Lab 11: Macros, Tail Recursion, Regular Expressions

Due by 11:59pm on Tuesday, August 3.

Starter Files

Download Inside the archive, you will find starter files for the questions in this lab, along with a copy of the Ok autograder.


Consult this section if you need a refresher on the material for this lab. It's okay to skip directly to the questions and refer back here should you get stuck.

So far we've been able to define our own procedures in Scheme using the define special form. When we call these procedures, we have to follow the rules for evaluating call expressions, which involve evaluating all the operands.

We know that special form expressions do not follow the evaluation rules of call expressions. Instead, each special form has its own rules of evaluation, which may include not evaluating all the operands. Wouldn't it be cool if we could define our own special forms where we decide which operands are evaluated? Consider the following example where we attempt to write a function that evaluates a given expression twice:

scm> (define (twice f) (begin f f))
scm> (twice (print 'woof))

Since twice is a regular procedure, a call to twice will follow the same rules of evaluation as regular call expressions; first we evaluate the operator and then we evaluate the operands. That means that woof was printed when we evaluated the operand (print 'woof). Inside the body of twice, the name f is bound to the value undefined, so the expression (begin f f) does nothing at all!

The problem here is clear: we need to prevent the given expression from evaluating until we're inside the body of the procedure. This is where the define-macro special form, which has identical syntax to the regular define form, comes in:

scm> (define-macro (twice f) (list 'begin f f))

define-macro allows us to define what's known as a macro, which is simply a way for us to combine unevaluated input expressions together into another expression. When we call macros, the operands are not evaluated, but rather are treated as Scheme data. This means that any operands that are call expressions or special form expression are treated like lists.

If we call (twice (print 'woof)), f will actually be bound to the list (print 'woof) instead of the value undefined. Inside the body of define-macro, we can insert these expressions into a larger Scheme expression. In our case, we would want a begin expression that looks like the following:

(begin (print 'woof) (print 'woof))

As Scheme data, this expression is really just a list containing three elements: begin and (print 'woof) twice, which is exactly what (list 'begin f f) returns. Now, when we call twice, this list is evaluated as an expression and (print 'woof) is evaluated twice.

scm> (twice (print 'woof))

To recap, macros are called similarly to regular procedures, but the rules for evaluating them are different. We evaluated lambda procedures in the following way:

  • Evaluate operator
  • Evaluate operands
  • Apply operator to operands, evaluating the body of the procedure

However, the rules for evaluating calls to macro procedures are:

  • Evaluate operator
  • Apply operator to unevaluated operands
  • Evaluate the expression returned by the macro in the frame it was called in.

Recall from lecture that Scheme supports tail-call optimization. The example we used was factorial (shown in both Python and Scheme):
# Python
def fact(n):
    if n == 0:
        return 1
    return n * fact(n - 1)

# Scheme
(define (fact n)
    (if (= n 0)
        (* n (fact (- n 1)))))

Notice that in this version of factorial, the return expressions both use recursive calls, and then use the values for more "work." In other words, the current frame needs to sit around, waiting for the recursive call to return with a value. Then the current frame can use that value to calculate the final answer.

As an example, consider a call to fact(5) (Shown with Scheme below). We make a new frame for the call, and in carrying out the body of the function, we hit the recursive case, where we want to multiply 5 by the return value of the call to fact(4). Then we want to return this product as the answer to fact(5). However, before calculating this product, we must wait for the call to fact(4). The current frame stays while it waits. This is true for every successive recursive call, so by calling fact(5), at one point we will have the frame of fact(5) as well as the frames of fact(4), fact(3), fact(2), and fact(1), all waiting for fact(0).

(fact 5)
(* 5 (fact 4))
(* 5 (* 4 (fact 3)))
(* 5 (* 4 (* 3 (fact 2))))
(* 5 (* 4 (* 3 (* 2 (fact 1)))))
(* 5 (* 4 (* 3 (* 2 (* 1 (fact 0))))))
(* 5 (* 4 (* 3 (* 2 (* 1 1)))))
(* 5 (* 4 (* 3 (* 2 1))))
(* 5 (* 4 (* 3 2)))
(* 5 (* 4 6))
(* 5 24)

Keeping all these frames around wastes a lot of space, so our goal is to come up with an implementation of factorial that uses a constant amount of space. We notice that in Python we can do this with a while loop:

def fact_while(n):
    total = 1
    while n > 0:
        total = total * n
        n = n - 1
    return total

