However, the adversary is not permitted to change the names in the initial basis.

Related reading: Examples of foldl and foldr are in sections 2.8.1 and 2.8.2 starting on page 129. You may also find it helpful to study the implementations of foldl and foldr in section 2.8.3, which starts on page 131; the implementations are at the end. Information on lambda can be found in section 2.7, on pages 122 to 127.

Laws: These functions must not be recursive, should not begin with case analysis, and do not return functions. Therefore, no algebraic laws are needed.

38. Functions as values. Do exercise 38 on page 189 of Build, Prove, and Compare. You cannot represent these sets using lists. If any part of your code to construct or to interrogate a set uses cons, car, cdr, or null?, you are doing the problem wrong.

Do all four parts:

  • Parts (a) and (b) require no special instructions.

  • In part (c), your add-element function must take two parameters: the element to be added as the first parameter and the set as the second parameter.

    When you code part (c), compare values for equality using the equal? function.

    To help you design part (c), put comments in your source code that complete the right-hand sides of the following properties:

    (member? x (add-element x s)) == ...
    (member? x (add-element y s)) == ..., where (not (equal? y x))
    (member? x (union s1 s2))     == ...
    (member? x (inter s1 s2))     == ...
    (member? x (diff  s1 s2))     == ...

    The properties are not quite algorithmic, but they should help anyway.

  • In part (d), when you code the third approach to polymorphism, write a function set-ops-from which places your set functions in a record. To define record functions, use the syntactic sugar described in the book in Section 2.13.6 on page 169. In particular, be sure your code includes this record definition:

    (record set-ops [emptyset member? add-element union inter diff])

    Code your solution to part (d) as a function set-ops-from, which will accept one argument (an equality predicate) and will return a record created by calling make-set-ops. Your function might look like this:

    (define set-ops-from (eq?)
      (let ([emptyset    ...]
            [member?     ...]
            [add-element ...]
            [union       ...]
            [inter       ...]
            [diff        ...])
        (make-set-ops emptyset member? add-element union inter diff)))

    Fill in each ... as appropriate. Each implementation is like the one given in the book or the one you wrote in part (c), except instead of using the predefined equal?, it uses the parameter eq?—that is what is meant by “the third approach to polymorphism.”

    No additional laws are needed for part (d).

To help you get part (d) right, we recommend that you use these unit tests:

(check-assert (function? set-ops-from))
(check-assert (set-ops? (set-ops-from =)))

And to write your own unit tests for the functions in part (d), you may use these definitions:

(val atom-set-ops     (set-ops-from =))
(val atom-emptyset    (set-ops-emptyset    atom-set-ops))
(val atom-member?     (set-ops-member?     atom-set-ops))
(val atom-add-element (set-ops-add-element atom-set-ops)) 
(val atom-union       (set-ops-union       atom-set-ops))
(val atom-inter       (set-ops-inter       atom-set-ops))
(val atom-diff        (set-ops-diff        atom-set-ops))

Hint: The recitation for this unit includes an “arrays as functions” exercise. Revisit it.

Related reading: For functions as values, see the examples of lambda in the first part of section 2.7 on page 122, and also the array exercise from recitation. For function composition and currying, see section 2.7.2. For polymorphism, see section 2.9, which starts on page 133.

Laws: Complete the right-hand sides of the properties listed above. These properties say what happens when member? is applied to any set created with any of the other functions. No other laws are needed.

A function that returns a function

F. The third lesson in program design (“Higher-order functions”) mentions a higher-order function flip, which can convert < into >, among other tricks. Using your algebraic law or laws from the comprehension questions, define flip. Don’t forget unit tests.

Related reading: Seven Lessons in Program Design, lesson 3.

Laws: Use your law or laws from the comprehension questions.

Calculational reasoning about functions

M. Reasoning about higher-order functions. Using the calculational techniques from Section 2.5.7, which starts on page 116, prove that

    (o ((curry map) f) ((curry map) g)) == ((curry map) (o f g))

To prove two functions equal, prove that when applied to equal arguments, they return equal results.

Related reading: Section 2.5.7. The definitions of composition and currying in section 2.7.2. Example uses of map in section 2.8.1. The definition of map in section 2.8.3.

