Question

In what sense do the following three steps not constitute an algorithm? Step 1: Draw a circle with center coordinates \((2,5)\) and radius 3 . Step 2: Draw a circle with center coordinates \((6,5)\) and radius \(5 .\) Step 3: Draw a line segment whose endpoints are at the intersections of the previous two circles.

Short answer

The steps lack guaranteed termination and clarity, particularly in handling no intersection scenario in Step 3.

Step-by-step solution

Understanding the Problem

To determine if these steps make up an algorithm, first recall the definition of an algorithm. An algorithm is a finite sequence of well-defined instructions to solve a specific problem or perform a specific task, which should eventually terminate.

Analyzing Step 1

Step 1 asks to draw a circle with given center and radius. This is a clear and concise instruction and fits the criteria of a step in an algorithm. It is explicit and well-defined.

Analyzing Step 2

Step 2 involves drawing a second circle with its own specified center and radius. Like Step 1, this instruction is specific and well-defined. It also fits the criteria of being a part of an algorithm.

Analyzing Step 3

Step 3's instruction is to draw a line segment between the intersections of the previously drawn circles. However, the feasibility of this step is not guaranteed as there may be one, two, or no intersection points between the two circles. This introduces ambiguity and uncertainty, which disqualifies Step 3 from being a definitive algorithmic step.

Evaluating Termination

An algorithm must always reach a conclusion upon execution. Since Step 3 could potentially result in no intersection points, the task may not be able to complete, failing the termination requirement of an algorithm.

Key concepts

Finite Sequence

When defining an algorithm, one of the essential characteristics is that it is a finite sequence. This means that the algorithm consists of a series of steps that are clearly defined and limited in number. Each step is executed in order, leading to a specific outcome without going on indefinitely. In the given exercise, steps for drawing circles and a line segment are included. However, to determine if this series is finite, the algorithm must have a starting point, a clear pathway through all instructions, and a definitive ending.
Without the certainty of a conclusion, as indicated by the potential lack of intersections in Step 3, the steps risk not forming a finite sequence. Therefore, a finite sequence requires that each step leads logically and systematically to the next, ultimately resolving the task as outlined without loops or recursion that could cause perpetuation.

Well-defined Instructions

In algorithms, each instruction must be clear and unambiguous, providing a precise action to follow. Well-defined instructions ensure that anyone executing the algorithm can do so without additional interpretation. It means there should be no room for doubt about what needs to be done. In the exercise, Steps 1 and 2 involve drawing circles based on specified coordinates and radii. These directions are well-defined because they explicitly state what action to take and with which parameters.
However, Step 3 is where the complications begin. The instruction to draw a line between the intersections of the circles assumes that intersections exist. This introduces ambiguity because such intersections are only possible in specific geometric conditions. If the instructions do not cater to all potential outcomes, including the case with no intersections, they cannot be considered well-defined. Thus, for an algorithm to be effective, each step must be crystal clear to eliminate confusion and variability.

Termination

One fundamental requirement for an algorithm is that it must terminate. This means that upon completion of the steps, a final result is achieved without the algorithm running indefinitely. For the given sequence, the algorithm must conclude with a defined outcome such as a completed drawing or finished computation. In Steps 1 and 2, the process of drawing a circle brings the tasks to end points, inherently providing termination.
However, the instruction to "draw a line segment whose endpoints are at the intersections" can lead to a lack of termination. This is because if no intersections exist between the two circles, the sequence cannot be completed as intended, causing a halt or requiring additional instruction to address unforeseen outcomes. Therefore, an algorithm must be structured in such a way that it always concludes properly, indicating successful task execution, regardless of intermediary conditions.

Ambiguity in Instructions

Ambiguity refers to a lack of clarity or precision in the instructions provided within an algorithm. When instructions are ambiguous, they leave too much room for interpretation or unforeseen circumstances that can prevent the successful execution of the task. In the discussed exercise, the first two steps are precise and leave no doubt about the task at hand—drawing circles at given coordinates with specific radii.
On the other hand, Step 3 introduces ambiguity by assuming that intersections between circles necessarily exist. This step fails to account for potential scenarios where no intersections occur, leading to structural ambiguity in performing or completing the task. For a coherent and functional algorithm, it is critical that every instruction is unequivocal and inclusive of all potential variations in conditions to ensure the task can be carried out as envisioned.