However, Scheme doesn't have for and while constructs. No problem! Everything that can be written with while and for loops and also be written recursively. Instead of a variable, we introduce a new parameter to keep track of the total.

def fact(n):
    def fact_optimized(n, total):
        if n == 0:
            return total
        return fact_optimized(n - 1, total * n)
    return fact_optimized(n, 1)

(define (fact n)
    (define (fact-optimized n total)
        (if (= n 0)
            (fact-optimized (- n 1) (* total n))))
    (fact-optimized n 1))

Why is this better? Consider calling fact(n) on based on this definition:

(fact 5)
(fact-optimized 5   1)
(fact-optimized 4   5)
(fact-optimized 3  20)
(fact-optimized 2  60)
(fact-optimized 1 120)
(fact-optimized 0 120)

Because Scheme supports tail-call optimization (note that Python does not), it knows when it no longer needs to keep around frames because there is no further calculation to do. Looking at the last line in fact_optimized, we notice that it returns the same thing that the recursive call does, no more work required. Scheme realizes that there is no reason to keep around a frame that has no work left to do, so it just has the return of the recursive call return directly to whatever called the current frame.

Therefore the last line in fact_optimized is a tail-call.

Regular Expressions

Regular expressions are a way to describe sets of strings that meet certain criteria, and are incredibly useful for pattern matching.

The simplest regular expression is one that matches a sequence of characters, like aardvark to match any "aardvark" substrings in a string.

However, you typically want to look for more interesting patterns. We recommend using an online tool like for trying out patterns, since you'll get instant feedback on the match results.

Character classes

A character class makes it possible to search for any one of a set of characters. You can specify the set or use pre-defined sets.

Class Description
[abc] Matches a, b, or c
[a-z] Matches any character between a and z
[^A-Z] Matches any character that is not between A and Z.
\w Matches any "word" character. Equivalent to [A-Za-z0-9_]
\d Matches any digit. Equivalent to [0-9].
[0-9] Matches a single digit in the range 0 - 9. Equivalent to \d
\s Matches any whitespace character (spaces, tabs, line breaks).
. Matches any character besides new line.

Character classes can be combined, like in [a-zA-Z0-9].

Combining patterns

There are multiple ways to combine patterns together in regular expressions.

Combination Description
AB A match for A followed immediately by one for B. Example: x[.,]y matches "x.y" or "x,y"
A⎮B Matches either A or B. Example: \d+⎮Inf matches either a sequence containing 1 or more digits or "Inf"

A pattern can be followed by one of these quantifiers to specify how many instances of the pattern can occur.

Quantifier Description
* 0 or more occurrences of the preceding pattern. Example: [a-z]* matches any sequence of lower-case letters or the empty string.
+ 1 or more occurrences of the preceding pattern. Example: \d+ matches any non-empty sequence of digits.
? 0 or 1 occurrences of the preceding pattern. Example: [-+]? matches an optional sign.
{1,3} Matches the specified quantity of the preceding pattern. {1,3} will match from 1 to 3 instances. {3} will match exactly 3 instances. {3,} will match 3 or more instances. Example: \d{5,6} matches either 5 or 6 digit numbers.


Parentheses are used similarly as in arithmetic expressions, to create groups. For example, (Mahna)+ matches strings with 1 or more "Mahna", like "MahnaMahna". Without the parentheses, Mahna+ would match strings with "Mahn" followed by 1 or more "a" characters, like "Mahnaaaa".


  • ^

    • Matches the beginning of a string. Example: ^(I⎮You) matches I or You at the start of a string.
  • $

    • Normally matches the empty string at the end of a string or just before a newline at the end of a string. Example: (\.edu|\.org|\.com)$ matches .edu, .org, or .com at the end of a string.
  • \b

    • Matches a "word boundary", the beginning or end of a word. Example: s\b matches s characters at the end of words.

Special characters

The following special characters are used above to denote types of patterns:

\ ( ) [ ] { } + * ? | $ ^ .

That means if you actually want to match one of those characters, you have to escape it using a backslash. For example, \(1\+3\) matches "(1 + 3)".

Using regular expressions in Python

Many programming languages have built-in functions for matching strings to regular expressions. We'll use the [Python re module] in 61A, but you can also use similar functionality in SQL, JavaScript, Excel, shell scripting, etc.

The search method searches for a pattern anywhere in a string:"(Mahna)+", "Mahna Mahna Ba Dee Bedebe")

That method returns back a match object, which is considered truth-y in Python and can be inspected to find the matching strings.

For more details, please consult the re module documentation or the re tutorial.

