Hierarchical Data and the Closure Property

As we have seen, pairs provide a primitive "glue" that we can use to construct compound data objects. Figure 2-2 shows a standard way to visualize a pair---in this case, the pair formed by (cons 1 2). In this representation, which is called box-and-pointer notation , each object is shown as a pointer to a box. The box for a primitive object contains a representation of the object. For example, the box for a number contains a numeral. The box for a pair is actually a double box, the left part containing (a pointer to) the car of the pair and the right part containing the cdr.

Figure 2.2: Box-and-pointer representation of (cons 1 2).

We have already seen that cons can be used to combine not only numbers but pairs as well. (You made use of this fact, or should have, in doing Exercise 2-2 and Exercise 2-3.) As a consequence, pairs provide a universal building block from which we can construct all sorts of data structures. Figure 2-3 shows two ways to use pairs to combine the numbers 1, 2, 3, and 4.

Figure 2.3: Two ways to combine 1, 2, 3, and 4 using pairs.

The ability to create pairs whose elements are pairs is the essence of list structure's importance as a representational tool. We refer to this ability as the closure property of cons. In general, an operation for combining data objects satisfies the closure property if the results of combining things with that operation can themselves be combined using the same operation.6 Closure is the key to power in any means of combination because it permits us to create hierarchical structures---structures made up of parts, which themselves are made up of parts, and so on.

From the outset of Chapter 1, we've made essential use of closure in dealing with procedures, because all but the very simplest programs rely on the fact that the elements of a combination can themselves be combinations. In this section, we take up the consequences of closure for compound data. We describe some conventional techniques for using pairs to represent sequences and trees, and we exhibit a graphics language that illustrates closure in a vivid way.7

2.2.1 Representing Sequences

Such a sequence of pairs, formed by nested conses, is called a list , and Scheme provides a primitive called list to help in constructing lists.8 The above sequence could be produced by (list 1 2 3 4). In general,

is equivalent to

Lisp systems conventionally print lists by printing the sequence of elements, enclosed in parentheses. Thus, the data object in Figure 2-4 is printed as (1 2 3 4):

Be careful not to confuse the expression (list 1 2 3 4) with the list (1 2 3 4), which is the result obtained when the expression is evaluated. Attempting to evaluate the expression (1 2 3 4) will signal an error when the interpreter tries to apply the procedure 1 to arguments 2, 3, and 4.

We can think of car as selecting the first item in the list, and of cdr as selecting the sublist consisting of all but the first item. Nested applications of car and cdr can be used to extract the second, third, and subsequent items in the list.9 The constructor cons makes a list like the original one, but with an additional item at the beginning.

The value of nil, used to terminate the chain of pairs, can be thought of as a sequence of no elements, the empty list . The word nil is a contraction of the Latin word nihil, which means "nothing".10

List operations

The use of pairs to represent sequences of elements as lists is accompanied by conventional programming techniques for manipulating lists by successively "cdring down" the lists. For example, the procedure list-ref takes as arguments a list and a number n and returns the nth item of the list. It is customary to number the elements of the list beginning with 0. The method for computing list-ref is the following:

• For n = 0, list-ref should return the car of the list.

• Otherwise, list-ref should return the (n - 1)st item of the cdr of the list.

Often we cdr down the whole list. To aid in this, Scheme includes a primitive predicate null?, which tests whether its argument is the empty list. The procedure length, which returns the number of items in a list, illustrates this typical pattern of use:

The length procedure implements a simple recursive plan. The reduction step is:

• The length of any list is 1 plus the length of the cdr of the list.

This is applied successively until we reach the base case:

• The length of the empty list is 0.

We could also compute length in an iterative style:

Another conventional programming technique is to "cons up" an answer list while cdring down a list, as in the procedure append, which takes two lists as arguments and combines their elements to make a new list:

append is also implemented using a recursive plan. To append lists list1 and list2, do the following:

• If list1 is the empty list, then the result is just list2.