Laws: In this problem you don’t write new laws; you reuse existing ones. You may use any law in the Basic Laws handout, which includes laws for o, curry, and map. (If it simplifies your proof, you may also introduce new laws, provided that you prove each new law is valid.)

Ordered lists

O. Ordered lists. Like natural numbers, the forms of a list can be viewed in different ways. In almost all functions, we examine just two ways a list can be formed: '() and cons. But in some functions, we need a more refined view. Here is a problem that requires us to divide a list of values into three forms.

Define a function ordered-by? that takes one argument—a comparison function that represents a transitive relation—and returns a predicate that tells if a list of values is totally ordered by that relation. Assuming the comparison function is called precedes?, here is an inductive definition of a list that is ordered by precedes?:

  • The empty list of values is ordered by precedes?.

  • A singleton list containing one value is ordered by precedes?.

  • A list of values in the form (cons x (cons y zs)) is ordered by precedes? if the following properties hold:

    • x is related to y, which is to say (precedes? x y).

    • List (cons y zs) is ordered by precedes?.

Here are some examples. Note the parentheses surrounding the calls to ordered-by?.

-> ((ordered-by? <) '(1 2 3))
-> ((ordered-by? <=) '(1 2 3))
-> ((ordered-by? <) '(3 2 1)) 
-> ((ordered-by? >=) '(3 2 1))
-> ((ordered-by? >=) '(3 3 3))
-> ((ordered-by? =) '(3 3 3)) 


  • The entire 9-step software-design process applies, and it starts with the three forms of data in the definition of “list ordered by” above.

  • For the code itself, we recommend letrec. You can emulate the example of the polymorphic sort function on page 137.

  • We recommend that your submission include the following unit tests, which help ensure that your function has the correct name and takes the expected number of parameters.

    (check-assert (function? ordered-by?))
    (check-assert (function? (ordered-by? <)))
    (check-error (ordered-by? < '(1 2 3)))

Related reading: Section 2.9, which starts on page 133. Especially the polymorphic sort in section 2.9.2—the lt? parameter to that function is an example of a transitive relation.

Laws: Write algebraic laws for ordered-by?, including at least one law for each of the three forms of data used in the definition of “list ordered by” above.

A domain-specific language for input validation

In this part of the homework, you will use higher-order functions to develop an embedded domain-specific language for validating web-form submissions.4 For example, here is a version of the web form on the CS 105 regrades page (but this version is not connected to anything):

I’ve submitted a new photograph.

I was marked absent from recitation, but I attended this recitation:

  • Date and time:
  • Led by:

I submitted the wrong PDF (or the wrong files) for assignment

I want someone to review my grade for problem on assignment

What do we need to fix?

In a real form, clicking on the Submit button sends data to the server, and that data has to be validated. For example, if the form asks for a grade to be reviewed, it must specify an assignment. (To try our validator for yourself, go to the course regrade form at https://www.cs.tufts.edu/comp/105/regrade, and click the Submit button without filling in the form.) Our validator checks the form’s input using higher-order functions, and for problem V below, you will define similar functions.

Representation of form data and validation results

When form data reaches the server, it is turned into an association list, which we call a response. In the response, each key names a field of the form; the key is represented by a symbol. The key is bound to the value the user responded with. If the user did not supply a value, the key is bound to the value #f. Here’s an example response, representing a student who wants something regraded because they accidentally submitted the wrong PDF:

(val sample-response
  '([why badsubmit]
    [badsubmit_asst scheme]
    [info (I accidentally submitted the opsem PDF again.)]
    [badgrade_asst ...]
    [problem #f]))

But what if they say they submitted the wrong PDF, but they forgot to say what assignment? The validator needs to identify such faults. For our purposes, a fault is simply a μScheme symbol that identifies a bad input. And a validator is a function that takes a response and returns a list of faults (without duplicates):

input and output of a single validator
input and output of a single validator

A language of validators

The key idea behind the validator functions—what makes them a “domain-specific language” and not just another API—is composition: you define functions that build validators from other validators. The functions are specified using these metavariables:

Here are the specifications:

An example validator

To illustrate the use of the validator functions, here is some code for the regrade validator. It will help to know that it uses these faults:

If the regrade validator were written in μScheme, it might look like this:

