An Overview of the. Computably Enumerable Sets. Abstract. The purpose of this article is to summarize some of the results on the

Transcription

1 An Overview of the Computably Enumerable Sets Robert I. Soare Abstract The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987, particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; ). We do not study the partial ordering of the c.e. degrees under Turing reducibility, although a number of the results here relate the algebraic structure of a c.e. set A to its (Turing) degree in the sense of the information content of A. We consider here various properties of E: (1) denable properties; (2) automorphisms; (3) invariant properties; (4) decidability and undecidability results; miscellaneous results. This is not intended to be a comprehensive survey of all results in the subject since 1987, but we give a number of references in the bibliography to other results. 1 A Brief History of C.E. Sets G del 1934 introduced the denition of a (general) recursive function, and Church 1936 proposed that this be taken as the denition of a computable function. Independently, Turing 1936 proposed that the computable functions be dened as those which could be computed by an automatic machine (now called a Turing machine). G del and most others accepted Turing's denition and analysis of computability as the most persuasive. 1 In addition to dening computable functions, there was in interest in dening computable generated sets. Church 1936 and Kleene1936 dened a set of positive integers to be recursively enumerable if it is the range of a recursive function. Little more was done with these sets until Post 1943 proposed a formal system for generating sets rather than computing their characteristic functions. Post introduced his production systems and in a restricted form his normal (production) systems and dened a normal set to be one generated by a normal system. Post The author was supported by National Science Foundation Grant DMS We cite references by their number in the order listed in our bibliography but also in the usual convention by author and year, e.g., [Post, 1944] or simply Post 1944, with the year in italics. To save space we omit from our bibliography some references which appear in Soare 1987, and we cite them as there by year. For several publications by an author in a single year, we list 1970 (=1970a), 1970b, 1970c etc. 1

2 showed that the normal sets are exactly the recursively enumerable (r.e.) sets, providing the empty set is added as an r.e. set. Post, however, thought not so much in formal systems as in informal terms and described the corresponding informal concept of eectively enumerable set or generated set. Post 1944 wrote, Suce it to say that each element of the set is at some time written down, and earmarked as belonging to the set, as a result of predetermined eective processes. It is understood that once an element is placed in the set, it stays there. Post then [1944, p. 286] restated his thesis from 1943 that every generated set of positive integers is recursively enumerable. In his paper 1944 Post did not use the formalism of recursive functions. He used either his informal notion of generated set, or when formalism was required, his own formal denition of normal set. Nevertheless, he accepted the terminology of recursively enumerable which Kleene and Church had proposed. In this paper we use the terminology computably enumerable (c.e.) to stress the intensional concepts and for the reasons explained in Soare A function is Turing computable if it is denable by a Turing machine, as dened by Turing 1936, see Soare [1987, p. 11]. A set A is computably enumerable (c.e.) if A is ; or is the range of a Turing computable function. By Turing's Thesis (T.T.) 1936 (which we accept) the (informal) class of computable functions is the same as the (formal) class of Turing computable functions. Hence, we shall use the term computable for either class of functions, and likewise we shall use the term computably enumerable for either class of corresponding sets. With its informal style Post's paper 1944 gave new excitement to computably enumerable sets and indeed the whole subject of computability. Post stated his famous Post's Problem, Does there exist a c.e. set A which is noncomputable but incomplete? He initiated Post's Program, the study of the relationship between the structure of A as a set and its information content, usually measured by its degree under Turing reducibility, dened by Turing Post proposed three properties he hoped might guarantee incompleteness, simple, h-simple, and hh-simple, but none succeeded. Post's Problem was solved by Friedberg [1957a] and Muchnik [1956a] with the introduction of the priority method. Meanwhile, Myhill 1956 had rediscovered the fact [Post, 1943 ] that the c.e. sets form a lattice E under inclusion, and he asked whether there is a maximal set (i.e., ua coatom of the quotient lattice E of E modulo nite sets) since this implies nondensity of E. Let fw e g e2! be the standard enumeration of the c.e. sets. We let E denote this lattice E = (fw e g e2! ; ; [ ; \ ; ;;!) as a lattice under union, intersection, with least and greatest elements. Often we let E denote just the partial ordering (fw e g e2! ; ) because the other four are denable from inclusion and because the lattice properties are rarely used in our paper. The solution of Post's Problem stimulated more research into Post's Program. In the 1960's using ever stronger forms of the priority method Sacks, Yates, Martin, Lachlan, and others obtained deeper results about Post's Program, particularly about maximal sets. For example, Martin 1966b showed that the degrees of maximal 2

3 sets are exactly the high c.e. degrees, and Lachlan 1968c extended this to all hhsimple sets. By the end of the 1960's the fascination with c.e. sets increased as Lachlan 1970a noted that all known constructions can be viewed as a game between two players each placing nitely many balls (integers) into buckets (c.e. sets) on each of! many moves in the game. At the end the second player wins if the sets satisfy certain prearranged requirements stated as E conditions. Lachlan suggested that this reduced the emphasis on computability to merely nding winning strategies. We may think of the denability results here and the automorphism results [Harrington- Soare, 1996c] as a contest between two players the denability player (RED) and the automorphism player (BLUE) such that any particular theorem for one or the other is a Lachlan-type game. Soare 1974 introduced machinery for generating automorphisms of E and showed that all maximal sets are automorphic. This machinery was then turned to the question of Rogers 1967 of whether all creative sets are automorphic, but here the automorphism method merely led to several false proofs in the 1970's that the creative sets did not form an orbit. In the mid 1980's Harrington, analyzing the failure of these attempted automorphisms viewed as games, distilled the obstacle and from it a winning strategy for the denability player. Harrington proved [Soare, 1987, p. 339] that there is an E-denably property, CRE(A), such that A is creative if and only if E j= CRE(A), a surprising fact considering how creativity appears to be so closely tied to a productive function, which does not seem to be preserved under automorphisms of E. This advance led eventually to the second denable property Q(A) in 2.2. Echoing Post's Program, Sacks asked in his book on degrees [1963, p. 172, Q(3)] whether there is a property in the style of Post of a noncomputable c.e. set A which guarantees its incompleteness. Marchenkov 1976 proved that -maximal semi-recursive c.e. sets are incomplete, and D. Miller 1981 later showed (see [36, p. 232]) they are all low 2. Soare [1987, p. 73] gave a property characterizing low c.e. sets, and Ambos-Spies and Nies 1992 gave a property characterizing c.e. sets whose degrees are cappable (i.e. halves of a minimal pair). However, these three properties are all non E-denable by [15, Theorems 3.1 and 3.2], respectively. The question for E-denable properties remained open and several partial negative results were obtained using automorphisms (see Downey-Stob 1992 ) in an unsuccessful attempt to show that every noncomputable c.e. set is automorphic to a complete set. Finally, by analyzing the failure of these automorphism attempts Harrington and Soare 1991 showed that the answer to the Post-Sacks question is actually yes; there exists an E-denable property Q(A) which guarantees that A is noncomputable and incomplete. In the automorphism direction, if one interprets the Post-Sacks question as a property of A alone, namely a denable property of L(A) = (fw : W Ag; ), then the answer is no because Cholak 1995 has shown that for every noncomputable c.e. set A and every high c.e. degree d there is a B 2 d such that L(A) = L(B). Convention. From now on all sets will be c.e. sets unless specically stated otherwise. 3