• Otherwise, append the cdr of list1 and list2, and cons the car of list1 onto the result:

Mapping over lists

One extremely useful operation is to apply some transformation to each element in a list and generate the list of results. For instance, the following procedure scales each number in a list by a given factor:

We can abstract this general idea and capture it as a common pattern expressed as a higher-order procedure, just as in section 1.3. The higher-order procedure here is called map. map takes as arguments a procedure of one argument and a list, and returns a list of the results produced by applying the procedure to each element in the list:12

Now we can give a new definition of scale-list in terms of map:

map is an important construct, not only because it captures a common pattern, but because it establishes a higher level of abstraction in dealing with lists. In the original definition of scale-list, the recursive structure of the program draws attention to the element-by-element processing of the list. Defining scale-list in terms of map suppresses that level of detail and emphasizes that scaling transforms a list of elements to a list of results. The difference between the two definitions is not that the computer is performing a different process (it isn't) but that we think about the process differently. In effect, map helps establish an abstraction barrier that isolates the implementation of procedures that transform lists from the details of how the elements of the list are extracted and combined. Like the barriers shown in Figure 2-1, this abstraction gives us the flexibility to change the low-level details of how sequences are implemented, while preserving the conceptual framework of operations that transform sequences to sequences. Section 2.2.3 expands on this use of sequences as a framework for organizing programs.

Recursion is a natural tool for dealing with tree structures, since we can often reduce operations on trees to operations on their branches, which reduce in turn to operations on the branches of the branches, and so on, until we reach the leaves of the tree. As an example, compare the length procedure of section 2.2.1 with the count-leaves procedure, which returns the total number of leaves of a tree:

To implement count-leaves, recall the recursive plan for computing length:

• length of a list x is 1 plus length of the cdr of x.
• length of the empty list is 0.

count-leaves is similar. The value for the empty list is the same:

• count-leaves of the empty list is 0.

But in the reduction step, where we strip off the car of the list, we must take into account that the car may itself be a tree whose leaves we need to count. Thus, the appropriate reduction step is

• count-leaves of a tree x is count-leaves of the car of x plus count-leaves of the cdr of x.

Finally, by taking cars we reach actual leaves, so we need another base case:

• count-leaves of a leaf is 1.

To aid in writing recursive procedures on trees, Scheme provides the primitive predicate pair?, which tests whether its argument is a pair. Here is the complete procedure:13

Mapping over trees

Just as map is a powerful abstraction for dealing with sequences, map together with recursion is a powerful abstraction for dealing with trees. For instance, the scale-tree procedure, analogous to scale-list of section 2.2.1, takes as arguments a numeric factor and a tree whose leaves are numbers. It returns a tree of the same shape, where each number is multiplied by the factor. The recursive plan for scale-tree is similar to the one for count-leaves:

Another way to implement scale-tree is to regard the tree as a sequence of sub-trees and use map. We map over the sequence, scaling each sub-tree in turn, and return the list of results. In the base case, where the tree is a leaf, we simply multiply by the factor:

Many tree operations can be implemented by similar combinations of sequence operations and recursion.

In section 1.3 we saw how program abstractions, implemented as higher-order procedures, can capture common patterns in programs that deal with numerical data. Our ability to formulate analogous operations for working with compound data depends crucially on the style in which we manipulate our data structures. Consider, for example, the following procedure, analogous to the count-leaves procedure of section 2.2.2, which takes a tree as argument and computes the sum of the squares of the leaves that are odd:

On the surface, this procedure is very different from the following one, which constructs a list of all the even Fibonacci numbers $\text{Fib}(k)$, where $k$ is less than or equal to a given integer $n$:

Despite the fact that these two procedures are structurally very different, a more abstract description of the two computations reveals a great deal of similarity. The first program

• enumerates the leaves of a tree;

• filters them, selecting the odd ones;

• squares each of the selected ones; and

• accumulates the results using +, starting with 0.

The second program

• enumerates the integers from 0 to n;

• computes the Fibonacci number for each integer;

• filters them, selecting the even ones; and

• accumulates the results using cons, starting with the empty list.

A signal-processing engineer would find it natural to conceptualize these processes in terms of signals flowing through a cascade of stages, each of which implements part of the program plan, as shown in Figure 2-7. In sum-odd-squares, we begin with an enumerator , which generates a "signal" consisting of the leaves of a given tree. This signal is passed through a filter , which eliminates all but the odd elements. The resulting signal is in turn passed through a map , which is a "transducer" that applies the square procedure to each element. The output of the map is then fed to an accumulator , which combines the elements using +, starting from an initial 0. The plan for even-fibs is analogous.