(val regrade-validator  ;; example for the regrade form
  (faults/switch 'why
    (bind         'photo
      (bind       'badsubmit
                   (faults/both (faults/equal 'badsubmit_asst '...)
                                (faults/equal 'info #f))
        (bind     'badgrade
                      (faults/equal 'badgrade_asst '...)
                         (faults/equal 'info #f)
                         (faults/equal 'problem #f)))
          (bind   'recitation
                      (faults/equal 'date #f)
                      (faults/equal 'leaders #f))

            (bind '#f
                  (faults/always 'nobutton)

For example, if I submit a photo for regrade, faults/switch figures out that no more validation is required. But if I select “review my grade” (badgrade), faults/switch figures out that we need an assignment (badgrade_asst), an explanation (info) and a problem number (problem). If I leave all these fields blank, the validator will return a fault list containing symbols problem, badgrade_asst, and info, like this one:

'(problem badgrade_asst info)

If I leave only one or two of these fields blank, the validator will return a fault list identifying those fields.

You can use this validator to test your functions with some integration tests.

At last, your assigned problem

V. Implement validation functions faults/none, faults/always, faults/equal, faults/both, and faults/switch. As usual, each function must be accompanied by algebraic laws and unit tests. The five functions in the solutions total less than 20 lines of μScheme.

Additional expectations and other notes:

  • Every validator must treat the response as an abstraction. A validator must never interrogate a response about its form of data; the validator should restrict itself to function find and possibly function bound-in?:

    (define bound-in? (key pairs)
      (if (null? pairs)
          (|| (= key (alist-first-key pairs))
              (bound-in? key (cdr pairs)))))
  • Most of your algebraic laws, like the algebraic laws for curry and flip, will need to specify what happens when the result of calling a function is itself applied. If written well, your laws will be clearer and easier to follow than the informal descriptions above. Which is one reason why we write them.

  • Function faults/none is not itself a validator; it is a function that is called with no arguments and returns a validator. This design choice makes the design seem more regular: to get a validator, you always call a function.

  • The implementation of faults/switch is simplest if key k is bound in the response to a value v that is bound in the validator table. When you write your algebraic laws, make this simplifying assumption. And when you write your code, you may simply assume that k is bound in the response. But if v is unbound in the validator table, the validator must halt with a checked run-time error. (Any error will do.)

If you like you can start with the template below, which provides a couple of algebraic laws, plus local bindings for faults/both. The local bindings will enable your fault code to avoid conflicts with the set code in exercise 38.

(val the-fault list1) ; build a singleton fault set
(val no-faults ’())   ; an empty fault set

; ((faults/none) response) == no-faults
(define faults/none ()

; ((faults/always f) response) == (the-fault f)
(define faults/always (f)

; ... law or laws for faults/equal ...
(define faults/equal (key value)

; ... law or laws for faults/both ...
(val faults/both
    (let* ([member?  (lambda (x s) (exists? ((curry =) x) s))]
           [add-elem (lambda (x s) (if (member? x s) s (cons x s)))]
           [union    (lambda (faults1 faults2)
                        (foldr add-elem faults2 faults1))])

; ... law or laws for faults/switch ...
(define faults/switch (key validators)

Related reading:

  • For a review of association lists and find, consult section 2.3.8, which starts on page 107.

  • All the functions in this problem are “function factories”: each one takes in some arguments and returns a function. The main function factories in the book are curry and o (“compose”); to review them, study section 2.7.2, which starts on page 127.

  • For faults/both, revisit the combine and divvy examples from the lecture on lambda. (These functions are intended to be called conjoin and disjoin.)

Extra credit

VX. For extra credit, answer the questions below.

  1. Which of the following equations are valid properties of the fault-validation functions?

    • (faults/both (faults/none) V) V?

    • (faults/both (faults/always F) V) (faults/always F)?

  2. For each of the equations in part (a),

    • If the equation is a valid property, present a calculational proof using your laws from problem V.

    • If the equation is not a valid property, present a counterexample. That is, present examples of V, F (if necessary), and a response such that, when applied to the response, the two validators produce different fault sets.

What and how to submit

You must submit three files:

If you complete the extra-credit exercise VX, you must submit it separately:

As soon as you have the files listed above, run submit105-hofs to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.

Avoid common mistakes

Listed below are some common mistakes, which we encourage you to avoid.