4 2 Denable Properties of C.E. Sets Denition 2.1 If fx s g s2! and fy s g s2! are recursive enumerations of c.e. sets X and Y dene X n Y = fz : (9s)[z 2 X s? Y s ]g; the elements enumerated in X before Y, and X & Y = (X n Y ) \ Y; the elements enumerated rst in X and later in Y. All the denable properties could be written just in the language L() over E because the functions [, \, and constants ; and! are denable in L() but for convenience we use the former also in order to improve readability. Likewise, we use the E-denability of the properties nite and computable [36, p. 179]. 2.1 Creative Sets Post 1944 dened a c.e. set C to be creative if there is a partial computable function such that (8e)[W e C =) (e)# 2 C? W e ] It follows by Myhill's Theorem [36, p. 43] that C is creative i C is m-complete, i.e., W e m C for every e, or equivalently K m C for the complete set K. Although these properties of C at rst appear to be very far from being E-denable, Harrington [36, p. 339] exhibited the following E-denable property CRE(A) which denes C being creative The Dening Property for Creative Sets Theorem 2.2 [Harrington] A c.e. set A is creative i (1) (2) CRE(A): (9C A)(8B C)(9R)[R is computable & R \ C is noncomputable & R \ A = R \ B]; where all variables range over E. We may represent the property CRE(A) as a two person game in the sense of Lachlan 1970 between the 9-player (called RED, the denability player) who plays the c.e. sets C, R (the red sets) and the 8-player (called BLUE, the automorphism player) who plays the c.e. set B (the blue set). Theorem 2.3 (Blue) If CRE(A) then K m A so A is creative. Proof. Suppose CRE(A). We may visualize R as dividing the universe! into two halves and on the R half we visualize in the Venn diagram the following states (corresponding roughly to e-states) 1 = R \ C, 2 = R \ C? B, 3 = R \ B? A, 4 = R \ A. The static condition CRE(A) forces certain dynamic properties of the sets as follows. The condition that R \ C is noncomputable means that R? C is 4

5 not c.e. so there must be an innite c.e. set of elements, say fx n g n2!, which move from state 1 to state 2. Dene (n) = x n. If n enters K, then enumerate (n) in B, from which the second conjunct of (2) eventually forces that (n) 2 A, so x n passes from 1 to 2 to 3 to 4 in that order. If n 2 K then x n remains in 2 forever. Hence, K m A via. This strategy succeeds if BLUE knows an index W e = R while he denes B, but in general he must play against all possible sets W i, i 2!, simultaneously, and dene a possible function i for each i 2!. To prevent some i < e from interfering with the minimal index e for R, we dene a new value i;s+1 (x) only if (3) (8y < x)[ i;s (y)[y 2 K s () i;s (y) 2 A s ]: Thus, except for nitely many elements taken for i < e the strategy for e succeeds as before. The condition (3) is a 0 2 condition on i so the construction may be viewed as being done on a tree with e corresponding to the rst innite node and hence the node along the true path. (For more details see [36, p. 339].) Lemma 2.4 (Red) If A is creative then CRE(A) holds. Proof. Since all creative sets are computably (recursively) isomorphic we can choose A to be a specic creative set. Dene A = fhx; yi : x 2 K & hx; yi 2 W y g: Now K 1 A because if we choose W y0 =! then x 2 K i hx; y 0 i 2 A. Next dene C = f hx; yi : x 2 K g. Given B c.e. and B C choose y 0 such that W y0 = B. Dene R = f hx; y 0 i : x 2! g. Now R \ C = f hx; y 0 i : x 2 K g which is not computable, and because B C. R \ A = R \ B = f hx; y 0 i : x 2 K & hx; y 0 i 2 W y0 = B g; Remarks on the Method Notice that this property CRE(A), and all later E-denable properties P (A) presented here, begin with (9C A). It is well known that in constructing a c.e. set A it is the complement A which matters during the construction because once an element x has entered A there is no further action with it. All the action in these constructions will be on C and particularly on C? A, where the two players will contest exactly which and how quickly elements will enter A. Thus, C? A provides a convenient arena for this contest, and the properties will imply that C? A is innite. Second, notice that all four results in this section show that a newly discovered E-denable property P (A) implies a well-known c.e. set condition C(A) like creativeness, completeness, incompleteness, or non-lowness. In every case, after the denition of P (A) we prove two theorems. The rst theorem by BLUE shows that P (A) implies C(A). We call this a BLUE theorem because if A satises the property P (A) then BLUE can play 5