Figure 2.7: The signal-flow plans for the procedures sum-odd-squares (top) and even-fibs (bottom) reveal the commonality between the two programs.

Unfortunately, the two procedure definitions above fail to exhibit this signal-flow structure. For instance, if we examine the sum-odd-squares procedure, we find that the enumeration is implemented partly by thenull? and pair? tests and partly by the tree-recursive structureof the procedure. Similarly, the accumulation is found partly in the tests and partly in the addition used in the recursion. In general, there are no distinct parts of either procedure that correspond to the elements in the signal-flow description. Our two procedures decompose the computations in a different way, spreading the enumeration over the program and mingling it with the map, the filter, and the accumulation. If we could organize our programs to make the signal-flow structure manifest in the procedures we write, this would increase the conceptual clarity of the resulting code.

Sequence Operations

The key to organizing programs so as to more clearly reflect the signal-flow structure is to concentrate on the "signals" that flow from one stage in the process to the next. If we represent these signals as lists, then we can use list operations to implement the processing at each of the stages. For instance, we can implement the mapping stages of the signal-flow diagrams using the map procedure from section 2.2.1:

Filtering a sequence to select only those elements that satisfy a given predicate is accomplished by

Accumulations can be implemented by

All that remains to implement signal-flow diagrams is to enumerate the sequence of elements to be processed. For even-fibs, we need to generate the sequence of integers in a given range, which we can do as follows:

To enumerate the leaves of a tree, we can use14

Now we can reformulate sum-odd-squares and even-fibs as in the signal-flow diagrams. For sum-odd-squares, we enumerate the sequence of leaves of the tree, filter this to keep only the odd numbers in the sequence, square each element, and sum the results:

For even-fibs, we enumerate the integers from 0 to n, generate the Fibonacci number for each of these integers, filter the resulting sequence to keep only the even elements, and accumulate the results into a list:

The value of expressing programs as sequence operations is that this helps us make program designs that are modular, that is, designs that are constructed by combining relatively independent pieces. We can encourage modular design by providing a library of standard components together with a conventional interface for connecting the components in flexible ways.

Modular construction is a powerful strategy for controlling complexity in engineering design. In real signal-processing applications, for example, designers regularly build systems by cascading elements selected from standardized families of filters and transducers. Similarly, sequence operations provide a library of standard program elements that we can mix and match. For instance, we can reuse pieces from the sum-odd-squares and even-fibs procedures in a program that constructs a list of the squares of the first $n + 1$ Fibonacci numbers:

We can rearrange the pieces and use them in computing the product of the odd integers in a sequence:

We can also formulate conventional data-processing applications in terms of sequence operations. Suppose we have a sequence of personnel records and we want to find the salary of the highest-paid programmer. Assume that we have a selector salary that returns the salary of a record, and a predicate programmer? that tests if a record is for a programmer. Then we can write

These examples give just a hint of the vast range of operations that can be expressed as sequence operations.15

Sequences, implemented here as lists, serve as a conventional interface that permits us to combine processing modules. Additionally, when we uniformly represent structures as sequences, we have localized the data-structure dependencies in our programs to a small number of sequence operations. By changing these, we can experiment with alternative representations of sequences, while leaving the overall design of our programs intact. We will exploit this capability in section 3.5, when we generalize the sequence-processing paradigm to admit infinite sequences.

Nested Mappings

We can extend the sequence paradigm to include many computations that are commonly expressed using nested loops.18 Consider this problem: Given a positive integer n, find all ordered pairs of distinct positive integers i and j, where 1 <= j< i<= n, such that i + j is prime. For example, if n is 6, then the pairs are the following: \begin{array}{c|ccc} i & 2 & 3 & 4 & 4 & 5 & 6 & 6 \\ j & 1 & 2 & 1 & 3 & 2 & 1 & 5 \\ \hline \\ i+j & 3 & 5 & 5 & 7 & 7 & 7 & 11 \end{array}

A natural way to organize this computation is to generate the sequence of all ordered pairs of positive integers less than or equal to $n$, filter to select those pairs whose sum is prime, and then, for each pair $(i, j)$ that passes through the filter, produce the triple $(i,j,i+j)$.

