Many-one reduction

In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction[1]) is a reduction which converts instances of one decision problem into instances of a second decision problem where the instance reduced to is in the language if the initial instance was in its language and is not in the language if the initial instance was not in its language . Thus if we can decide whether instances are in the language , we can decide whether instances are in its language by applying the reduction and solving . Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that reduces to if, in layman's terms is harder to solve than . That is to say, any algorithm that solves can also be used as part of a (otherwise relatively simple) program that solves .

Many-one reductions are a special case and stronger form of Turing reductions.[1] With many-one reductions, the oracle (that is, our solution for B) can be invoked only once at the end, and the answer cannot be modified. This means that if we want to show that problem A can be reduced to problem B, we can use our solution for B only once in our solution for A, unlike in Turing reduction, where we can use our solution for B as many times as needed while solving A.

This means that many-one reductions map instances of one problem to instances of another, while Turing reductions compute the solution to one problem, assuming the other problem is easy to solve. The many-one reduction is more effective at separating problems into distinct complexity classes. However, the increased restrictions on many-one reductions make them more difficult to find.

Many-one reductions were first used by Emil Post in a paper published in 1944.[2] Later Norman Shapiro used the same concept in 1956 under the name strong reducibility.[3]

DefinitionsEdit

Formal languagesEdit

Suppose   and   are formal languages over the alphabets   and  , respectively. A many-one reduction from   to   is a total computable function   that has the property that each word   is in   if and only if   is in  .

If such a function   exists, we say that   is many-one reducible or m-reducible to   and write

 

If there is an injective many-one reduction function then we say A is 1-reducible or one-one reducible to   and write

 

Subsets of natural numbersEdit

Given two sets   we say   is many-one reducible to   and write

 

if there exists a total computable function   with   If additionally   is injective we say   is 1-reducible to   and write

 

Many-one equivalence and 1-equivalenceEdit

If   we say   is many-one equivalent or m-equivalent to   and write

 

If   we say   is 1-equivalent to   and write

 

Many-one completeness (m-completeness)Edit

A set   is called many-one complete, or simply m-complete, iff   is recursively enumerable and every recursively enumerable set   is m-reducible to  .

Many-one reductions with resource limitationsEdit

Many-one reductions are often subjected to resource restrictions, for example that the reduction function is computable in polynomial time, logarithmic space, by   or   circuits, or polylogarithmic projections where each subsequent reduction notion is weaker than the prior; see polynomial-time reduction and log-space reduction for details.

Given decision problems   and   and an algorithm N which solves instances of  , we can use a many-one reduction from   to   to solve instances of   in:

  • the time needed for N plus the time needed for the reduction
  • the maximum of the space needed for N and the space needed for the reduction

We say that a class C of languages (or a subset of the power set of the natural numbers) is closed under many-one reducibility if there exists no reduction from a language in C to a language outside C. If a class is closed under many-one reducibility, then many-one reduction can be used to show that a problem is in C by reducing a problem in C to it. Many-one reductions are valuable because most well-studied complexity classes are closed under some type of many-one reducibility, including P, NP, L, NL, co-NP, PSPACE, EXP, and many others. It is known for example that the first four listed are closed up to the very weak reduction notion of polylogarithmic time projections. These classes are not closed under arbitrary many-one reductions, however.

PropertiesEdit

  • The relations of many-one reducibility and 1-reducibility are transitive and reflexive and thus induce a preorder on the powerset of the natural numbers.
  •   if and only if  
  • A set is many-one reducible to the halting problem if and only if it is recursively enumerable. This says that with regards to many-one reducibility, the halting problem is the most complicated of all recursively enumerable problems. Thus the halting problem is r.e. complete. Note that it is not the only r.e. complete problem.
  • The specialized halting problem for an individual Turing machine T (i.e., the set of inputs for which T eventually halts) is many-one complete iff T is a universal Turing machine. Emil Post showed that there exist recursively enumerable sets that are neither decidable nor m-complete, and hence that there exist nonuniversal Turing machines whose individual halting problems are nevertheless undecidable.

Karp reductionsEdit

A polynomial-time many-one reduction from a problem A to a problem B (both of which are usually required to be decision problems) is a polynomial-time algorithm for transforming inputs to problem A into inputs to problem B, such that the transformed problem has the same output as the original problem. An instance x of problem A can be solved by applying this transformation to produce an instance y of problem B, giving y as the input to an algorithm for problem B, and returning its output. Polynomial-time many-one reductions may also be known as polynomial transformations or Karp reductions, named after Richard Karp. A reduction of this type is denoted by   or  .[4][5]

ReferencesEdit

  1. ^ a b Abrahamson, Karl R. (Spring 2016). "Mapping reductions". CSCI 6420 – Computability and Complexity. East Carolina University. Retrieved 2021-11-12.
  2. ^ E. L. Post, "Recursively enumerable sets of positive integers and their decision problems", Bulletin of the American Mathematical Society 50 (1944) 284–316
  3. ^ Norman Shapiro, "Degrees of Computability", Transactions of the American Mathematical Society 82, (1956) 281–299
  4. ^ Goldreich, Oded (2008), Computational Complexity: A Conceptual Perspective, Cambridge University Press, pp. 59–60, ISBN 9781139472746
  5. ^ Kleinberg, Jon; Tardos, Éva (2006). Algorithm Design. Pearson Education. pp. 452–453. ISBN 978-0-321-37291-8.