Reduction (Recursion Theory)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In computability theory, many reducibility relations (also called reductions, reducibilities, and notions of reducibility) are studied. They are motivated by the question: given sets A and B of natural numbers, is it possible to effectively convert a method for deciding membership in B into a method for deciding membership in A? If the answer to this question is affirmative then A is said to be reducible to B. The study of reducibility notions is motivated by the study of decision problems. For many notions of reducibility, if any noncomputable set is reducible to a set A then A must also be noncomputable. This gives a powerful technique for proving that many sets are noncomputable.