Here is a way to generate the sequence of pairs: For each integer $i <= n$, enumerate the integers $j<i$, and for each such $i$ and $j$ generate the pair $(i,j)$. In terms of sequence operations, we map along the sequence (enumerate-interval 1 n). For each $i$ in this sequence, we map along the sequence (enumerate-interval 1 (- i 1)). For each $j$ in this latter sequence, we generate the pair (list i j). This gives us a sequence of pairs for each $i$. Combining all the sequences for all the $i$ (by accumulating with append) produces the required sequence of pairs:19

The combination of mapping and accumulating with append is so common in this sort of program that we will isolate it as a separate procedure:

Now filter this sequence of pairs to find those whose sum is prime. The filter predicate is called for each element of the sequence; its argument is a pair and it must extract the integers from the pair. Thus, the predicate to apply to each element in the sequence is

Finally, generate the sequence of results by mapping over the filtered pairs using the following procedure, which constructs a triple consisting of the two elements of the pair along with their sum:

Combining all these steps yields the complete procedure:

Nested mappings are also useful for sequences other than those that enumerate intervals. Suppose we wish to generate all the permutations of a set $S$; that is, all the ways of ordering the items in the set. For instance, the permutations of $\{1,2,3\}$ are $\{1,2,3\}$, $\{1,3,2\}$, $\{2,1,3\}$, $\{2,3,1\}$, $\{3,1,2\}$, and $\{3,2,1\}$. Here is a plan for generating the permutations of $S$: For each item $x$ in $S$, recursively generate the sequence of permutations of $S - x$,20 and adjoin $x$ to the front of each one. This yields, for each x in S, the sequence of permutations of S that begin with $x$. Combining these sequences for all $x$ gives all the permutations of $S$:21

Notice how this strategy reduces the problem of generating permutations of S to the problem of generating the permutations of sets with fewer elements than S. In the terminal case, we work our way down to the empty list, which represents a set of no elements. For this, we generate (list nil), which is a sequence with one item, namely the set with no elements. The remove procedure used in permutations returns all the items in a given sequence except for a given item. This can be expressed as a simple filter:

This section presents a simple language for drawing pictures that illustrates the power of data abstraction and closure, and also exploits higher-order procedures in an essential way. The language is designed to make it easy to experiment with patterns such as the ones in Figure 2-9, which are composed of repeated elements that are shifted and scaled.22 In this language, the data objects being combined are represented as procedures rather than as list structure. Just as cons, which satisfies the closure property, allowed us to easily build arbitrarily complicated list structure, the operations in this language, which also satisfy the closure property, allow us to easily build arbitrarily complicated patterns.

Figure 2.9: Designs generated with the picture language.

The picture language

When we began our study of programming in section 1.1, we emphasized the importance of describing a language by focusing on the language's primitives, its means of combination, and its means of abstraction. We'll follow that framework here.

Part of the elegance of this picture language is that there is only one kind of element, called a painter . A painter draws an image that is shifted and scaled to fit within a designated parallelogram-shaped frame. For example, there's a primitive painter we'll call wave that makes a crude line drawing, as shown in Figure 2-10. The actual shape of the drawing depends on the frame---all four images in Figure 2-10 are produced by the same wave painter, but with respect to four different frames. Painters can be more elaborate than this: The primitive painter called rogers paints a picture of MIT's founder, William Barton Rogers, as shown in Figure 2-11.23 The four images in Figure 2-11 are drawn with respect to the same four frames as the wave images in Figure 2-10.

To combine images, we use various operations that construct new painters from given painters. For example, the beside operation takes two painters and produces a new, compound painter that draws the first painter's image in the left half of the frame and the second painter's image in the right half of the frame. Similarly, below takes two painters and produces a compound painter that draws the first painter's image below the second painter's image. Some operations transform a single painter to produce a new painter. For example, flip-vert takes a painter and produces a painter that draws its image upside-down, and flip-horiz produces a painter that draws the original painter's image left-to-right reversed.

Figure 2.10: Images produced by the wave painter, with respect to four different frames. The frames, shown with dotted lines, are not part of the images.

Figure 2.11: Images of William Barton Rogers, founder and first president of MIT, painted with respect to the same four frames as in Figure 2-10 (original image reprinted with the permission of the MIT Museum).

Figure 2-12 shows the drawing of a painter called wave4 that is built up in two stages starting from wave:

Figure 2.12: Creating a complex figure, starting from the wave painter of Figure 2-10.

