Undecidable Problems Chapter Notes | AP Computer Science Principles - Grade 9 PDF Download

Introduction

Undecidable problems are a fascinating concept in AP Computer Science Principles, highlighting the limitations of what computers can solve. This chapter explores the difference between decidable and undecidable problems, focusing on decision problems that require yes or no answers. It introduces the famous halting problem as a classic example of an undecidable problem, pioneered by Alan Turing. Understanding undecidable problems helps programmers recognize the boundaries of computational problem-solving.

Decidable and Undecidable

• A decidable problem is a decision problem (one with a yes or no answer) that can be solved by an algorithm for all possible inputs.
• An undecidable problem is a decision problem where no algorithm can always provide a correct yes or no answer for every input.
• Undecidable problems may be solvable in some cases, but not in all cases.

The Halting Problem

• The halting problem is a well-known undecidable problem introduced by Alan Turing in 1936.
• It asks whether an algorithm can determine if a given program will stop (halt) or run forever for a specific input.
• No algorithm can solve the halting problem for all programs and inputs, making it undecidable.

Conclusion

Some problems, called undecidable problems, cannot be solved by computers in all cases. While specific instances of these problems might have solutions, no general algorithm exists to solve them universally.

Key Terms