6 certain blue sets to take control of certain red sets or A and can achieve C(A). For example, in CRE(A), the condition (2) that A \ R = B \ R means that on R the control of A has been turned over to B in the sense that on R: (i) A n B = ;, because any x 2 A n B could be withheld by BLUE from B forever contradicting (2); and (ii) that A must copy B in the sense that any element in B must eventually enter A. The second theorem by RED shows that there exists a set A with property P (A) (and sometimes with additional properties) so that the rst theorem is nontrivial. In the case of CRE(A) the second theorem showed that there is a creative set A satisfying P (A), and hence that every creative set C satises P(C). 2.2 Incomplete Sets After unsuccessful attempts to give a negative answer to the Post-Sacks question using automorphisms, Harrington and Soare produced the following E-denable property Q(A) which guarantees that A is incomplete but noncomputable. Denition 2.5 (i) A subset A C is a major subset of C(written A m C) if C? A is innite and for all e, C W e =) A W e : (Note that if A m C then both A and C are nonrecursive.) (ii) A < B if there exists C such that AtC = B (i.e. A[C = B and A\C = ;) The Property Q(A) Guaranteeing Incompleteness Theorem 2.6 (Harrington-Soare [13]) There is a property Q(A) which guarantees that A < T K and which holds of some noncomputable set. Dene the property Q(A) by: Q(A) : (9C) AmC (8B C)(9D C)(8S) S<C [ (4) (5) [B \ (S? A) = D \ (S? A)] =) (9T )[C T & A \ (S \ T ) = B \ (S \ T )]]: Theorem 2.7 (Blue) If Q(A) then A is incomplete ( i.e., A T K). Proof. (Sketch only, see Harrington-Soare 1991 for details.) We may visualize the property Q(A) as a two person game in the sense of Lachlan 1970 between the 9-player (RED) who plays the c.e. sets A, C, D and T and the 8-player (BLUE) who plays the c.e. sets B and S. For simplicity ignore all the sets but C, D, B, and A, since the others are necessary only to give us a suitable domain on which to play the following strategy. Visualize C D B A, and let 1 ; 2 ; : : : 5 denote the dierences of c.e. sets (called d.c.e. sets):!? C, C? D, D? B, B? A, A respectively, but viewed dynamically like e-states, so an element can pass from i to j, i < j. The oversimplied Q(A) property now asserts that if BLUE plays: 6

7 (4) 0 D = B on A, then RED will play: (5) 0 B = A. In particular, if both players are following their best strategies, then for an element x to enter A, it must pass through the -states in the order 1 ; 2 ; : : : 5 as proved in [13]. However, the set B acts like a wall of restraint, like the minimal pair restraint of Lachlan and Yates in [36, p. 153]. When presented with an x 2 D? B, BLUE may hold x as long as he likes, but must eventually put x into B at which point RED is free to put x into A but not before. This implies that A is tardy (i.e., not of promptly simple degree, see Ambos-Spies, Jockusch, Shore, and Soare in [36, p. 284] or [36, Chap XIII]), so A is incomplete. Furthermore, Harrington and Soare 1966d have recently discovered that Q(A) imposes a much stronger tardiness property on A (called 2-tardy) which helps us classify those sets which can be coded into any nontrivial orbit. Theorem 2.8 (Red) There exists a c.e. set A satisfying property Q(A). Proof. (Sketch only, see Harrington-Soare 1991 for details.) This proof is very similar to the standard proof (see Soare [1987, p. 194]) that every noncomputable c.e. set C has a small major subset A (A sm C) to which the reader should now refer. (See below for small sets.) Let C be any noncomputable c.e. set. (If we choose C simple (maximal) then A will be simple (r-maximal).) To make A m C it suces to meet for every e the requirement, P e : C W e =) A W e : Replace W e by V e = S s2! V e;s dened by (6) x 2 V e;s =) x 2 V e;s?1 _ [x 2 (W e;s? C s ) & (8y x)[y 2 W e;s [ C s ]]: Note that C & V i = ;, for every i. Dene the e-state (e; x; s) = fi : i e & x 2 V i;s g; with the usual ordering of e-states. Let C = f(!) for f a 1:1 computable function, and let c i = f(i): Let C s?a s = fd s 0 ; ds 1 ; g, in the ordering induced by fc 0; c 1 ; g. (Hence, if x = d s i = dt j for t > s, then j i.) The strategy for P e is as follows. If i e, and j > i is minimal such that (e? 1; d s i ; s) = (e? 1; ds j ; s), ds i 62 V e;s, and d s j 2 V e;s then P e wants to enumerate into A all the elements fd s k : i k < jg (but subject to the negative restraint by N i, i e, as described below). Let fb i : i 2!g and f(s j ; ^S j ) : j 2!g be an eective listing of all c.e. sets and all pairs of c.e. sets respectively. Let RED play D i against B i, D i C, and also construct T i;j to meet (5) if BLUE satises (4). Let = hi; ji; and let D, B, S, ^S, and T denote D i, B i, S j, ^S j, and T i;j respectively. For each the conjunction of the matrix of (4) for (B ; D ; S ) with the conditions B C, and S t ^S = C is a 0 2 relation F (). Let fz g 2! be an c.e. array of c.e. sets such that F () holds i jz j = 1. Dene T by (7) x 2 T ;s =) x 2 T ;s?1 _ [x 2 C s & x jz ;s j]: 7

8 The negative requirement N on A asserts that if (4) holds for (B ; D ; S ) then (5) holds for (B ; S ; T ). The strategy for N is this. If x 2 T & C, then N restrains x from A until x 2 S t ^S. (If the latter never occurs then F () fails so Z and T are nite, and only nitely many such x 2 T \ C are permanently restrained by N.) If x 2 ^S then N imposes no further restraint on x. If x 2 S, and while x 2 A, x is enumerated in B n D then N restrains x from both D and A forever (unless some P e, e <, enumerates x in A). (If N successfully keeps x 2 D \ A then x violates (4) so F () fails and Z and T are nite.) Otherwise, suppose that B D & B holds on S? A. If x 2 S and some P e, e, wants to enumerate x in A then N rst enumerates x in D and then restrains x from A until x is enumerated in B at which time N releases x (forever). (If x remains in B then x 2 (D?B )\(S?A) so x violates (4) and again T is nite.) Hence, in any of the three cases N permanently restrains at most nitely many elements. We now combine the P e and N -strategies to sketch the full construction of A. If x 2 C s, choose the least e (if any) such that P e wants to enumerate x in A. First P e waits until x 2 S t ^S for all e. Next let 0 ; 1 ; ; n be a listing of all e such that x 2 S and D 6= D for all <, with x 2 S. For each k, 0 k n, we make x pass through the N k -strategy above (also called the N k -gate) in the order N n ; : : : ; N 1 ; N 0. Hence, for example, when x is released by the N 2 -gate by being enumerated in B 2, RED then enumerates x 2 D 1 and waits for x to be released by N 1 by being enumerated in B 1. When x is released by the N 0 -gate RED enumerates x in A. Now it is easy to check that every requirement P e and N is satised. (See [13].) Small Sets and Remarks on the Q(A) Property Lachlan 1968d introduced small sets in his program to construct canonical examples of certain diagrams and then rule out possible extensions so as to give a decision procedure for the?! 8?! 9 -theory of the lattice of c.e. sets. The following denition is clearly equivalent to the standard denition as in [36, Denition 4.10, p. 193]. Denition 2.9 A subset A C is a small subset of C (written A s C) if A 1 C and for all X and Y, if (i) X \ (C? A) Y; then (ii) (X? C) [ Y is c.e., namely (ii) 0 (9Z) ZX [Z (X? C) & (Z \ C) Y ]; so Z = (X? C) [ Y and is c.e. If A is both a small subset and major subset of C we say it is a small major subset and write A sm C. Harrington and Soare 1996 showed that A sm C i the following dynamic property small-tardy(a; C) holds, namely (8f)(9T )[C T & (8x)[x 2 (T \ C) at s =) x 62 A f(s)]]: It is easy to show [36, p. 194] that: (1) if A C and either A or C is computable then A s C; (2) if A s C and C is noncomputable then A [ C is noncomputable. 8