In building up a complex image in this manner we are exploiting the fact that painters are closed under the language's means of combination. The beside or below of two painters is itself a painter; therefore, we can use it as an element in making more complex painters. As with building up list structure using cons, the closure of our data under the means of combination is crucial to the ability to create complex structures while using only a few operations.

Once we can combine painters, we would like to be able to abstract typical patterns of combining painters. We will implement the painter operations as Scheme procedures. This means that we don't need a special abstraction mechanism in the picture language: Since the means of combination are ordinary Scheme procedures, we automatically have the capability to do anything with painter operations that we can do with procedures. For example, we can abstract the pattern in wave4 as

and define wave4 as an instance of this pattern:

We can also define recursive operations. Here's one that makes painters split and branch towards the right as shown in figures Figure 2-13 and Figure 2-14:

Figure 2.13: Recursive plans for right-split and corner-split.

We can produce balanced patterns by branching upwards as well as towards the right (see Exercise 2-44 and figures Figure 2-13 and Figure 2-14):

Figure 2.14: The recursive operations right-split and corner-split applied to the painters wave and rogers. Combining four corner-split figures produces symmetric square-limit designs as shown in Figure 2-9.

By placing four copies of a corner-split appropriately, we obtain a pattern called square-limit, whose application to wave and rogers is shown in Figure 2-9:

Higher-order operations

In addition to abstracting patterns of combining painters, we can work at a higher level, abstracting patterns of combining painter operations. That is, we can view the painter operations as elements to manipulate and can write means of combination for these elements---procedures that take painter operations as arguments and create new painter operations.

For example, flipped-pairs and square-limit each arrange four copies of a painter's image in a square pattern; they differ only in how they orient the copies. One way to abstract this pattern of painter combination is with the following procedure, which takes four one-argument painter operations and produces a painter operation that transforms a given painter with those four operations and arranges the results in a square. Tl, tr, bl, and br are the transformations to apply to the top left copy, the top right copy, the bottom left copy, and the bottom right copy, respectively.

Then flipped-pairs can be defined in terms of square-of-four as follows:24

and square-limit can be expressed as25

Frames

Before we can show how to implement painters and their means of combination, we must first consider frames. A frame can be described by three vectors---an origin vector and two edge vectors. The origin vector specifies the offset of the frame's origin from some absolute origin in the plane, and the edge vectors specify the offsets of the frame's corners from its origin. If the edges are perpendicular, the frame will be rectangular. Otherwise the frame will be a more general parallelogram.

Figure 2-15 shows a frame and its associated vectors. In accordance with data abstraction, we need not be specific yet about how frames are represented, other than to say that there is a constructor make-frame, which takes three vectors and produces a frame, and three corresponding selectors origin-frame, edge1-frame, and edge2-frame (see Exercise 2-47).

Figure 2.15: A frame is described by three vectors -- an origin and two edges.

We will use coordinates in the unit square $(0 ≤ x,y ≤ 1)$ to specify images. With each frame, we associate a frame coordinate map , which will be used to shift and scale images to fit the frame. The map transforms the unit square into the frame by mapping the vector v = (x,y) to the vector sum $$ \text{Origin}(\text{Frame}) + x \cdot \text{Edge}_1(\text{Frame}) + y \cdot \text{Edge}_2(\text{Frame}) $$

For example, (0,0) is mapped to the origin of the frame, (1,1) to the vertex diagonally opposite the origin, and (0.5,0.5) to the center of the frame. We can create a frame's coordinate map with the following procedure:26

Observe that applying frame-coord-map to a frame returns a procedure that, given a vector, returns a vector. If the argument vector is in the unit square, the result vector will be in the frame. For example,

returns the same vector as

Painters

A painter is represented as a procedure that, given a frame as argument, draws a particular image shifted and scaled to fit the frame. That is to say, if p is a painter and f is a frame, then we produce p's image in f by calling p with f as argument.

The details of how primitive painters are implemented depend on the particular characteristics of the graphics system and the type of image to be drawn. For instance, suppose we have a procedure draw-line that draws a line on the screen between two specified points. Then we can create painters for line drawings, such as the wave painter in Figure 2-10, from lists of line segments as follows:27

The segments are given using coordinates with respect to the unit square. For each segment in the list, the painter transforms the segment endpoints with the frame coordinate map and draws a line between the transformed points.

