Many pairs one reduction

With the many pairs one reduction (たたいいちかんげん, many-one reduction), it is the name of the certain reduction operation in theory of computation and the computational complexity theory. I have work to convert some kind of decision problems into other decision problems.

It is a special case of the Turing reduction, and the many pairs one reduction is weaker than Turing reduction (the claim that many pairs one is reducible is stronger than Turing reduction possibility). Only, and finally, in the many pairs one reduction, the use (cf. oracle machine) of the Oracle is permitted only once.

The many pairs one reduction was introduced only after I stopped at the Emil post in 1944. In 1956, Norman Shapiro applied the same concept by the name of strong reducibility.

Table of contents

Definition

Abstract language employing precise grammatical constructions

It will be said that it is the abstract language employing precise grammatical constructions that A and B were written as set Σ of the alphabet on Γ each. Whole computability function f which meets the next property with the many pairs one reduction from A to B: I point to Σ^* → Γ^*. A property: "A necessary and sufficient condition to have individual word w in A (i.e.${\displaystyle A=f^{-1}(B)}$ It is).

If such function f exists, A says to B that many pairs one reduction possibility or m-is reducible and writes it as follows.

${\displaystyle A\leq _{m}B.}$ 

If there is the many pairs one reduction that is an injection, I say that A can return 1-to B or one to one is reducible and write it as follows.

${\displaystyle A\leq _{1}B.}$ 

Subset of the natural number

Two meetings ${\displaystyle A,B\subseteq \mathbb {N}}$  But, it is said that there is it. Some kind of whole computability functions ${\displaystyle f}$  But, exist ${\displaystyle A=f^{-1}[B]}$  であるとき,${\displaystyle A}$  は ${\displaystyle B}$  I say and write that it is に 多対一還元可能 as follows.

${\displaystyle A\leq _{m}B.}$ 

In addition to this and ${\displaystyle f}$  But, when it is an injection,${\displaystyle A}$  は ${\displaystyle B}$  I say and write that it is に 1- reduction possibility as follows.

${\displaystyle A\leq _{1}B.}$ 

A many pairs all members level and 1-equivalent

${\displaystyle A\leq _{m}B\,\mathrm {and} \,B\leq _{m}A}$ であるとき, ${\displaystyle A}$  は ${\displaystyle B}$  I say and write that it is に 多対一同値 or the m-equivalent as follows.

${\displaystyle A\equiv _{m}B.}$ 

${\displaystyle A\leq _{1}B\,\mathrm {and} \,B\leq _{1}A}$ であるとき, ${\displaystyle A}$  は ${\displaystyle B}$  I say and write that it is the に 1- equivalent as follows.

${\displaystyle A\equiv _{1}B.}$ 

Many pairs one integrity (m-integrity)

When inductive countable set B exists, and all inductive countable set A can reduce m-to B, I say that B is many pairs one perfection or m-perfection.

Resources limit many pairs one reduction with it

The many pairs one reduction is often discussed in conjunction with the limit of calculation resources. For example, can the reduction function calculate in multinomial expression time and a logarithm domain? Specifically, refer to reduction and logarithm domain reduction at multinomial expression time.

There are decision problem A and B and does B again when there is solved algorithmic N. I apply N and can solve A if I can return A to B many pairs one then, but the cost of this time is as follows.

• Time necessary for time + reduction necessary to carry out N
• Domain necessary for biggest domain + reduction necessary to carry out N

When I cannot return a language not to be included in C to a language included in C many pairs one about class C of some kind of languages (or a set of the natural number), C says, "it is closed under the many pairs one reduction". It may be said the problem of the reduction cause being included in C when I come to other problems by many pairs one reduction by a problem included in C if C is closed under the many pairs one reduction. The reason why many pairs one reduction is convenient is that the most of the well known computational complexity are closed under some kind of many pairs one reduction. Such a class includes P, NP, L, NL, co-NP, PSPACE, EXPTIME and, besides, exists a lot. However, I do not close these classes under any many pairs one reduction either.

Property

• Many pairs one reduction and the one-on-one reduction are transitional and reflex and therefore make half order on a power set of the natural number.
• ${\displaystyle A\leq _{m}B}$  The の necessary and sufficient condition ${\displaystyle \mathbb {N} \setminus A\leq _{m}\mathbb {N} \setminus B}$  である.
• The necessary and sufficient condition that a certain meeting becomes reducible many pairs one for halting problem is that it is an inductive countable set. As for many pairs one reduction, this means that halting problem is the most complicated in every inductive countable set. Therefore, as for the halting problem, many pairs one is complete.
• The necessary and sufficient condition that halting problem (set of the input that, i.e., T finally stops) specialized in individual Turing machine T is complete many pairs one is that T is all-around Turing machine. I showed that the inductive countable set that was not m-perfection existed even if the Emil post could be decided. Therefore, the Turing machine which is not the almighty that inherent halting problem cannot decide exists. (c.f. simplicity meeting)

References

• E. L. Post, "Recursively enumerable sets of positive integers and their decision problems", Bulletin of the American Mathematical Society 50 (1944) 284-316
• Norman Shapiro, "Degrees of Computability", Transactions of the American Mathematical Society 82, (1956) 281-299

Footnote

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