Required Questions


Q1: WWSD: Macros

One thing to keep in mind when doing this question, builtins get rendered as such:

scm> +
scm> list

If evaluating an expression causes an error, type SchemeError. If nothing is displayed, type Nothing.

Use Ok to test your knowledge with the following "What Would Scheme Display?" questions:

python3 ok -q wwsd-macros -u
scm> +
scm> list
scm> (define-macro (f x) (car x))
scm> (f (2 3 4)) ; type SchemeError for error, or Nothing for nothing
scm> (f (+ 2 3))
scm> (define x 2000)
scm> (f (x y z))
scm> (f (list 2 3 4))
scm> (f (quote (2 3 4)))
scm> (define quote 7000)
scm> (f (quote (2 3 4)))
scm> (define-macro (g x) (+ x 2))
scm> (g 2)
scm> (g (+ 2 3))
scm> (define-macro (h x) (list '+ x 2))
scm> (h (+ 2 3))
scm> (define-macro (if-else-5 condition consequent) `(if ,condition ,consequent 5))
scm> (if-else-5 #t 2)
scm> (if-else-5 #f 3)
scm> (if-else-5 #t (/ 1 0))
scm> (if-else-5 #f (/ 1 0))
scm> (if-else-5 (= 1 1) 2)

Q2: Scheme def

Implement def, which simulates a python def statement, allowing you to write code like (def f(x y) (+ x y)).

The above expression should create a function with parameters x and y, and body (+ x y), then bind it to the name f in the current frame.

Note: the previous is equivalent to (def f (x y) (+ x y)).

Hint: We strongly suggest doing the WWPD questions on macros first as understanding the rules of macro evaluation is key in writing macros.

(define-macro (def func args body)

Use Ok to test your code:

python3 ok -q scheme-def

Tail Recursion

Q3: Replicate

Write a tail-recursive function that returns a list with x repeated n times.

scm> (tail-replicate 3 10)
(3 3 3 3 3 3 3 3 3 3)
scm> (tail-replicate 5 0)
scm> (tail-replicate 100 5)
(100 100 100 100 100)
(define (tail-replicate x n)

Use Ok to test your code:

python3 ok -q tail-replicate

Q4: Exp

We want to implement the exp procedure. So, we write the following recursive procedure:

(define (exp-recursive b n)
  (if (= n 0)
      (* b (exp-recursive b (- n 1)))))

Try to evaluate

(exp-recursive 2 (exp-recursive 2 10))

You will notice that it will cause a maximum recursion depth error. To fix this, we need to use tail recursion! Implement the exp procedure using tail recursion:

(define (exp b n)
  ;; Computes b^n.
  ;; b is any number, n must be a non-negative integer.

Use Ok to test your code:

python3 ok -q exp

Regular Expressions

Q5: Roman Numerals

Write a regular expression that finds any string of letters that resemble a Roman numeral and aren't part of another word. A Roman numeral is made up of the letters I, V, X, L, C, D, M and is at least one letter long.

import re

def roman_numerals(text):
    Finds any string of letters that could be a Roman numeral
    (made up of the letters I, V, X, L, C, D, M).

    >>> roman_numerals("Sir Richard IIV, can you tell Richard VI that Richard IV is on the phone?")
    ['IIV', 'VI', 'IV']
    >>> roman_numerals("My TODOs: I. Groceries II. Learn how to count in Roman IV. Profit")
    ['I', 'II', 'IV']
    >>> roman_numerals("I. Act 1 II. Act 2 III. Act 3 IV. Act 4 V. Act 5")
    ['I', 'II', 'III', 'IV', 'V']
    >>> roman_numerals("Let's play Civ VII")
    >>> roman_numerals("i love vi so much more than emacs.")
    >>> roman_numerals("she loves ALL editors equally.")
    return re.findall(__________, text)

Use Ok to test your code:

python3 ok -q roman_numerals

Q6: Calculator Ops

Write a regular expression that parses strings written in the 61A Calculator language and returns any expressions which have two numeric operands, leaving out the parentheses around them.

import re

def calculator_ops(calc_str):
    Finds expressions from the Calculator language that have two
    numeric operands and returns the expression without the parentheses.

    >>> calculator_ops("(* 2 4)")
    ['* 2 4']
    >>> calculator_ops("(+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))")
    ['* 2 4', '+ 3 5', '- 10 7']
    >>> calculator_ops("(* 2)")
    return re.findall(__________, calc_str)

Use Ok to test your code:

python3 ok -q calculator_ops


Make sure to submit this assignment by running:

python3 ok --submit