Representing painters as procedures erects a powerful abstraction barrier in the picture language. We can create and intermix all sorts of primitive painters, based on a variety of graphics capabilities. The details of their implementation do not matter. Any procedure can serve as a painter, provided that it takes a frame as argument and draws something scaled to fit the frame.28

Transforming and combining painters

An operation on painters (such as flip-vert or beside) works by creating a painter that invokes the original painters with respect to frames derived from the argument frame. Thus, for example, flip-vert doesn't have to know how a painter works in order to flip it---it just has to know how to turn a frame upside down: The flipped painter just uses the original painter, but in the inverted frame.

Painter operations are based on the procedure transform-painter, which takes as arguments a painter and information on how to transform a frame and produces a new painter. The transformed painter, when called on a frame, transforms the frame and calls the original painter on the transformed frame. The arguments to transform-painter are points (represented as vectors) that specify the corners of the new frame: When mapped into the frame, the first point specifies the new frame's origin and the other two specify the ends of its edge vectors. Thus, arguments within the unit square specify a frame contained within the original frame.

Here's how to flip painter images vertically:

Using transform-painter, we can easily define new transformations. For example, we can define a painter that shrinks its image to the upper-right quarter of the frame it is given:

Other transformations rotate images counterclockwise by 90 degrees29

or squash images towards the center of the frame:30

Frame transformation is also the key to defining means of combining two or more painters. The beside procedure, for example, takes two painters, transforms them to paint in the left and right halves of an argument frame respectively, and produces a new, compound painter. When the compound painter is given a frame, it calls the first transformed painter to paint in the left half of the frame and calls the second transformed painter to paint in the right half of the frame:

Observe how the painter data abstraction, and in particular the representation of painters as procedures, makes beside easy to implement. The beside procedure need not know anything about the details of the component painters other than that each painter will draw something in its designated frame.

Levels of language for robust design

The picture language exercises some of the critical ideas we've introduced about abstraction with procedures and data. The fundamental data abstractions, painters, are implemented using procedural representations, which enables the language to handle different basic drawing capabilities in a uniform way. The means of combination satisfy the closure property, which permits us to easily build up complex designs. Finally, all the tools for abstracting procedures are available to us for abstracting means of combination for painters.

We have also obtained a glimpse of another crucial idea about languages and program design. This is the approach of stratified design , the notion that a complex system should be structured as a sequence of levels that are described using a sequence of languages. Each level is constructed by combining parts that are regarded as primitive at that level, and the parts constructed at each level are used as primitives at the next level. The language used at each level of a stratified design has primitives, means of combination, and means of abstraction appropriate to that level of detail.

Stratified design pervades the engineering of complex systems. For example, in computer engineering, resistors and transistors are combined (and described using a language of analog circuits) to produce parts such as and-gates and or-gates, which form the primitives of a language for digital-circuit design.31 These parts are combined to build processors, bus structures, and memory systems, which are in turn combined to form computers, using languages appropriate to computer architecture. Computers are combined to form distributed systems, using languages appropriate for describing network interconnections, and so on.

As a tiny example of stratification, our picture language uses primitive elements (primitive painters) that are created using a language that specifies points and lines to provide the lists of line segments for segments->painter, or the shading details for a painter like rogers. The bulk of our description of the picture language focused on combining these primitives, using geometric combiners such as beside and below. We also worked at a higher level, regarding beside and below as primitives to be manipulated in a language whose operations, such as square-of-four, capture common patterns of combining geometric combiners.

Stratified design helps make programs robust , that is, it makes it likely that small changes in a specification will require correspondingly small changes in the program. For instance, suppose we wanted to change the image based on wave shown in Figure 2-9. We could work at the lowest level to change the detailed appearance of the wave element; we could work at the middle level to change the way corner-split replicates the wave; we could work at the highest level to change how square-limit arranges the four copies of the corner. In general, each level of a stratified design provides a different vocabulary for expressing the characteristics of the system, and a different kind of ability to change it.

(cadr <arg>) = (car (cdr <arg>))

(define flipped-pairs
  (square-of-four identity flip-vert identity flip-vert))

Based on Structure and Interpretation of Computer Programs, a work at https://mitpress.mit.edu/sicp/.