Discussion 11: Interpreters
Calculator
An interpreter is a program that understands other programs. Today, we will explore how to build an interpreter for Calculator, a simple language that uses a subset of Scheme syntax.
The Calculator language includes only the four basic arithmetic operations: +
, -
, *
, and /
. These operations can be nested and can take any numbers of arguments. A few examples of calculator expressions and their corresponding values are shown below.
calc> (+ 2 2)
4
calc> (- 5)
-5
calc> (* (+ 1 2) (+ 2 3))
15
The reader component of an interpreter parses input strings and represents them as data structures in the implementing language. In this case, we need to represent Calculator expressions as Python objects. To represent numbers, we can just use Python numbers. To represent the names of the arithmetic procedures, we can use Python strings (e.g. '+'
).
To represent Scheme lists in Python, we will use the Pair
class. A Pair
instance holds exactly two elements. Accordingly, the Pair
constructor takes in two arguments, and to make a list we must nest calls to the constructor and pass in nil
as the second element of the last pair. Note that in the Python code, nil
is bound to a special user-defined object that represents an empty list, whereas nil
in Scheme is actually an empty list.
>>> Pair('+', Pair(2, Pair(3, nil)))
Pair('+', Pair(2, Pair(3, nil)))
Each Pair
instance has two instance attributes: first
and rest
, which are bound to the first and second elements of the pair respectively.
>>> p = Pair('+', Pair(2, Pair(3, nil)))
>>> p.first
'+'
>>> p.rest
Pair(2, Pair(3, nil))
>>> p.rest.first
2
Pair
is very similar to Link
, the class we developed for representing linked lists -- they have the same attribute names first
and rest
and are represented very similarly.
Here's an implementation of what we described:
class Pair:
"""Represents the built-in pair data structure in Scheme."""
def __init__(self, first, rest):
self.first = first
if not scheme_valid_cdrp(rest):
raise SchemeError("cdr can only be a pair, nil, or a promise but was {}".format(rest))
self.rest = rest
def map(self, fn):
"""Maps fn to every element in a list, returning a new
Pair.
>>> Pair(1, Pair(2, Pair(3, nil))).map(lambda x: x * x)
Pair(1, Pair(4, Pair(9, nil)))
"""
assert isinstance(self.rest, Pair) or self.rest is nil, \
"rest element in pair must be another pair or nil"
return Pair(fn(self.first), self.rest.map(fn))
def __repr__(self):
return 'Pair({}, {})'.format(self.first, self.rest)
class nil:
"""Represents the special empty pair nil in Scheme."""
def map(self, fn):
return nil
def __getitem__(self, i):
raise IndexError('Index out of range')
def __repr__(self):
return 'nil'
nil = nil() # this hides the nil class *forever*
Questions
Q1: Using Pair
Answer the following questions about a Pair
instance
representing the Calculator expression (+ (- 2 4) 6 8)
.
Write out the Python expression that returns a Pair
representing the given expression:
What is the operator of the call expression?
If the Pair
you constructed in the previous part was bound to the name p
,
how would you retrieve the operator?
What are the operands of the call expression?
If the Pair
you constructed was bound to the name p
, how would
you retrieve a list containing all of the operands?
How would you retrieve only the first operand?
Q2: New Procedure
Suppose we want to add the //
operation to our Calculator interpreter. Recall from Python that //
is the floor division operation, so we are looking to add a built-in procedure //
in our interpreter such that (// dividend divisor)
returns dividend // divisor. Similarly we handle multiple inputs as illustrated in the following example (// dividend divisor1 divisor2 divisor3)
evaluates to (((dividend // divisor1) // divisor2) // divisor3). For this problem you can assume you are always given at least 1 divisor. Also for this question do you need to call calc_eval
inside floor_div
? Why or why not?
calc> (// 1 1)
1
calc> (// 5 2)
2
calc> (// 28 (+ 1 1) 1)
14
Run in 61A Code
Q3: New Form
Suppose we want to add handling for comparison operators >
,
<
, and =
as well as and
expressions to our
Calculator interpreter. These should work the same way they do in Scheme.
calc> (and (= 1 1) 3)
3
calc> (and (+ 1 0) (< 1 0) (/ 1 0))
#f
i. Are we able to handle expressions containing the comparison operators
(such as <
, >
, or =
) with the existing implementation of calc_eval
?
Why or why not?
ii. Are we able to handle and
expressions with the existing
implementation of calc_eval
? Why or why not?
Hint: Think about the rules of evaluation we've implemented in
calc_eval
. Is anything different aboutand
?
iii. Now, complete the implementation below to handle and
expressions. You may assume the conditional operators (e.g. <
, >
,
=
, etc) have already been implemented for you.
Q4: Saving Values
In the last few questions we went through a lot of effort to add operations so we can do most arithmetic operations easily. However it's a real shame we can't store these values. So for this question let's implement a define
special form that saves values to variable names. This should work like variable assignment in Scheme; this means that you should expect inputs of the form(define <variable_name> <value>)
and these inputs should return the symbol corresponding to the variable name.
calc> (define a 1)
a
calc> a
1
This is a more involved change. Here are the 4 steps involved:
- Add a
bindings
dictionary that will store the names and correspondings values of variables as key-value pairs of the dictionary. - Identify when the define form is given to
calc_eval
. - Allow variables to be looked up in
calc_eval
. - Write the function
eval_define
which should actually handle adding names and values to the bindings dictionary.
We've done step 1 for you. Now you'll do the remaining steps in the code below.
Run in 61A CodeQ5: Counting Eval and Apply
How many calls to calc_eval
and calc_apply
would it
take to evaluate each of the following Calculator expressions?
scm> (+ 1 2)
For this particular prompt please list out the inputs to calc_eval
and calc_apply
.
scm> (+ 2 4 6 8)
scm> (+ 2 (* 4 (- 6 8)))
scm> (and 1 (+ 1 0) 0)
Q6: From Pair to Calculator
Write out the Calculator expression with proper syntax that
corresponds to the following Pair
constructor calls.
>>> Pair('+', Pair(1, Pair(2, Pair(3, Pair(4, nil)))))
>>> Pair('+', Pair(1, Pair(Pair('*', Pair(2, Pair(3, nil))), nil)))