How Scheme Evaluates Expressions (take 1)

Summary

We consider the algorithm and structures that the Scheme interpreter users to evaluate Scheme expressions.

Introduction

To many Scheme programmers, the Scheme environment is a bit of a “black box”: You type an expression, the Scheme interpreter chugs along for a few milliseconds (or perhaps longer) and then spits out an answer. It is certainly possible to use Scheme without knowing anything more about the Scheme interpreter. However, we have found that students use Scheme much better when they know a little bit about the way in which Scheme interprets expressions. As importantly, when things go wrong, knowing a bit about the interpreter can help you resolve problems.

Clearly, it’s not appropriate to cover every aspect of the Scheme interpreter this early in your experience of Scheme. Hence, we will start with a simple description of the Scheme interpreter and, when you learn new parts of Scheme, we’ll add more information about the interpreter.

Right now, you know how to enter four kinds of expressions in Scheme:

• Simple values: The building blocks of more complex expressions. Things like 2.4 and "hello".
• Procedure calls: Computations that involves applying some procedure (such as addition) to some values (such as 1, 2, and 3).
• Definitions: The association of a name, such as homework1, with a value, such as 91, using the define keyword.
• Names of values: A name given by a prior definition, like the homework1 just described, or built-in to Scheme, such as pi.

We will consider how Scheme evaluates each in turn.

Structures

When the Scheme interpreter evaluates expressions, it relies on a data structure to keep track of information. In particular, a dictionary associates names with values. The Scheme interpreter uses the dictionary to look up each name that it encounters. The Scheme interpreter updates the dictionary when it encounters a define expression.

Evaluating the Four Kinds of Expressions

Evaluating Simple Values

The evaluation strategy for simple expressions is fairly straightforward: The Scheme interpreter simply converts the human-readable expression to some internal representation. If the simple expression is all that’s been typed, it then converts that internal representation back to human-readable form and prints it out.

How does the Scheme interpreter know that it has a simple value to evaluate? If the interpreter sees a digit (0-9) or a minus-sign, it knows that the value is a number, and reads the remaining parts of the number. If the interpreter sees a quotation mark (“), it knows that it has a string and reads everything else until the ending quotation mark. Since those are the only kinds of simple values we know about right now, those are the only ones we discuss here.

Evaluating Procedure Calls

Evaluating procedure calls is clearly a bit more difficult. In general, the Scheme interpreter can’t apply a procedure to some arguments until it has evaluated the arguments. Hence, it evaluates all the arguments (using the same algorithm) and then applies the procedure.

For example, to evaluate (+ (* 2 3) (* 4 5)), the interpreter must first evaluate (* 2 3) and (* 4 5). The results of those evaluations are 6 and 20, which it can then add, giving 26. If the addition was the outermost operation, it can then print the result. If the addition expression is part of a larger expression, such as (* 0.25 (+ (* 2 3) (* 4 5))), it uses the 26 in evaluating the enclosing expression.

How does the interpreter know that it has a compound expression to evaluate? It sees an open parenthesis. Unless the first thing after the open parenthesis is the keyword define (or one of a few other special keywords you’ll learn about later), it assumes that the first thing after the open parenthesis is a procedure, the remaining things are the arguments to that procedure.

Evaluating Definitions

Definitions are a special case of compound expressions in which the procedure is define. To evaluate a definition, the interpreter must evaluate the associated expression. Once it is done evaluating that expression, it puts the name and the value of the expression into the dictionary. For example, given (define x (+ 2 3)), the interpreter evaluates (+ 2 3), giving 5, and then puts the entry (x . 5) in the dictionary. (We’ll represent dictionary entries with an open parenthesis, the name, a period, the value, and a close parenthesis.)

Evaluating Names

As the definition of the dictionary suggests, the evaluation strategy for names is pretty straightforward: The Scheme interpreter simply looks them up in the dictionary.

Some Examples

Let’s consider a few examples. Suppose we start with the empty dictionary and evaluate the expression (+ 22 3).

• The expression begins with an open parenthesis, so it is either a procedure call or a definition.
• The next thing is not the keyword define, so the whole expression is a procedure call. The procedure will we apply is +, the addition procedure.
• We need to evaluate each argument.
  • The first argument begins with 2. 2 is a digit, and therefore starts a number. We read the number 22, and we’re done with that argument.
  • The second argument begins with 3. 3 is a digit, and therefore starts a number. We read the number 3, and we’re done with that argument.
  • The next thing is a close parenthesis, so we know that we’re done with the argument list.
• We now apply the procedure (that is, +) to the two arguments, (that is, 22 and 3). The result is 25.
• We print the result: 25

Now, let’s suppose we had a somewhat more complex expression, such as (* pi (expt (/ diameter 2) 2)). As you might note, that expression computes the area of a circle given its diameter.

