reducibilities

Reducibility is a fundamental concept in computability theory and complexity theory. It describes a relationship between two problems, typically computational problems. Problem A is said to be reducible to problem B if a solution to problem B can be used to solve problem A. This doesn't mean that A is equivalent to B, but rather that A is no harder than B. If A can be solved, and A is reducible to B, then B can also be solved. Conversely, if B is unsolvable, then A must also be unsolvable.

There are different types of reducibilities, depending on the power of the "reduction" process allowed. For instance, in computability theory, we often talk about Turing reducibility, where the reduction involves a Turing machine that can call an oracle for problem B. If a problem is Turing reducible to the Halting Problem, for example, it means that if we had a way to solve the Halting Problem, we could also solve the original problem.

In complexity theory, reducibilities are used to classify problems based on their difficulty. Polynomial-time reducibility is a common type, where the reduction can be performed in polynomial time. This is crucial for defining complexity classes like P and NP. If problem A is polynomial-time reducible to problem B, and B is in NP, then A is also in NP. This implies that if we find a polynomial-time algorithm for B, we can use it to efficiently solve A. Reducibility is a key tool for proving that certain problems are NP-complete, meaning they are among the hardest problems in NP.