Higher-Order Functions

COMP 105 Assignment

Due Monday, October 12, 2020 at 11:59PM

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

Overview

Higher-order functions are a cornerstone of functional programming. And 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.

Setup

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 (10 percent)

Answer these questions before starting the rest of the assignment. If you submit the Reading Comprehension quenstions by Sunday night (by 11:59 PM), you will earn extra credit. As usual, you can download the questions.

  1. The first step in this assignment is to learn the standard higher-order functions on lists, which you will use a lot. Suppose you need a list, or a Boolean, or a function—what can you call?

    Review sections 2.7.2, 2.8.1, and 2.8.2. Now consider each of the following functions:

      map  filter  exists?  all?  curry  uncurry  foldl  foldr

    Put each function into exactly one of the following four categories:

    (B) Always returns a Boolean
    (F) Always returns a function
    (L) Always returns a list
    (A) Can return anything (including a Boolean, a function, or a list)

    After each function, write (B), (F), (L), or (A):

     map  
    
     filter  
    
     exists?  
    
     all?  
    
     curry  
    
     uncurry  
    
     foldl  
    
     foldr
  2. Here are the same functions again:

      map  filter  exists?  all?  curry  uncurry  foldl  foldr

    For each function, say which of the following five categories best describes it. Pick the most specific category (e.g., (S) is more specific than (L) or (M), and all of these are more specific than (?)).

    (S) Takes a list & a function; returns a list of exactly the same size
    (L) Takes a list & a function; returns a list of at least the same size
    (M) Takes a list & a function; returns a list of at most the same size
    (?) Might return a list
    (V) Never returns a list

    After each function, write (S), (L), (M), (?), or (V):

     map  
    
     filter  
    
     exists?  
    
     all?  
    
     curry  
    
     uncurry  
    
     foldl  
    
     foldr
  3. Here are the same functions again:

      map  filter  exists?  all?  curry  uncurry  foldl  foldr

    Put each function into exactly one of the following categories. Always pick the most specific category (e.g. (F2) is more specific than (F)).

    (F) Takes a single argument: a function
    (F2) Takes a single argument: a function that itself takes two arguments
    (+) Takes more than one argument

    After each function, write (F), (F2), or (+):

     map  
    
     filter  
    
     exists?  
    
     all?  
    
     curry  
    
     uncurry  
    
     foldl  
    
     foldr

    You are now ready to tackle most parts of exercise 29.

  4. Review the difference between foldr and foldl in section 2.8.1. You may also find it helpful to look at their implementations in section 2.8.3, which starts on page 131; the implementations are at the end.

    1. Do you expect (foldl + 0 '(1 2 3)) and (foldr + 0 '(1 2 3)) to be the same or different?

    2. Do you expect (foldl cons '() '(1 2 3)) and (foldr cons '() '(1 2 3)) to be the same or different?

    3. Look at the initial basis, which is summarized on 99. Give one example of a function, other than + or cons, that can be passed as the first argument to foldl or foldr, such that foldl always returns exactly the same result as foldr.

    4. Give one example of a function, other than + or cons, that can be passed as the first argument to foldl or foldr, such that foldl may return a different result from foldr.

    You are now ready to tackle all parts of exercises 29 and 30.

  5. Read the third Lesson in Program Design: Higher-Order Functions. The lesson mentions a higher-order function flip, which can convert < into >, among other tricks. Write as many algebraic laws as are needed to specify flip:

  6. Review function composition and currying, as described in section 2.7.2, which starts on page 127. Then judge the proposed properties below, which propose equality of functions, according to these rules:

    • Assume that names curry, o, <, *, cons, even?, and odd? have the definitions you would expect, but that m may have any value.

    • Each property proposes to equate two functions. If the functions are equal—which is to say, when both sides are applied to an argument, they always produce the same result—then mark the property Good. But if there is any argument on which the left-hand side produces different results from the right, mark the property Bad.

    Mark each property Good or Bad:

    ((curry <) m)     == (lambda (n) (< m n))
    
    ((curry <) m)     == (lambda (n) (< n m))
    
    ((curry cons) 10) == (lambda (xs) (cons 10 xs))
    
    (o odd?  (lambda (n) (* 3 n))) == odd?
    
    (o even? (lambda (n) (* 4 n))) == even?

    You are now ready to tackle the first three parts of exercise 38, as well as problem M below.

  7. Read about association lists in section 2.3.8, which starts on page 107. Given the definition

    (val mascots 
       '((Tufts Jumbo) (MIT Beaver) (Northeastern Husky) (BU Terrier)))

    Say what is the value of each of these expressions:

    (find 'Tufts   mascots)
    (find 'MIT     mascots)
    (find 'Harvard mascots)
    (find 'MIT (bind 'MIT 'Engineer mascots))

    You are ready to use association lists in the V family of problems below.

Programming and Proof (90 percent)

Overview

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:

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?:

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

-> ((ordered-by? <) '(1 2 3))
#t
-> ((ordered-by? <=) '(1 2 3))
#t
-> ((ordered-by? <) '(3 2 1)) 
#f
-> ((ordered-by? >=) '(3 2 1))
#t
-> ((ordered-by? >=) '(3 3 3))
#t
-> ((ordered-by? =) '(3 3 3)) 
#t

Hints:

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 COMP 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?

The web version of this homework displays an example web form for COMP 105 regrades. 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).

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 appears on the next page.)

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
                  (faults/none)
      (bind       'badsubmit
                   (faults/both (faults/equal 'badsubmit_asst '...)
                                (faults/equal 'info #f))
        (bind     'badgrade
                  (faults/both
                      (faults/equal 'badgrade_asst '...)
                      (faults/both
                         (faults/equal 'info #f)
                         (faults/equal 'problem #f)))
          (bind   'recitation
                  (faults/both
                      (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:

Related reading:

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 four files:

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
Structure

• 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
Correctness

• 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).

Performance

• 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

• 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.