Passing unnecessary parameters. In this assignment, a very common mistake is to pass unnecessary parameters to a nested helper function. Here’s a silly example:

    (define sum-upto (n)
      (letrec ([sigma (lambda (m n) ;;; UGLY CODE
                         (if (> m n) 0 (+ m (sigma (+ m 1) n))))])
         (sigma 1 n)))

The problem here is that the n parameter to sigma never changes, and it is already available in the environment. To eliminate this kind of problem, don’t pass the parameter:

    (define sum-upto (n)
      (letrec ([sum-from (lambda (m) ;;; BETTER CODE
                         (if (> m n) 0 (+ m (sum-from (+ m 1)))))])
         (sum-from 1)))

The name of the internal function is different, but the only other things that are different is that the second formal parameter from the lambda is gone and as is the second actual parameter from the call sites. You can still use n in the body of sum-from; it’s visible from the definition.

An especially good place to avoid this mistake is in your definition of ordered-by? in problem O.

Another common mistake is to fail to redefine predefined functions like map and filter in exercise 30. Yes, we really want you to provide new definitions that replace the existing functions, just as the exercise says.

How your work will be evaluated

Structure and organization

The criteria in the general coding rubric apply. As always, we emphasize contracts and naming. In particular, unless the contract is obvious from the name and from the names of the parameters, an inner function defined with lambda and a let form needs a contract. (An anonymous lambda that is returned from a function like faults/both does not need a contract—the behavior of that lambda is part of the contract of the function that returns it.)

There are a few new criteria related to let, lambda, and the use of basis functions. The short version is use the functions in the initial basis; except when we specifically ask you to, don’t redefine initial-basis functions.

Exemplary Satisfactory Must Improve

• Short problems are solved using simple anonymous lambda expressions, not named helper functions.

• When possible, inner functions use the parameters and let-bound names of outer functions directly.

• The initial basis of μScheme is used effectively.

• Most short problems are solved using anonymous lambdas, but there are some named helper functions.

• An inner function is passed, as a parameter, the value of a parameter or let-bound variable of an outer function, which it could have accessed directly.

• Functions in the initial basis, when used, are used correctly.

• Most short problems are solved using named helper functions; there aren’t enough anonymous lambda expressions.

• Functions in the initial basis are redefined in the submission.

Functional correctness

In addition to the usual testing, we want appropriate list operations to take constant time.

Exemplary Satisfactory Must Improve

• Your code passes every one of our stringent tests.

• Testing shows that your code is of high quality in all respects.

• Testing reveals that your code demonstrates quality and significant learning, but some significant parts of the specification may have been overlooked or implemented incorrectly.

• Testing suggests evidence of effort, but the performance of your code under test falls short of what we believe is needed to foster success.

• Testing reveals your work to be substantially incomplete, or shows serious deficiencies in meeting the problem specifications (serious fault).

• Code cannot be tested because of loading errors, or no solutions were submitted (No Credit).


• Empty lists are distinguished from non-empty lists in constant time.

• Distinguishing an empty list from a non-empty list might take longer than constant time.

Proofs and inference rules

For your calculational proof, use induction correctly and exploit the laws that are proved in the book.

Exemplary Satisfactory Must Improve

• Proofs that involve predefined functions appeal to their definitions or to laws that are proved in the book.

• Proofs that involve inductively defined structures, including lists and S-expressions, use structural induction exactly where needed.

• Each proof by induction states the induction hypothesis explicitly.

• Proofs involve predefined functions but do not appeal to their definitions or to laws that are proved in the book.

• Proofs that involve inductively defined structures, including lists and S-expressions, use structural induction, even if it may not always be needed.

• A proof by induction does not state the induction hypothesis explicitly, but course staff can easily figure out what it is.

• A proof that involves an inductively defined structure, like a list or an S-expression, does not use structural induction, but structural induction is needed.

• There is a proof by induction, but course staff cannot easily figure out what part of the proof is the induction hypothesis.

  1. Case analysis may be happening, but on this problem, it will be happening inside functions like map and foldr, not in any code that you write.

  2. Case analysis may be happening, but on this problem, it will be happening inside functions like map and foldr, not in any code that you write.

  3. In your README, please identify this credit as EXACT-EXISTS.

  4. It’s actually just a library, but a library like this is called a “language” because of the way functions compose. The composition of library functions resembles the syntactic composition of expressions in an actual programming language.