Question

Suppose we evaluate the expression (list 1 (list 2 (list 3 4))). Give the result printed by the interpreter, the corresponding box-and-pointer structure, and the interpretation of this as a tree (as in figure 2.6).

Short answer

The result printed is [1, [2, [3, 4]]]; box-and-pointer and tree depict nested lists with 1, 2, and 3, 4.

Step-by-step solution

Evaluating the Expression

The given expression is \((\text{list} \ 1 \ (\text{list} \ 2 \ (\text{list} \ 3 \ 4)))\). This creates a nested list structure where each \(\text{list}\) function call produces a new list: the outer \(\text{list}\) wrapping the number 1 and another list that wraps the number 2 and another list containing 3 and 4.

Printing the Result

The interpreter prints the result of the expression: \([1, [2, [3, 4]]] \). This indicates a list containing 1 and another list that contains 2 and another list with the numbers 3 and 4. It shows the nested list structure clearly.

Constructing Box-and-Pointer Diagram

In the box-and-pointer diagram, we visualize the nested list: a single box with a pointer to another box, and this continues until the base elements (1, 2, 3, 4) are reached. For example, the first box points to 1 and the second box which contains a pointer to 2 and another box.

Converting into a Tree

The tree structure interpretation translates nested lists into branches and leaf nodes of a tree. The root is 1 with a single child containing 2, which itself has a child containing 3 and 4. The tree branches at each \(\text{list}\) call, showing the hierarchy, with 1 at the top-level with subsequent nested structures.

Key concepts

Understanding Box-and-Pointer Diagrams

Box-and-pointer diagrams are a visual way to represent list structures. They help to understand how elements and sub-elements are connected. Each list is shown as a box, and inside there are pointers leading to either another box (another list) or a terminal value (like a number). For the expression
\[ (\text{list} \ 1 \ (\text{list} \ 2 \ (\text{list} \ 3 \ 4))) \], each level of list nesting has its own box:

• The first box contains a pointer to 1 and to another box.
• The second box has a pointer to 2 and leads to a third box.
• This third box points to both 3 and 4 at its conclusion.

This diagrammatic approach aids in visualizing the links and hierarchies within nested lists easily. The process of drawing them helps to clarify the complexity often seen in list structures, making it simpler to track how components relate to each other.

Tree Structures and List Transformation

Transforming list structures into tree diagrams provides a hierarchical perspective of nested lists. Trees consist of nodes representing list elements and branches connecting them based on their nested relationships. Considering the expression
\[ (\text{list} \ 1 \ (\text{list} \ 2 \ (\text{list} \ 3 \ 4))) \], the tree structure represents these relationships:

• The root node is 1, at the top.
• It has a child node 2.
• This child node further branches to a node containing both 3 and 4, representing the deepest nesting level.

Tree structures highlight how each list element is related, clarifying the hierarchy. Using this tree representation makes it easier to visualize and analyze the nested nature of lists effectively.

List Evaluation in Expressions

Evaluating expressions in list structures involves traversing through nested lists and identifying their compositions. In the expression
\[ (\text{list} \ 1 \ (\text{list} \ 2 \ (\text{list} \ 3 \ 4))) \], initial evaluation starts with identifying the numbers and incrementally enclosing them within list functions:

• The inner list creates \([3, 4]\).
• The middle list builds \([2, [3, 4]]\).
• The outermost list wraps it all to form \([1, [2, [3, 4]]] \).

The task of evaluating nested list expressions rests on identifying each layer of lists and their corresponding values. By breaking down the structure into simpler parts, the overall outcome becomes clear. This method provides a straightforward way to interpret complex expressions through step-by-step unveiling of their nested components.