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We saw how functions can be manipulated as data using higher-order functions. We also saw how data can be endowed with behavior using message passing and an object system. We have also studied techniques for organizing large programs, such as functional abstraction, data abstraction, class inheritance, and generic functions. These core concepts constitute a strong foundation upon which to build modular, maintainable, and extensible programs. This chapter focuses on the third fundamental element of programming:

## Chapter 3: The Structure and Interpretation of Computer Programs

We saw how functions can be manipulated as data using higher-order functions. We also saw how data can be endowed with behavior using message passing and an object system. We have also studied techniques for organizing large programs, such as functional abstraction, data abstraction, class inheritance, and generic functions. These core concepts constitute a strong foundation upon which to build modular, maintainable, and extensible programs. This chapter focuses on the third fundamental element of programming: A Python program is just a collection of text.

Only through the process of interpretation do we perform any meaningful computation based on that text. A programming language like Python is useful because we can define an interpreter, a program that carries out Python’s evaluation and execution procedures. It is no exaggeration to regard this as the most fundamental idea in programming, that an interpreter, which determines the meaning of expressions in a programming language, is just another program.

To appreciate this point is to change our images of ourselves as programmers. We come to see ourselves as designers of languages, rather than only users of languages designed by others. For example, the constraint propagator from the previous chapter has its own primitives and means of combination. The constraint language was quite specialized: As another example, the object system implemented in the previous chapter created a new language for expressing class and inheritance relationships.

While we have been designing languages of a sort already, the material of this chapter will greatly expand the range of languages we can interpret.

Programming languages vary widely in their syntactic structures, features, and domain of application. Among general purpose programming languages, the constructs of function definition and function application are pervasive. On the other hand, powerful languages exist that do not include an object system, higher-order functions, assignment, or even control constructs like while and for statements.

To illustrate just how different languages can be, we will introduce Scheme as an example of a powerful and expressive programming language that includes few built-in features. The subset of Scheme introduced here does not allow mutable values at all. In this chapter, we study the design of interpreters and the computational processes that they create when executing programs. The prospect of designing an interpreter for a general programming language may seem daunting.

After all, interpreters are programs that can carry out any possible computation, depending on their input. However, many interpreters have an elegant common structure: The first evaluates expressions in environments; the second applies functions to arguments. These functions are recursive in that they are defined in terms of each other: We covered recursive functions in Chapter 1, and in the next section of this chapter, we will focus on the evolution of recursive processes.

We will then turn to recursive data structures, which will prove essential to understanding the design of an interpreter. The end of the chapter focuses on three new languages and the task of implementing interpreters for them.

It specifies how each stage of the process is built upon the previous stage. We would like to be able to make statements about the overall behavior of a process whose local evolution has been specified by one or more functions. This analysis is very difficult to do in general, but we can at least try to describe some typical patterns of process evolution. In Chapter 1, we saw some common “shapes” for processes generated by simple functions, such as linear and tree recursion.

In this section, we will investigate the rates at which these processes consume the important computational resources of time and space. In Chapter 1, we saw multiple implementations of a function to compute Fibonacci numbers. The recursive version was as follows: However, consider the pattern of computation that results from evaluating fib 6 , shown below. To compute fib 6 , we compute fib 5 and fib 4. To compute fib 5 , we compute fib 4 and fib 3.

This recursive implementation is a terribly inefficient way to compute Fibonacci numbers because it does so much redundant computation. Notice that the entire computation of fib 4 — almost half the work — is duplicated. To get an idea of how bad this is, one can show that the value of fib n grows exponentially with n.

The function above uses a number of steps that grows exponentially with the input. We have also seen an iterative implementation of Fibonacci numbers, repeated here for convenience. Implicitly, the for statement also keeps track of the iteration count. This definition does not reflect the standard mathematical definition of Fibonacci numbers as clearly as the recursive approach. However, the amount of computation required in the iterative implementation is only linear in n, rather than exponential.

Even for small values of n, this difference can be enormous. One should not conclude from this difference that tree-recursive processes are useless. When we consider processes that operate on hierarchically structured data rather than numbers, we will find that tree recursion is a natural and powerful tool. Furthermore, tree-recursive processes can often be made more efficient through memoization, a powerful technique for increasing the efficiency of recursive functions that repeat computation.

