Decision problem

Definition

A decision problem is a question that can be posed as a yes-or-no query based on specific conditions or criteria. It is significant in computer science and mathematics, particularly when discussing the limits of computation and decidability. Decision problems help in understanding which problems can be effectively solved by algorithms and which cannot, thus connecting to broader theories like the Church-Turing thesis.

5 Must Know Facts For Your Next Test

1. Decision problems are foundational in computer science as they help categorize problems into those that are solvable (decidable) and those that are not (undecidable).
2. The Church-Turing thesis posits that any computation performed by an algorithm can also be performed by a Turing machine, solidifying the importance of decision problems in understanding computational limits.
3. Many classic decision problems, such as the halting problem, are shown to be undecidable, highlighting the boundaries of what can be computed.
4. In computational complexity theory, decision problems are often classified into complexity classes like P (problems solvable in polynomial time) and NP (nondeterministic polynomial time).
5. Decision problems form the basis of many important areas in theoretical computer science, including logic, automata theory, and complexity theory.

Review Questions

• How do decision problems help in distinguishing between decidable and undecidable problems?
  • Decision problems are critical for distinguishing between decidable and undecidable problems because they provide a framework for determining whether an algorithm can resolve a given problem with a definitive yes or no answer. If an algorithm exists that can always produce such answers for all possible inputs of a decision problem, it is considered decidable. Conversely, if no such algorithm can exist—like in the case of the halting problem—the decision problem is classified as undecidable.
• Discuss how the Church-Turing thesis relates to decision problems and their significance in computational theory.
  • The Church-Turing thesis states that any function that can be computed by an algorithm can be computed by a Turing machine. This thesis directly relates to decision problems as it implies that any decision problem we can formulate has an equivalent representation in terms of Turing machines. This relationship emphasizes the significance of decision problems in computational theory, as it allows us to explore the limits of computation and understand which types of problems are effectively solvable within these frameworks.
• Evaluate the implications of having undecidable decision problems on computational theory and practice.
  • The existence of undecidable decision problems has profound implications for both computational theory and practical computing applications. In theory, it shows us that there are inherent limitations to what algorithms can achieve; certain questions cannot be answered regardless of computational power or time. In practice, this insight informs programmers and researchers about the boundaries of problem-solving capabilities and guides them towards focusing on decidable problems where effective solutions exist. It also impacts fields like cryptography and complexity theory by highlighting areas where certain computations may remain forever elusive.

Related terms

decidability: Decidability refers to whether a decision problem can be resolved algorithmically, meaning there exists a procedure that can determine a yes or no answer for every possible input.

Turing machine: A Turing machine is a theoretical computational model that can simulate any algorithm's logic, used to explore the limits of what can be computed.

algorithm: An algorithm is a step-by-step procedure or formula for solving a problem, often used in the context of computing and mathematics.