• The expression begins with an open parenthesis, so it is either a procedure call or a definition.
• The next thing is not the keyword define, so the whole expression is a procedure call. The procedure we will apply is *.
• We need to evaluate each argument.
  • The first argument begins with a letter. It is therefore a name of some sort. We read the whole name, which is pi.
  • We look up pi in the definitions table. If we are lucky, it’s been predefined. Let’s see … it’s defined as 3.14159....
  • The next argument begins with an open parenthesis, so it is either a procedure call or a definition.
  • The next thing after the open parenthesis is not the keyword define (and it probably shouldn’t be a definition in the middle of an expression), so the subexpression is a procedure call. The procedure we will apply is expt.
  • We need to evaluate the arguments to expt.
    • The first argument begins with an open parenthesis, so it is either a procedure call or a definition.
    • The next thing is not the keyword define, so the whole subexpression is a procedure call. The procedure we will apply is / (division).
    • We need to evaluate the arguments.
      • The first argument begins with a letter. It is therefore a name. In this case, it is the name diameter.
      • We look up diameter in the table. Hmmm … it’s not a predefined value (which is probably good, since it is likely that programs will want to use only one diameter). But we haven’t defined it either. Boom! It is clearly impossible to continue evaluating the expression, so we stop.
      • It’s frustrating to do all the preparatory work for no result, isn’t it? That’s one of the reasons computers are malicious.

Well, it seems we should have defined diameter first. Let’s do so. (define diameter 10)

• The expression begins with an open parenthesis. It is either a definition or a procedure call.
• The next thing to appear is the keyword define. It’s a definition. The name we will be defining is diameter.
• We now need to evaluate the expression.
  • The expression begins with the digit 1, so it must be a number. We read the number 10, and we’re done.
• We add the entry (diameter . 10) to the dictionary.

Okay, let’s go back and explore the previous expression, (* pi (expt (/ diameter 2) 2)).

• The expression begins with an open parenthesis, so it is either a procedure call or a definition.
• The next thing is not the keyword define, so the whole expression is a procedure call. The procedure we will apply is *.
• We need to evaluate each argument.
  • The first argument begins with a letter. It is therefore a name of some sort. We read the whole name, which is pi.
  • We look up pi in the definitions table. If we are lucky, it’s been predefined. Let’s see … it’s defined as 3.14159....
  • The next argument begins with an open parenthesis, so it is either a procedure call or a definition.
  • The next thing after the open parenthesis is not the keyword define, so the whole subexpression is a procedure call. The procedure we will apply is expt.
  • We need to evaluate the arguments to expt.
    • The first argument begins with an open parenthesis, so it is either a procedure call or a definition.
    • The next thing is not the keyword define, so the whole expression is a procedure call. The procedure we will apply is / (division).
    • We need to evaluate the arguments.
      • The first argument begins with a letter. It is therefore a name. In this case, it is the name diameter.
      • We look up diameter in the table. Since we just put (diameter . 10) into the dictionary, we can be fairly confident that we’ll find that entry, and can therefore use the value 10.
      • The next argument begins with 2, which is a digit. We read the complete number, with is 2.
      • The next thing to appear is a close parenthesis. We must be done with the arguments to the / procedure..
    • We’re now ready to divide 10 by 2. The result is 5.
    • As you may recall, we’ve been processing the arguments to expt. Do any remain?
    • The next thing we see is a digit, 2. We read the rest of the number, which is 2.
    • The next thing we see is a close parenthesis. We’re done with the arguments. (That’s a good thing, since expt only has two parameters.)
  • We compute 5 squared, which is 25.
  • Okay, we’re now back to processing the parameter list to the outermost multiplication. So far, we’ve seen pi, which we found was 3.14159, and an expression, which we just found was 25.
  • The next thing left is a close paren, which marks the end of the expression.
• We can finally do the multiplication: 3.14159 * 25 is 78.53975

The Evaluation Algorithm

Here’s a simplified version of the algorithm that the Scheme interpreter seems to follow.

Look at the next non-space character
If the next non-space character is an open parenthesis
  Look at the next thing.
  If the next thing is the keyword define
    Read the next thing (a name to define)
    Read the next thing (an expression)
    Evaluate the expression, giving a value
    Add a (name . value) entry to the dictionary
  Otherwise the next thing must be a function
    Read and evaluate each argument
    Apply the function to the evaluated arguments
Otherwise, see if the next non-space character is a digit (or + or -)
  Read all the parts of a number
  The number is the result
Otherwise, see if the next non-space character is a letter
  Read everything up to the next space (or close paren).  
  Look up the thing just read in the dictionary.
  If it's not there, crash and burn
  Otherwise, the result is the value found in the dictionary

A Slight Complexity: Naming Functions

Of course, things are not quite as simple as the discussion above suggests (and it’s not even clear that the discussion above suggests things are simple). For example, there are a host of kinds of Scheme expressions that we have not yet explored. We will consider those in future discussions of Scheme’s evaluation strategies.

However, we’ve left off one thing that’s relevant to the Scheme you already know. As you may have seen, Scheme lets you write definitions for functions, such as (define multiply *). What does that do to our evaluation strategy? Primarily, it means that we have to check if the thing we see after an open parenthesis is a defined name and, if so, get its definition from the dictionary.

In fact, Scheme looks up the procedure, even if you’re using one of the standard procedures. Hence, if you use + or sqrt, it still looks it up in the table, where it finds the internal representation of the corresponding procedure. You can tell that Scheme looks up these standard procedures by redefining them. For example, (define + -) will make your program behave very strangely thereafter.

Self Checks

Check 1: Handling Expressions

a. What does Scheme do to evaluate a simple value, such as a number?

b. What does Scheme do evaluate a name, such as pi?

c. What two steps does Scheme need to do to evaluate a procedure call, such as (quotient 7 2)?

d. What two steps does Scheme need to do to evaluate a definition, such as (define today "Tuesday")?

Check 2: Evaluation Algorithm

What are the roughly seven steps Scheme will take to evaluate the following expression?

(define average (/ (+ 7 8 9) 3))