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?W H A T I S . . .

Turing Reducibility?Martin Davis

Intuitively, to say that something is computable

means that there is an algorithm for computing

it. Computability theory makes this concept pre-

cise. So, in particular, we are enabled to define

with full rigor what it means to say that a function

defined on the natural numbers and with natural

number values is computable. Likewise a set of

natural numbers is computable if its characteristic

function (defined to be 1 for members and 0 for

non-members) is computable. Having such defini-

tions makes it possible to prove that certain objects

are not computable. A fundamental result is that

there is a computable function whose range (theset of all values that it assumes) is not computable.

Sets that are the range of a computable function

are called recursively enumerable. (The empty set

is also considered to be recursively enumerable.)

The fact that there is a recursively enumerable

set that is not computable is a special case of a

more general result that will be explained later in

this article. It isthe use of thisfact that has made it

possible to prove that a numberof importantmath-

ematical problems are unsolvable, that algorithms

that mathematicians had been seeking simply do

not exist. Among these problems are Hilberts 10th

Problem (to decide whether a given Diophantineequation has solutions), the word problem for

groups (to decide whether a given product of gen-

erators and their inverses is the identity element

of a group defined by a finite set of equations

between such products), and the homeomorphy

problem (to decide whether the topological spaces

Martin Davis is professor emeritus of mathematics and

computer science, New York University, and is a visiting

scholar in mathematics at the University of California,

Berkeley. His email addressismartin@eipye.com .

defined by a given pair of simplicial complexes are

homeomorphic).

Theconcept ofTuring reducibilityhastodowith

the question: can one non-computable set be more

non-computable than another? In a rather inciden-

talaside to the main topic of Alan Turings doctoral

dissertation (the subject of Soloman Fefermans ar-

ticle in this issue of the Notices), he introduced the

idea of a computation with respect to an oracle. An

oracle for a particular set of natural numbers maybe visualized as a black box that will correctlyan-

swerquestions aboutwhether specific numbers be-

longto thatset.We can thenimagine an oraclealgo-rithmwhoseoperationscanbeinterruptedtoquery

such an oracle with its further progress dependent

on the reply obtained. Then for sets A, B of natu-

ral numbers, A is said to be Turing reducible to B

if there is an oracle algorithm for testing member-

ship in A having full recourse to an oracle for B.

The notation used is: A t B. Of course, if B is it-

selfacomputableset,thennothingnewhappens;in

suchacaseA t Bjustmeans thatA is computable.But ifB is non-computable, then interesting things

happen.

As the notation suggests,Turing reducibilityis a

partial order. If sets A, B are each Turing reducibleto the other, they are said to be Turing equivalent,

written A t B. And ifA is Turing reducible to B

but not conversely we write A

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, the set provides an example of a recursive-ly enumerable set that is not computable. Iteratingthe jump operation, one obtains the sequence ofmoreandmoreunsolvableproblems,,, . . ..

The relation of Turing equivalence is, naturally,an equivalence relation, and the equivalence class-es are called Turing degrees. One speaks of the de-gree of a set of natural numbers to mean the equiv-alenceclassto which it belongs. Thedegrees inheritthepartialorder,andthejumpoperationisadegreeinvariant. All of the computable sets form a singledegree written 0 which is at the bottom of the par-tial order. That is, 0 a for every Turing degree a.Also, a < a. Are there degrees between a and a?Kleene and Post were able to show that the order-ing of degrees is complicated and messy.For exam-ple, for any degree a, they showed how to obtaindegrees b, c such that a < b < a, a < c < a, but bandcareincomparable: neither is less than theoth-er. They also found densely ordered degrees; thatis, they showed that for a given degreea, an infinite

linearly ordered setWof degrees between a and a

can befound such that ifb, c W,thereisadegreedWbetweenbandc.

The degree of every recursively enumerableset is 0. There is a sense in which the typicalmathematical problems that have been proved to

be unsolvable are of degree 0. For example, if weenumerate all polynomial Diophantine equationswith integer coefficients in some standard way, thedegree of the set of natural numbers n such thatthe nth equation has a solution in natural numbersis exactly 0. So we can say that Hilberts tenthproblem is not only unsolvable but has exactlythe degree of unsolvability 0. A degree is calledrecursively enumerable if it contains a recursivelyenumerable set. 0 and 0 are both recursively enu-merable degrees with0 < 0. In a classic paper Postraised the question of the existence of other recur-sivelyenumerable degrees, andthis became knownas Posts Problem. It required a new combinatorialtechnique, known as the priority method, to settlethe question. The idea was to list a countable infin-ity of requirements that the desired objects wouldneed to satisfy, and to mediate among conflictingrequirements in a manner that would result in allof them being ultimately satisfied. By using thistechnique, it was shown that not only are there

recursively enumerable degrees strictly between0 and 0, but indeed that pairs of such degreescan be found that are mutually incomparable. Theuse and refinement of the priority method hasmade it possible to prove a number of strikingfacts about the recursively enumerable degrees.For example, the Sacks Density Theorem statesthat for given recursively enumerable degreesa < b, there is a recursively enumerable degree csuchthata < c < b.

References

[1] Hartley Rogers, Theory of recursive functions

and effective computability, New York: McGraw-Hill

1967.

[2] Joseph Shoenfield, Recursion theory, Lecture

Notes in Logic 1, Berlin, New York: Springer-Verlag,

1993.

[3] Robert Soare, Recursively enumerable sets and

degrees: a study of computable functions andcomputably generated sets, Berlin, New York:

Springer-Verlag, 1987.

November2006 Notices of theAMS 1219