9 Also if A s C and C is noncomputable then C? A is innite. To see the latter, suppose Y = C?A is nite; then (i) of Denition 2.9 is satised with Y = C?A and W =!; choose c.e. Z satisfying (ii) 0 ; then Z? Y is c.e., but Z? Y = C, contrary to C being noncomputable. Hence, the intuition is that A s C guarantees among other things that the A boundary is far below the C boundary. Small sets will be used again in the next section. Because of the similarity of the proof of Theorem 2.8 to the small major subset construction, it is natural to ask whether the Q-like property Q(A) b : (9C)[A sm C] guarantees A < T K. This is false, but Q(A) b implies that A is not a promptly simple set. This is easy to see from the property small-tardy(a, C) because if small-tardy(a, C) then A is not promptly simple [14]. 2.3 Complete Sets The property CRE(A) of 2.1 gives an orbit consisting of only complete sets, but all these are creative and therefore computably isomorphic, so there is no diversity including, for example, simple sets. In this section we present an E-denable property T (A) which implies that A is Turing complete, but which also allows A to be simple, even promptly simple, and to have other standard properties, such as being r-maximal or being a major subset of C. This exibility allows us to refute various automorphism conjectures and to limit the power of the automorphism building player (BLUE). For example, since the 1970's it has been known that one way to build an automorphism between certain promptly simple sets A and B is to rst construct an isomorphism on their complements guaranteeing that L(A) = L(B), and then to use prompt simplicity of A to ensure that A covers B and dually that B co-covers A in the sense of the automorphism machinery [36, p. 352]. Maass showed 1982 [36, p. 377] that this works if A and B are low and promptly simple. With this phenomenon in mind, Cholak raised the following question. Question 1. For all promptly simple high degrees h and for all promptly simple sets A is there a c.e. set B 2 h such that A ' 0 3 B? The present property T (A) negatively answers this question, and indeed shows that it is false with such A for every degree h 6= 0 0. Theorem 2.10 (Harrington-Soare [18]) There is an E-denable property T satised by a promptly simple set A such that for all W, T (W ) implies that K T W. Dene T (A) by: (8) (9) T (A) : (9C A)(8B C)(9 a computable set R) (R?C not c.e.) [B s C : =) : A (B \ R) & [(C? A) \ R is not co-c.e.] (We only sketch the proof which appears in [18].) To understand the intuition behind T (A) in an slightly oversimplied setting, assume that BLUE knows the true set R satisfying T (A). On R consider the states C, C? (B [ A), B? A, and A, denoted by 1, 2, 3, and 4, respectively. To show that T (A) implies 9

10 K T A, BLUE denes a Turing reduction A = K. When he denes A (n), he simultaneously denes the use function A (n) to be an element x n of (C \ R)? (B [ A), namely in state 2. Since A C and B C we may assume A n C = ; and B n C = ;. Hence, the clause R? C not c.e. guarantees that innitely many elements pass (ow) from 1 to 2. Furthermore, B s C implies that 2 is innite by the remarks in 2.2.2, so there will be a distinct position for each (n), and (n) eventually settles on some x n in state 2 if n 62 K. If n later enters K then BLUE puts x n into B but A B \ R by the second clause of (9), so x n eventually enters A allowing (n) to be redened. While n 2 K, if x n prematurely enters A, then (n) becomes undened and waits for a new element x 0 n distinct from the other (m). The nal clause of (9) guarantees that innitely many such elements exist so for every n 2 K we see that (n) comes to rest on some x 2 C? A. To prove the First Theorem for T (A), that If T (A) then K T A, BLUE takes given sets A and C satisfying T (A) and constructs the c.e. set B, Turing reductions i, and c.e. sets Z i, i 2!, to satisfy the following positive requirements, P i, for showing A = K as above, and negative requirements N i for showing B s C, for i 2!. P i : If T (A) and i is the minimal 0 -index of R satisfying (9) then i(a) = K; N i : Y i X i \ (C? B) =) Z i = (X i? C) [ Y i (which is therefore c.e.): To prove the second theorem for T (A) that there is a promptly simple set A satisfying T (A), the red player constructs C and A and the blue player a set B. Let R e =! [e] = f hx; yi : y = e g a computable decomposition of! into disjoint, innite, computable sets. Since RED does not know which set B BLUE is playing he considers all c.e. sets f W e g e2!, and plays against the candidate B = W e on the computable set R e. For convenience we now x e and drop the subscript e. We let B s denote W e;s. To achieve the clause A B \ R of (9) strategy commits to putting every element x 2 B \ R into A. Since A C and B C, we may assume that the enumerations satisfy A n C = ; and B n C = ;. The main task of is to arrange the nal clause of (9) that (C? A) \ R is not co-c.e., namely we must meet for every i the requirement, Q i : (C? A) \ R 6= W i : To achieve this will try to choose some element x 2 R s?c s, wait for x 2 W i;s, then enumerate x 2 C and restrain x from A, so that either x 2 (C? A) \ W i or else x 2 (C? A) \ W i, and either way requirement Q i is satised. The problem is that once x enters C s, then can restrain x from A only if x 62 B because must play A (B \ R). To force that x 62 B, assumes that B s C, namely that every requirement N i, i 2!. Decompose R into an innite disjoint union of innite computable sets, R = t j2! S j. Now the -strategy is as follows. In phase 1 we begin to enumerate the red sets X i and Y i so that X i includes an initial segment of C s, and Y i includes an initial segment of C s?b s until BLUE enumerates suciently many elements in one of his sets Z i;j,j 2!, in reply, attempting to make Z i;j = (X i? C) [ Y i for at least one j. In phase 2 we choose the least pair hx; ji (if it exists) such that 10

