Proc. Natl. Acad. Sci. USA Vol. 92, pp. 617-621, January 1995 Mathematics Algebraic aspects of the computably enumerable degrees (computability theory/recursive function theory/computably enumerable sets/Turing computability/degrees of unsolvability) THEODORE A. SLAMAN AND ROBERT I. SOARE Department of Mathematics, University of Chicago, Chicago, IL 60637 Communicated by Robert M. Solovay, University of California, Berkeley, CA, July 19, 1994 ABSTRACT A set A of nonnegative integers is computably equivalently represented by the set of Diophantine equations enumerable (c.e.), also called recursively enumerable (r.e.), if with integer coefficients and integer solutions, the word there is a computable method to list its elements. The class of problem for finitely presented groups, and a variety of other sets B which contain the same information as A under Turing unsolvable problems in mathematics and computer science. computability (<T) is the (Turing) degree ofA, and a degree is Friedberg (5) and independently Mucnik (6) solved Post's c.e. if it contains a c.e. set. The extension ofembedding problem problem by producing a noncomputable incomplete c.e. set. for the c.e. degrees Qk = (R, <, 0, 0') asks, given finite partially Their method is now known as the finite injury priority ordered sets P C Q with least and greatest elements, whether method. Restated now for c.e. degrees, the Friedberg-Mucnik every embedding ofP into Rk can be extended to an embedding theorem is the first and easiest extension of embedding result of Q into Rt. Many of the most significant theorems giving an for the well-known partial ordering of the c.e. degrees, R = (R, algebraic insight into Rk have asserted either extension or <, 0, O'). nonextension of embeddings. We extend and unify these THEOREM 1.1 (FRIEDBERG-MUCNIK). results and their proofs to produce complete and complemen- tary criteria and techniques to analyze instances of extension (3x)[0<x<O']. and nonextension. We conclude that the full extension of Sacks (7) introduced a stronger form of the finite injury embedding problem is decidable. priority method to prove the following splitting theorem. Let a V b denote the join (least upper bound) of degrees a and b which always exists in R. Section 1. Introduction THEOREM 1.2 (SACKS SPLITrING THEOREM). Godel's incompleteness theorem (1) and his subsequent 1934 > y & x V y = a]. work on computable functions (2) showed that undecidability (Va 0)(3x, y)[x (i.e., noncomputability) is resident in the most familiar math- Sacks (8) then developed a strong version of the infinite ematical settings, even in elementary number theory. Follow- injury priority method to prove the density theorem. ing Godel, there has been an intensive study of noncomputable THEOREM 1.3 (SACKS DENSITY THEOREM). sets arising in ordinary mathematics. The computably enumer- able (c.e.) sets are of particular interest here, because these are (Va,b)[a<b=(3x)[a< x< b]]. the sets which can be computably listed. The c.e. sets are the The Sacks splitting and density theorems demonstrate the most effectively presented sets beyond those which are com- most well-behaved algebraic properties of R, and they inspired putable (i.e., recursive). The facts that natural examples of c.e. Shoenfield (9) to conjecture a universal extension of embed- sets appear in most branches of mathematics and that there is ding property for R. Shoenfield conjectured that for all finite a noncomputable c.e. set are pivotal in the proofs of such partial orderings P C Q with join and least and greatest well-known results as Godel's incompleteness theorem, the elements, any embedding ofP into the usual upper semilattice unsolvability of the word problem for finitely presented (R, <, V, 0, 0') can be extended to an embedding of Q into the groups, and the unsolvability of Hilbert's tenth problem on same. However, Lachlan (10) and independently Yates (11) Diophantine equations. refuted Shoenfield's conjecture. Let a A b denote the infimum Turing (3) introduced a notion of relative computability. For (greatest lower bound) of a and b in R if it exists. sets of natural numbersA and B, we say thatA is Turing reducible THEOREM 1.4 (MINIMAL PAIR THEOREM, LACHLAN AND to B or computable in B if there is an algorithm to decide whether YATES). x EA when given answers to all questions ofthe form "Isy E B?" We will write A CT B to indicate that A is computable in B and (3a, b)[a b & a A b = 0]. A =TB ifA <TB and B ETA. The equivalence class ofA under -T is the (Turing) degree ofA and is written deg(A) = a. A degree A pair of degrees satisfying Theorem 1.4 is called a minimal is c.e. if it contains a c.e. set. Lowercase boldface letters a, b, c, pair. The method of proof is called the minimal pair method. ... x, y, z will denote c.e. degrees. Theorem 1.4 was the first nonextension of embedding result for Post (4) noted that the natural noncomputable c.e. set K R. It asserts that one cannot always extend an embedding ofthe arising from Godel's work is Turing complete in the sense that partial order in which a and b appear as incomparable W CT K for every c.e. set W Hence, the c.e. degrees form a elements to an embedding of a larger partial order for which partially ordered set RIt = (R, <, 0, 0') with least element 0 = there is an x so that x < a, b and 0 < x. deg(0) and greatest element 0' = deg(K). Post then posed this The density theorem method was applied by Robinson (12, famous question (Post's problem): Does there exist a c.e. 13) and others to produce stronger extension of embedding degree a which is not computable and not complete (i.e., 0 < results, while the minimal pair method was developed by a < 0')? If not, then there would be a single Turing degree Lachlan and others to produce nonextension results and embedding of various lattices into the c.e. degrees. The next The publication costs of this article were defrayed in part by page charge significant advance was the Lachlan nonsplitting theorem (14), payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. Abbreviation: c.e., computably enumerable. 617 Downloaded by guest on September 25, 2021 618 Mathematics: Slaman and Soare Proc. Natl. Acad Sci. USA 92 (1995) which asserts that the Sacks splitting and density theorems order embedding. We now describe conditions for deciding cannot be combined simultaneously. whether f can be extended to an embedding g: Q -- Rk. THEOREM 1.5 (LACHLAN NONSPLITrING THEOREM). Definition 2.1: For S a subset of Q, we define -,(Va,b)[b<a (3x,y)[b<x,y<a&a = xVy]]. A(S) = {ala EP & for allxinS,a -x} Lachlan's result was significant first because it demon- i(S)S={bjbEP & forallxinS,x-b}. strated a new kind of nonextension phenomenon, and second because its proof introduced a powerful technology for con- At(S) and 3(S) are the filter and ideal in P determined by structing c.e. sets, which is now referred to as the O0-priority S. We will write sA(x) for L({x}) and 2(x) for R({x}). We also method [see Soare (15), Chap. XIV]. write x : 21 if x : b for all b E 213 and similarly x . s4 ifx c During the 1970s and 1980s this method was further devel- a for all a C si. Typically, we will use a, b, c, and d to denote oped and applied to prove a number of deep results about R elements ofP and x, y, z, u, and v to denote elements of Q. The such as the Harrington-Shelah theorem (16) that the elemen- two nonextension conditions are the following: tary theory of Rk is undecidable. These results demonstrated that 2R has a more complicated global structure than antici- (3x,y E Q)[x jy & 9a(si(y)) C (s(21(x)))], [1] pated by the results of the 1960s. However, it remained evident that 2k admits a considerable level of algebraic analysis. In this (3x E Q - P)[2(x) = 0 & R(AM(x) U 1(x))) X 013(x)], [2] vein, attention was directed toward embedding and extension of embedding problems for 2k. In fact, virtually all the major where algebraic results about R can be viewed as embedding, exten- Q-P & x>z & x) X 3(s(z))}. sion of embedding results, or nonextension of embedding !(x)=f{zlzE results for R viewed as either a partial ordering or sometimes In Section 3 we prove that if condition [1] holds, then we can with partial lattice structure. embed P into 2k so that there is no extension to an embedding The extension of embedding question was solved for certain of Q into 2R; the proof uses a combination of elements from the related structures. For example, Fejer and Shore (17) solved it of the minimal pair and splitting theorems. In Section for the c.e. tt-degrees and wtt-degrees. Slaman and Shore (18) proofs calculated the extension of embedding theory which is com- 5 we prove that if condition [2] holds, then we can construct mon to all principal ideals in 2k for which the top point is low2 a similar counterexample to the extension of embedding; the (i.e., a' = 0"). Roughly speaking, the method of proof in these proof uses a more complicated version of the Lachlan non- results is to rule out certain extensions by using the minimal splitting technology.