Turing reduction

Definition

Turing reduction is a way of comparing the computational power of different problems by transforming one problem into another using an algorithm. Specifically, it allows us to solve a problem A using an oracle that can solve another problem B in a single step, meaning that if we can decide B, we can decide A as well. This concept is crucial in understanding computational complexity and the relationships between various decision problems.

5 Must Know Facts For Your Next Test

1. Turing reduction is often denoted as A ≤_T B, meaning that problem A can be reduced to problem B via Turing reduction.
2. In Turing reduction, the algorithm for problem A can make multiple calls to the oracle for problem B, unlike many-one reductions which only allow a single call.
3. Turing reductions help classify problems into complexity classes, such as NP-complete and undecidable problems, based on their relationships with other problems.
4. If a problem can be reduced to another problem through Turing reduction and the second problem is solvable, then the first problem is also solvable.
5. Turing reductions illustrate the concept of relative computability, showing how the solvability of one problem might depend on the ability to solve another.

Review Questions

• How does Turing reduction demonstrate the relationship between two problems in terms of their solvability?
  • Turing reduction shows that if one problem can be transformed into another using an algorithm that calls an oracle for the second problem, then solving the second problem allows us to solve the first. This means if we can solve problem B, we can also solve problem A using B as a tool. It highlights how some problems depend on others and helps establish the hierarchy of problem difficulty in terms of decidability.
• Discuss how Turing reduction relates to complexity classes and what implications it has for classifying decision problems.
  • Turing reduction plays a crucial role in classifying decision problems within complexity classes. By showing how one problem can be transformed into another through these reductions, we can establish relationships among different classes. For example, if an NP-complete problem can be reduced to another problem via Turing reduction, it indicates that the latter has similar complexity characteristics. This classification helps us understand which problems are efficiently solvable and which are intractable.
• Evaluate the significance of Turing reductions in understanding undecidable problems and their implications for computer science.
  • The significance of Turing reductions lies in their ability to clarify the boundaries of computability and undecidability. When exploring undecidable problems, Turing reductions help us understand how certain problems relate to one another in terms of their unsolvability. For instance, by showing that a known undecidable problem can be reduced to another, we confirm that this new problem is also undecidable. This insight has profound implications for theoretical computer science, influencing areas like algorithm design, cryptography, and more.

Related terms

Decidability: Decidability refers to whether a problem can be solved by an algorithm in a finite amount of time, determining if there exists a procedure that will always give a correct yes or no answer.

Complexity class: A complexity class is a set of problems that can be solved by a computational model within a certain resource constraint, such as time or space.

Oracle machine: An oracle machine is a theoretical computational model that can solve specific decision problems instantaneously, serving as a 'black box' for certain queries.