11 1. x 2 W i;s 2. x 2 S j;s? C s, 3. x 2 Z i;j;s, and 4. :(9y)[y 2 (Z i;j;s \ B s )? Y i;s ]: Enumerate x in C s+1 ; restrain x from Y i ; and temporarily cease all enumeration of X i and Y i, but if x later enters B, then return to Phase 1. (Necessarily, x 62 (A s [ B s [ Y i;s ) because B n C = A n C = Y i n C = ;.) This completes the description of strategy. Now suppose B s C. Then every N i is met. Hence, for the sets X i and Y i above there must exist a Z i;j satisfying N i. Suppose that (C? A) \ R = W i. Then W i (S j? C) \ R, which will be innite and X i C. Hence, eventually an element x 2 S j \ Z i;j \ W i is enumerated in C and permanently restrained from Y i. Hence, x 62 B because j cannot be bad for N i. Hence, x 62 A because nothing other than x entering B will cause RED to put x into A. The eect of strategy for requirement N i is this. For every j we may imagine a movable marker? i;j resting on an element? i;j;s = x 2 (S i;j;s? C s ) \ R s until x 2 W i;s \ Z i;j;s, at which point x enters C. Now either x remains in B forever and lim s? i;j;s = x, or else x eventually enters B at which point? i;j is removed forever and never again attached to any y because j has been proved bad for N i. Now it is easy to construct a promptly simple set A such that T (A) in the usual fashion. 2.4 Nonlow Sets The previous property T (A) guaranteed such a rapid ow of elements from C? B into A that A was complete. The next property NL(A) is more subtle but guarantees a suciently large ow into A so that A is nonlow, but A can still be low 2 and hence incomplete. It is also compatible with NL(A) to make A promptly simple as for T (A). (Recall that a c.e. set A is low n if A (n) T ; (n) and high n if A (n) T ; (n+1), and similarly for c.e. degrees.) Dene NL(A) by: (10) (11) (12) NL(A) : (9C A)(8B 0 ; B 1 )[ [B 0 t B 1 = C] & [B 1 A] : =) (9 computable R)[ [A B 0 \ R] & [R? B 0 is not c.e. ]]]: Theorem 2.11 (i) (8W )[NL(W ) =) W is not low]: (ii) (9A)[NL(A) & A is promptly simple and low 2 & A is semi-low 1:5 ]. This refutes the appealing conjecture based on Maass 1983 as discussed in Note that in the statement of NL(A) we can weaken the rst clause of (11) to B 0 t B 1 = C: Suppose B 0 [ B 1 = C, and F = C? (B 0 [ B 1 ) is nite. Then bb 0 = B 0 [ F and B 1 satisfy (11) so R must exist satisfying (12) for b B0 and B 1, and therefore for B 0 and B 1. Proof. (Proof of Theorem 2.11, sketch only, see details in [18].) As for the preceding properties we prove the theorem with two theorems. 11

12 2.4.1 The First Theorem for N L(A) Theorem 2.12 (Blue) (8W )[ NL(W ) =) W is not low ]: The informal intuition behind NL(A) is this. BLUE will enumerate B 0 and B 1 to satisfy the hypotheses (10) and (11), and we let R be the reply by RED satisfying (12). Let all sets be restricted to R. Let 1 be R? C, 2 be C? (B 0 [ B 1 ), 3 be B 0? A, 4 be B 0 \ A, 5 be B 1? A, and 6 be B 1 \ A, again interpreted dynamically. The second clause of (12) guarantees that R? C is not c.e., so there is a ow of innitely many elements from state 1 to 2. When such an element x arrives in 2, BLUE can wait an arbitrarily long time but must eventually put x either into B 1 (providing x 2 A already because of the second clause of (11)) or into B 0 (state 3 ) from which RED must eventually move x into A (state 4 ) because of the rst clause of (12). (Note that there is no ow from 2 to 5 only to 6.) The Second Theorem for N L(A) The property N L(A) can be used to show that certain automorphisms do not exist. To get the maximum power we wish to construct such an A which has as many other lowness properties as possible other than low 1, because these lowness properties tend to facilitate building automorphic copies of A. In addition to low 2 another lowness property which has been studied by Maass, Soare, Stob, and others in connection with automorphisms is the property dened in [36, p. 230] of A being semi-low 1:5, namely that f i : jw i \ Aj = 1 g m Inf, where Inf = f i : jw i j = 1 g. Maass [27] proved that a coinnite set c.e. set A satises L (A) =eff E i A is semi-low 1:5, where =eff denotes moreover that the isomorphism is eective [36, p. 244]. Theorem 2.13 (Red) There exists a c.e. set A such that: (i) NL(A); (ii) A is promptly simple; (iii) A is low 2 ; and (iv) A is semi-low 1:5. We rst sketch the strategy to achieve each of these four properties and then assemble them on a tree. The strategy to guarantee N L(A). Player RED enumerates C and A. Let f (B0; e B e 1 ) g e2! be a computable enumeration of all disjoint pairs of c.e. sets (B 0 ; B 1 ) such that B 0 [ B 1 C. Next RED species the computable set R e =! [e]. To achieve NL(A) as in RED must enumerate A to meet for every e the requirement, Q e : [B e 0 t B e 1 = C] & [Be 1 A] : =) : [A B e 0 \ R e ] & [R e? B e 0 is not c.e.] To achieve that R e?b e 0 is not c.e. it suces to meet for all i 2! the subrequirement, M e;i : W i 6= R e? B e 0: Let R e;i = R [i] e, i 2!; a computable decomposition of R e into innitely many innite pieces. RED will guarantee that C? R e;i is innite for all e and i. To meet 12