A memoized function will store the return value for any arguments it has previously received. A second call to fib 4 would not evolve the same complex process as the first, but instead would immediately return the stored result computed by the first call. If the memoized function is a pure function, then memoization is guaranteed not to change the result.

Memoization can be expressed naturally as a higher-order function, which can also be used as a decorator. The definition below creates a cache of previously computed results, indexed by the arguments from which they were computed. The use of a dictionary will require that the argument to the memoized function be immutable. To compute fib 40 , the body of fib is executed 40 times, rather than ,, times in the unmemoized recursive case.

To understand the space requirements of a function, we must specify generally how memory is used, preserved, and reclaimed in our environment model of computation. In evaluating an expression, we must preserve all active environments and all values and frames referenced by those environments. An environment is active if it provides the evaluation context for some expression being evaluated.

For example, when evaluating fib, the interpreter proceeds to compute each value in the order shown previously, traversing the structure of the tree. To do so, it only needs to keep track of those nodes that are above the current node in the tree at any point in the computation.

The memory used to evaluate the rest of the branches can be reclaimed because it cannot affect future computation. In general, the space required for tree-recursive functions will be proportional to the maximum depth of the tree.

The diagram below depicts the environment created by evaluating fib 3. In the process of evaluating the return expression for the initial application of fib, the expression fib n-2 is evaluated, yielding a value of 0. Once this value is computed, the corresponding environment frame grayed out is no longer needed: Thus, a well-designed interpreter can reclaim the memory that was used to store this frame.

On the other hand, if the interpreter is currently evaluating fib n-1 , then the environment created by this application of fib in which n is 2 is active. In turn, the environment originally created to apply fib to 3 is active because its return value has not yet been computed. The number of entries in the cache dictionary grows linearly with the number of unique arguments passed to fib, which scales linearly with the input.

On the other hand, the iterative implementation requires only two numbers to be tracked during computation: Memoization exemplifies a common pattern in programming that computation time can often be decreased at the expense of increased use of space, or vis versa.

For some functions, we can exactly predict the number of steps in the computational process evolved by those functions. For most functions, we cannot exactly determine the number of steps or iterations they will require.

One convenient way to describe this difference is to use the notion of order of growth to obtain a coarse measure of the resources required by a process as the inputs become larger.

In general there are a number of properties of the problem with respect to which it will be desirable to analyze a given process. In computers that do only a fixed number of operations at a time, the time required to evaluate an expression will be proportional to the number of elementary machine operations performed in the process of evaluation.

Orders of growth provide only a crude description of the behavior of a process. There are certainly cases in which an order of growth analysis is too coarse a method for deciding between two possible implementations of a function.

However, order of growth provides a useful indication of how we may expect the behavior of the process to change as we change the size of the problem. For an exponential process, each increment in problem size will multiply the resource utilization by a constant factor. The next example examines an algorithm whose order of growth is logarithmic, so that doubling the problem size increases the resource requirement by only a constant amount.

Exponentiation Consider the problem of computing the exponential of a given number. Just as with factorial, we can readily formulate an equivalent linear iteration that requires a similar number of steps but constant space. The size of the exponent we can compute therefore doubles approximately with every new multiplication we are allowed. Thus, the number of multiplications required for an exponent of n grows about as fast as the logarithm of n base 2.

We showed that a pair can be implemented using a built-in tuple. The closure property of pairs indicated that either element of a pair could itself be a pair. This closure property allowed us to implement the recursive list data abstraction, which served as our first type of sequence.

Recursive lists are most naturally manipulated using recursive functions, as their name and structure would suggest. In this section, we discuss functions for creating and manipulating recursive lists and other recursive data structures.

We previously implemented recursive lists using functions, but at this point we can re-implement them using a class. Thus, these definitions will end up calling themselves. Recursive calls on the rest of the list are a ubiquitous pattern in recursive list processing. This class definition of a recursive list interacts properly with Python’s built-in sequence and printing operations. Instead, recursive lists are taken apart and constructed incrementally as a consequence of function application.

As a result, they have linear orders of growth in both the number of steps and space required.

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