Higher-Order Functions

COMP 105 Assignment

Due Tuesday, October 2, 2018 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 have migrated into 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 set you up with 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*.

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.

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. Submission 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. 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 14.

  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 133; 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 159. 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 14 and 15.

  5. Review the handout “Program Design with Higher-Order Functions”. The handout 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 128. 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 19, as well as problem M below.

Programming and Proof (90 percent)

Overview

For this assignment, you will do Exercises 14 (b-f,h,j), 15, and 19, from pages 212 to 216 of Build, Prove, and Compare, plus the exercises A, F, G1, G2, G3, M, and O below.

A summary of the initial basis can be found on page 159. A copy was handed out in class—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

14. Higher-order functions. Do exercise 14 on page 212 of Build, Prove, and Compare, parts (b) to (f), part (h), and part (j). Note which functions accept only nonempty lists, and code accordingly. You must not use recursion—solutions using recursion will receive No Credit. This restriction applies only to code you write. For example, gcd, which is defined in the initial basis, may use recursion.

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 sections 2.8.1 and 2.8.2 starting on page 131. For material on curry, see section 2.7.2, which starts on page 128.

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.

15. Higher-order functions. Do exercise 15 on page 214. You must not use recursion—solutions using recursion will receive No Credit. As above, this restriction applies only to code 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 credit2 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 131. You may also find it helpful to study the implementations of foldl and foldr in section 2.8.3, which starts on page 133; the implementations are at the end. Information on lambda can be found in section 2.7, on pages 121 to 124.

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

19. Functions as values. Do exercise 19 on page 216 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 (procedure? 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-empty 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 121, 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 135.

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.

Relating imperative code to functional code

A. Good functional style. The Impcore-with-locals function

    (define f-imperative (y) (locals x)
      (begin 
        (set x e)
        (while (p? x y) 
           (set x (g x y)))
        (h x y)))

is in a typical imperative style, with assignment and looping. Write an equivalent μScheme function f-functional that doesn’t use the imperative features begin (sequencing), while (goto), and set (assignment).

Hint #1: If you have trouble getting started, rewrite while to use if and goto. Now, what is like a goto?

Hint #2: (set x e) binds the value of e to the name x. What other ways do you know of binding the value of an expression to a name?

Don’t be confused about the purpose of this exercise. The exercise is a thought experiment. We don’t want you to write and run code for some particular choice of g, h, p?, e, x, and y. Instead, we want you write a function that works the same as f-imperative given any choice of g, h, p?, e, x, and y. So for example, if f-imperative would loop forever on some inputs, your f-functional must also loop forever on exactly the same inputs.

Once you get your mind twisted in the right way, this exercise should be easy. The point of the exercise is not only to show that you can program without imperative features, but also to help you develop a technique for eliminating such features.

Related reading: No part of the book bears directly on this question. You’re better off reviewing your experience with recursive functions and perhaps the solutions for the Scheme assignment.

Laws: This problem doesn’t need laws.

A function that returns a function

F. The handout “Program Design with 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.

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

Graph problems

From COMP 15, you should be familiar with graphs and graph algorithms. In the next few problems you will work with an immutable representation of directed graphs: a graph is represented by an association list in which each node is associated with a list of its immediate successors. This representation is called a successors map. (It is a close cousin to the widely used “adjacency list.”)
For example, the ASCII-art graph

    A --> B --> C
    |           ^
    |           |
    +-----------+

could be represented as a successors map by '([A (B C)] [B (C)] [C ()]).

The successors map, while it is an association list, a list of ordinary S-expressions, and an association list, is best treated as its own form of data. When writing algebraic laws, treat every successors map as one of two cases:

You can tell these two cases apart using null?. When you have the second case, you can extract node and successors using predefined functions alist-first-key and alist-first-attribute, and you can extract graph using cdr.

Note: The graph problems below can be solved using only first-order functions. But you will find the problems much easier if you use let, lambda, and either of the fold functions.

Related reading: The previous assignment. The definitions of equal? in section 2.3.1 (basic recursive functions on lists). Material on association lists in section 2.3.6.

Laws: In each of the graph problems, you must write algebraic laws for every function that does case analysis (and only for those functions). You must write such laws even for helper functions and for inner functions defined using lambda. Put all laws, even laws for inner functions, in comments above the entire function definition.

When you define laws for graph functions, you must treat the successors map as one of the two forms of data listed above. Writing cons in graph-function laws is not acceptable.

G1. List of edges. An edge is represented by a record

(make-edge N1 N2)

where N1 and N2 are nodes. Define make-edge using the following record definition:

(record edge [from to])

Function edge-list consumes a graph represented as a successors map and returns a list of all the edges in the graph. Edges may be listed in any order. For example, here are acceptable responses for a list of the edges in the graph pictured above:

(list3 (make-edge A B) (make-edge B C) (make-edge A C))
(list3 (make-edge A B) (make-edge A C) (make-edge B C))

Define function edge-list. Algebraic laws are optional, but unit tests are required.

Hints:

G2. Graph-building: adding an edge. Function add-edge takes two arguments: an edge made with make-edge and a graph that is represented as a successors map. It returns a new graph that is like the original, except that the new graph has had the given edge added to it. Depending on whether the from node already appears in the graph, it may have to be added. (Determine its appearance using equal?.) In the new graph, both the from and to nodes should be associated with their successors, even if the list of successors is empty.

For any edge e and graph g, function add-edge satisfies this algebraic law:

(permutation? (cons e (edge-list g))
              (edge-list (add-edge e g)))

Implement add-edge, and in addition, write the following:

You may include the implementation of permutation? from the solutions to the previous homework.

Here are our requirements for algebraic laws:

Hint: We know of at least two entirely different ways of coding add-edge:

We recommend coding the first way, and we recommend avoiding recursion.

G3. Graph update: removing a node. Calling (remove-node node graph) returns a new graph that is like graph, but with all references to node removed:

If the original graph does not mention node, then (remove-node node graph) returns a new graph that is equal? to the original.

Implement remove-node. For full credit, implement remove-node without using any recursion. If you do choose to use recursion, specify each recursive function by giving algebraic laws.

Calculational reasoning about functions

M. Reasoning about higher-order functions. Using the calculational techniques from Section 2.4.5, which starts on page 110, 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.

Take the following properties as given:

   ((o f g) x) == (f (g x))        ; apply-compose law
   (((curry f) x) y) == (f x y)    ; apply-curried law

Using these properties should keep your proof relatively simple.

Related reading: Section 2.4.5. 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 take as given any algebraic law in the textbook (they start on page 108), in recitation, or in the lecture notes. (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 135. Especially the polymorphic sort in section 2.9.2—the lt? parameter to that function is an example of a transitive relation. Section 2.7.2. Example uses of map in section 2.8.1. The definition of map in section 2.8.3.

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.

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)))

I’ve changed the name of the internal function, but the only other things that are different is that I have removed the formal parameter from the lambda and I have removed the second actual parameter from the call sites. I 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 functions length and so on in exercise 15. 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.

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’ll evaluate the correctness of your translation in problem A. We’ll also want appropriate list operations to take constant time.

Exemplary Satisfactory Must improve
Correctness

• The translation in problem A is correct.

• Your code passes every one of our stringent tests.

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

• The translation in problem A is almost correct, but an easily identifiable part is missing.

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

• The translation in problem A is obviously incorrect,

• Or course staff cannot understand the translation in problem A.

• 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.

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


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