13 M e;i, RED waits until a suciently large initial segment of R e;i;s? C s has been enumerated in W i;s and then selects some x 2 (W i;s \ R e;i;s )?C s and enumerates x into C s+1, and waits for BLUE to enumerate x in B e 0 or B e 1. If BLUE enumerates x into B e 1 before RED enumerates x into A, then RED restrains x from A forever so B e 1 6 A and the requirement Q e is automatically met. If BLUE enumerates x into B e 0 then B e 0 \ W i 6= ; so requirement M e;i is satised forever. The remaining case is that BLUE never enumerates x into B e 0 or Be 1 in which case Be 0 [ B e 1 6= C and again requirement Q e is met. The Strategy to Make A Promptly Simple. This is the standard strategy. The Strategy to Make A semi-low 1:5. To ensure A semi-low 1:5 RED denes an array f Z i g i2! and denes h by W h(i) = Z i so as to meet the requirement, S i : jw i \ Aj = 1 () jw h(i) j = 1: Assume we have enumerated 0 through k-1 in W h(i). We wait until there are at least g(i; k) many fresh members of W i;s? A s, where g(i; k) is a predetermined computable function. Put these members in a set V i;k, restrain them from A with priority S i;k, and enumerate k in W h(i);s+1. Positive requirements of higher priority may later enumerate some members of V i;k in A, but the size of g(i; k) and the construction must be arranged so that once k enters V i;k at least one element of V i;k remains forever in A. Hence, requirement S i will be satised. The outcome of S i is 2 if W h(i) is innite and 2 otherwise. These outcomes are denoted by 0 and 1, respectively. The Strategy to Make A low 2. X X e;s (y) = e;s (y) undened To simplify the proof dene if (8z y)[ X e;s (z)#] otherwise. Dene the use function, e;s (y) = ' e;s (y) if e;s(t) is dened, and let e;s (y) be X undened otherwise. Note that e is either total or has nite domain, and X e is total i X e is total. The rst outcome is the 2 -outcome for N e and the second is the 2 -outcome. N e can ensure its success against a single positive requirement P i of lower priority as follows. P i has two separate forms, Pi 1 which guesses that N e will have the 2 -outcome, and Pi 0 which guesses that N e will have the 2 -outcome. Whenever a new value appears for e;s, the strategy Pi 1 is completely reset. Since Pi 0 believes that e is total, it can aord to wait for arbitrarily many values e;s (y) to appear before dening its next value x e n for? e. Therefore, it can wait until e;s(y) # for all y n before choosing x e n greater than all these values. Hence, a value e;s (y) can be later destroyed by the entry of some smaller x e m into A only if m < y, and this can happen for a single m at most once. Thus, in the presence of only this single positive requirement P i, N e just notices that this strategy guarantees that A e is total () (8y)(8k)(9s > k)[ A s e;s (y)#]: 13

14 The right hand side is a 2 predicate so Tot A is reduced to a 2 question and hence to ; 00. Now requirement N e easily accommodates all lower priority requirements f P i g i>e in the same fashion. The Tree for the Construction. We convert these four strategies into a tree construction. Let the tree T be f 0; 1 g <!. Put the empty node on T. 1. If 2 T and jj = 4e, then associate with a version of the strategy N e, to make A low 2, and put the nodes b0 and b1 on T which represent the 2 and 2 -outcomes, respectively, of the strategy. 2. If 2 T and jj = 4e+1, then associate with a version of the strategy for S i to make A semi-low 1:5, and put the outcomes b0 and b1 on T representing the 2 and 2 -outcomes of the strategy. 3. If 2 T and jj = 4e + 2, then associate with the strategy P e for prompt simplicity of A, and put b1 on T. (In cases 3 and 4 the outcomes are only 2 so we put only one successor node on T.) 4. If 2 T and jj = 4k + 3, and k = he; ii, then associate with the strategy M e;i, for ensuring that A has the NL property. Nodes dened under the rst two cases are negative nodes, and under the second two cases positive nodes. Each positive node contributes at most one element to A. Hence, each negative node will be injured at most nitely often by nodes or < L and will restart its strategy after each such injury. The construction is roughly as described above except in addition the positive nodes must respect negative restraint of higher priority nodes as follows. If is a positive node, jj = 4j + 3 or 4j + 4, b0, or < L, then in addition to the above conditions, when chooses a witness x: if jj = 4e, then must wait until ;s (k) # for all k < j, and must choose x to exceed all these values; and if jj = 4e + 1, then must wait until denes V ;k for all k j and must choose x not in these sets. Thus, will have the k th component of its strategy (i.e., ;s (k) or V ;k;s ) disturbed by only nitely many nodes (those with jj < k), and at most once by each of these. On the other hand if f, the true path, then for every such that b0 the action of the k th component of the strategy will stabilize because the strategy has the 2 -outcome. Hence, there will be at most a nite obstacle of restraint for to overcome, but an innite reservoir of elements from which to choose a witness. Therefore, the strategy will succeed. This completes the proof of Theorem Applications of the Property N L(A) In a very inuential paper Maass [26] introduced the notion of a c.e. set being generic and proved that every generic c.e. set is low 1 and promptly simple. He also showed that if A and B are promptly simple and low 1 then A is automorphic to B (A ' B). Meanwhile, Soare [35] proved that if A is a coinnite c.e. set and A is 14

15 low 1 or even semi-low 1, then L (A) =eff E, where L (A) denotes the supersets of A under inclusion, and =e ff denotes an eective isomorphism. Maass [27] then sharpened the second result by weakening the hypothesis to semi-low 1:5 and proving that if A is a coinnite c.e. set then L (A) =eff E i A is semi-low 1:5. Combining these ideas of these results led one naturally to ask the following tempting question. Question 2. If A and B are both promptly simple and their complements are semi-low 1:5 then are A and B automorphic? In a similar vein Cholak raised the following question. The outer splitting property was dened by Maass and follows from the semi-low 1:5 property. Question 3. Let A be a promptly simple set and p a promptly simple degree. If A is semi-low 2 and has the outer splitting property then is there a c.e. set B with deg(b) p, such that A ' 0 3 B? The point of these questions is that lowness properties on A guarantee that L(A) is well behaved, and prompt simplicity properties on A guarantee covering. Hence, one should be able to put the two halves together to produce an automorphism, at least a 0 3 automorphism [16]. A plausibility argument for the assertion in Question 2 is the following. Since A and B are both semi-low 1:5 we know that L (A) =eff L (B), and we begin to build a permutation h from A to B which induces this isomorphism. During this process elements will fall from A s to A and from B s to B but as in the maximal set automophism [34], but using prompt simplicity of A and B we know that a stream of elements enters A promptly enough to cover the stream of elements entering B so as in the Extension Theorem [36, p. 352] we can cover the stream of elements entering B by a prompt stream entering A and conversely. To refute both questions choose a c.e. set B which is low 1 and promptly simple and choose A to have the properties of Theorem 2.13, namely: NL(A); A is promptly simple; A is low 2 ; and A is semi-low 1:5. Now B cannot satisfy property NL(B) by Theorem 2.12 because B is low 1. But the property NL(X) is E-denable and therefore preserved under all automorphisms of E. Hence, A and B cannot be automorphic even by a 0 3 automorphism. What goes wrong with the plausibility argument above? When we apply the plausibility argument to the case where A and B are as here we see that Maass's theorem guarantees that in the limit we have L (A) = L (B) but we do not necessarily get that B co-covers A in real time during the construction. (Maass even explicitly remarks this, but his remark was apparently largely overlooked.) Therefore, on the A side a certain pathology develops which the B side cannot duplicate but need not duplicate because it will pass into A leaving the complements properly matched. However, this pathology already ruins the possibility of an automorphism and cannot be repaired after entering A by the mere hypothesis of prompt simplicity. The deeper conclusion here is that to produce an automorphism from A to B we need much more than isomorphisms on the complements and promptly simple type properties to guarantee covering. We need to study in a much deeper the way the c.e. sets relate the complement A to A not merely how they behave in isolation. The negative answer to Question 1 by the property T (A) in reinforces this principle. 15

16 3 Automorphisms of E 3.1 Some Results on Automorphisms Automorphisms are useful for two reasons. First, if we are unable to exhibit an E denition of some property P (see L 1 in 4), then we may be able to produce an automorphism of E mapping some A with property P to some B with :P, thereby proving P is undenable in E. The second use is in the spirit of Klein's Erlanger Programm, which is to classify some mathematical object such as a geometry in terms of the properties left invariant under its automorphisms. The rst application of the automorphism method was to classify orbits of maximal sets following Klein's program. To answer a question of Martin and Lachlan, Soare 1974 produced a new method for constructing automorphisms of E and used it to prove that any two maximal sets are automorphic. The method begins by choosing an appropriate skeleton for the c.e. sets (i.e., one member U g(e) from each class W e ) and then, by a fairly complicated construction, building an automorphism of E which is eective in the sense that there is a computable function h(e) such that (U e ) = W h(e). An automorphism is 0 3 if there is a 0 3 permutation h of! such that (W e ) = W h(e) for all e 2!. Recently, Harrington and Soare [16], and simultaneously Cholak [3] building on some conversations with Harrington, combined the essence of the eective automorphism method with the tree method of Lachlan 1975 to produce a powerful new method for constructing 0 3-automorphisms, and used it to prove, for example, the following. Theorem 3.1 (Harrington-Soare [16], Cholak [3]) For every noncomputable c.e. set A there is a c.e. set B which is high ( i.e., deg(b 0 ) = 0 00 ) such that A is 0 3-automorphic to B. Theorem 3.1 asserts that every nontrivial orbit contains a high set. This has some interesting corollaries for noninvariant classes as we shall see in 4. Before considering this, let us consider the relation of Theorem 3.1 to Theorem 2.6. Theorem 3.1 implies that Q(A) which prevents A from being complete cannot be extended to cause A to be low or even nonhigh. It says if we are willing to extend the target set from the complete degree to the high degrees, then the automorphism builder (BLUE) wins. On the other hand, if we insist on mapping A to a complete set, what stronger hypotheses must we place on A? Theorem 3.2 (Harrington-Soare [16]) If A is any c.e. set which is prompt ( i.e., of promptly simple degree) or even if A is almost prompt then A is automorphic to a complete set. Cholak, Downey, and Stob [7] proved this result under the stronger hypothesis A is a promptly simple set, and Harrington and Soare extended this with the much weaker hypothesis of promptly simple degree. They then realized that the essence of the hypothesis is a much weaker promptness property still, which they named 16

17 almost promptly simple, and which is based on the following notion of n-c.e. (also called n-r.e.) sets. In related work Downey and Stob have proved that the class of sets known as HHM sets are all automorphic to complete sets. We discuss the relationship between HHM and almost prompt in [16, 12]. Also Harrington proved [9] that there is no fat orbit, i.e., one containing a set in every nonzero degree. Denition 3.3 (i) A set X T K is n-c.e. if X = lim s X s for some computable sequence f X s g s2! such that for all x, X 0 (x) = 0 and cardf s : X s (x) 6= X s+1 (x) g n: For example, the only 0-c.e. set is ;, the 1-c.e. sets are the usual c.e. sets, and the 2-c.e. sets are the d.c.e. sets (also called d.r.e.). (ii) Such a sequence fx s g s2! is called an n-c.e. presentation of X. It is well-known and easy to show [36, Exercise III.3.8., p. 38] that for n > 0, X is n-c.e. i [ [ [ (13) X = (W e1? W e2 ) (W e3? W e4 ) : : : We2k+1 ; or (14) [ [ [ X = (W e1? W e2 ) (W e3? W e4 ) : : : (W e2k+1? W e2k+2 ); according as n = 2k + 1 is odd or n = 2k + 2 is even. Denition 3.4 For n = 0 let X 0 0 = ;. For n > 0 and e = he 1; e 2 ; : : :e n i dene (15) X n e = (W e 1? W e2 ) [ : : : ; as in (13) or (14) according as n is odd or even. We say that hn; ei is an n-c.e. index for X n e. Let (16) X n e;s = (W e 1;s? W e2;s) [ : : : : Denition 3.5 Let A be a c.e. set and let fa s g s2! be a computable enumeration of A. We say A is almost prompt, abbreviated a.p., if there is a nondecreasing computable function p(s) such that for all n and e, (17) X n e = A =) (9x)(9s)[x 2 Xn e;s & x 2 A p(s) ]: (ii) We say A is very tardy if A is not almost prompt, namely if for every nondecreasing computable function p(s), the negation of (17) holds. In this case, if a xed n works uniformly for all such functions p then we say A is n-tardy. Note that for the case n = 2 this is equivalent to our denition of being 2-tardy [16] and [14] which is a very important special case. 17

18 The results here on almost prompt sets and on 2-tardy sets stress the very important, but previously often hidden, connection between dynamic properties on one hand, and denable properties and automorphisms on the other. Here is another unexpected connection. In Theorem 3.2 we put extra hypotheses on the set A so that it could be mapped to a complete set. Now we ask what hypotheses are necessary on a set D so that it can be computably coded into some set B in every nontrivial orbit. The answer involves the 2-tardy sets in an unexpected way. The orbit of A is the class [A] of all sets B automorphic to A, written A ' B. By a nontrivial orbit we mean the orbit of a noncomputable c.e. set A. Denition 3.6 (i) We say X can be coded into the orbit of A, denoted by X T [A], if X T B for some B 2 [A]. (ii) We say X is codable if X can be coded in every nontrivial orbit, namely if X T [A] for every A > T ;. Theorem 3.7 (Harrington-Soare, Coding Theorem [17]) If D is 2-tardy (say if Q(D) holds) then D is codable. Corollary 3.8 A set X is codable i X T X T D for some 2-tardy D. D for some D satisfying Q(D) i Proof. If X T D and D is 2-tardy, then X T D T [A] for every A > T ; by Theorem 3.7. If X T [A] for every A > T ; then X T C for some C and D such that C 2 [D] and Q(D), hence D is 2-tardy. Hence, Q(C) because Q is E-denable. Note that Q(D) implies that D is a major subset and hence D is high. Thus, Theorem 3.7 is a very strong generalization of Theorem 3.1 because by Theorem 3.7 if A is noncomputable and Q(D) holds, then there exists B 2 [A] such that D T B so D and B are both high. Thus, Q(D), and its associated property of D being 2-tardy, were originally introduced just to force the enumeration of elements into D to be suciently slow so that D would have to be incomplete. Now we see that it also forces a suciently slow enumeration so that the machinery building an automorphism (A) = B has time to code into B the fact that x enters D. This connection of the speed of elements entering D to its codability into orbits is so interesting that we explore it further for a moment. Corollary 3.9 If S is a promptly simple set (or even of promptly simple degree) then S is not codable. Proof. If S is promptly simple (or even of prompt, i.e., of promptly simple degree) and S T C then C is also prompt [36, Corollary XIII.1.9, p. 287], thus not tardy, thus not 2-tardy, thus :Q(C). Hence, by Corollary 3.8 S is not codable. Thus, codable sets can be high by Theorem 3.1, while noncodable sets can be low (choose a low promptly simple set in Corollary 3.9). Therefore, one of our main conclusions is that the question of whether a set X can be coded into an arbitrary orbit [A] depends more on the speed of enumeration of X (prompt or tardy) than on its information content (high or low). 18

19 The fact that K is not codable has more to do with the fact that K is prompt (i.e., of promptly simple degree) than that K has complete information content. For example, to show that K T B for some B 2 [A] we must do very rapid coding, but if Q(A) holds then Q(B) holds for every B ' A (because Q is E-denable although 2-tardy is not). Thus, B is 2-tardy, hence tardy, and hence incomplete. To code D into [A], the orbit of a given noncomputable set A, we must construct B ' A and a computable functional such that D = (B). We must dene the use function (n) to be a convenient element y not yet in B such that if n enters D; then we can gradually move y into B. The property of D being 2-tardy, as described later, will imply that there is a computable function g (played by BLUE) such that if n wants to enter D; it must rst declare that intention at some stage s and then wait for some stage t p(s) before doing so. Since the automorphism machinery imposes considerable delay in putting (n) into B after rst starting the process, BLUE arranges that when n declares its intention at stage s, BLUE starts (n) toward B immediately and makes p(s) so large that (x) has arrived in B by stage p(s) before n has arrived in D. The entire automorphism construction is more complicated and is played on a tree T of nodes. Thus, our actual coding procedure is a bit more complicated as it is performed repeatedly for several nodes 2 T. Perhaps, (n) begins in the region R (dened in [16]) with witness y < (n). Now in its journey toward D, element n passes through a series of gates G for 2 T with witnesses y, each time undergoing a delay as above with G in place of D. (In reality these sets G are simply dierent names for the set D at least for f, where f is the true path through T.) After each successful entry y into B, the use function is redened to some number above y where =?, the predecessor of and (n) passes from region R to region R. Best of all, to complete the proof we do not need to know anything about the machinery for generating 0 3-automorphisms of E. Rather, in [16, 7] we develop the full automorphism machinery and a coding theorem, which can be applied here without further proof. Further details on coding can be found in [17]. We give a brief sketch here. 3.2 A Sketch of the 0 3-Automorphism Method By [36, page 343] building an automorphism of E is equivalent to building one of E, the quotient lattice of E modulo the ideal F of nite sets. To do this we x two copies of the natural numbers! and ^!. We let variables x; y; : : : (^x; ^y; : : :) range over! (^!). Normally, we shall specify the denitions and action for only one side (usually the!-side) since those for the opposite side will be entirely dual. We view the construction of the automorphism as a game between two players in the sense of Lachlan [24]. Player 1 (whom we call RED ) produces two standard indexings fu n g n2! and fv n g n2! of the r.e. sets, where we view U n as being on the!-side and V n on the ^!-side. Player 2 (whom we call BLUE ) responds by building r.e. sets f b Un g n2! on the ^!-side and fb Vn g n2! on the!-side. The condition necessary to show that this correspondence (U n ) = b Un and b Vn =?1 (V n ) is an automorphism is best stated in terms of the following notion of full e-state. 19