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Chapter 3 Isomorphic recursive structures
Chapter 3 Isomorphic Recursive Structures ]C. J. Ash* I Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Recursive structures Recursive stability Intrinsically recursive relations Recursive categoricity A~ Back-and-forth relations Recursive infinitary formulae A~ Intrinsically E ~ relations Non-recursive structures c~-systems References
An early result about recursive structures is the following, attributed to S. Tennenbaum. There is a recursive linear ordering of order type co + co* having no infinite recursive ascending or descending sequence. There are also, of course, more obvious recursive orderings of this type, for example 0 , 2 , 4, ... , 5, 3, 1. An ordering of the first sort, which can be constructed by a priority argument, is classically isomorphic to the second, but cannot be recursively isomorphic since it has different recursive properties. Some other examples are mentioned in [7]: a recursive linear ordering of type w on which the successor relation is not recursive; a recursive vector space over any recursive infinite field having no recursive basis; a recursive algebraically closed field of any characteristic having no recursive transcendence basis. *Due to the untimely death of the author, the final version of this paper was prepared by Professor J. Knight. The editors of the volume express their gratitude to Professor Knight for this valuable assistance. HANDBOOK OF RECURSIVE MATHEMATICS Edited by Yu. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. R e m m e l ~) 1998 Elsevier Science B.V. All rights reserved.
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In each of these cases it is clear that within the isomorphism type of a single structure we have recursive structures which are not recursively isomorphic. In this article we discuss general results which have been obtained concerning this and related matters.
Recursive structures We consider structures of some fixed recursive similarity type (I, J, K, #, i.,) where I, J and K are recursive sets and # : I -+ w and I., : J --+ w are recursive functions. A structure of this type is a quadruple of the form 9/ -
(A, (R~)iei , ( f j ) / e j , (ak)keK)
where each Ri is a # ( i ) - a r y relation on A, each fi is a u ( j ) - a r y operation on A, and each ak C A. Such a structure 91 is a recursive structure if A is a recursive set, each Ri is a recursive relation on A, uniformly in i, each fj is a recursive operation on A, uniformly in j, and the function k ~-~ ak is recursive. An alternative definition is used by the Russian workers in this area. Define a constructivization a of an abstract structure 92, to be an onto function a : N --+ A, such that for atomic formulae ~ the relations {~]P.l ~ ~[a(6)]} are recursive, uniformly in ~. Then such a pair (9.1,a) is essentially a recursive structure isomorphic to 91 (together with a particular isomorphism). Indeed, this approach is closer to mathematical nature, where one thinks of a classical isomorphism type as being an abstract structure. But, in practice, it seems to reduce the number of symbols significantly to deal only with the recursive structures themselves, and we shall follow this line.
2
R e c u r s i v e stability
A structure with which the least can be done by way of different recursive copies is (w, < , S) where S is the successor relation on ~o. This structure is rigid, that is, has no automorphisms other than the identity function, and it is easily seen that for any two recursive structures, 91 and if3, which are isomorphic to (w, < , S), the unique isomorphism from 9,1 to if3 is recursive. This rather strong property, that all isomorphisms between recursive copies are recursive, has been called "recursive stability".
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D e f i n i t i o n 2.1 A recursive structure 92 is said to be recursively stable if for every recursive structure ~ -~ P.I, every f 9 ~ -~ 9.1 is recursive. It seems plausible to expect that this phenomenon will occur only when the elements of the structure are suitably definable. D e f i n i t i o n 2.2 An r.e. defining family for a structure 92 is an r.e. set S of existential formulae qp(v) of one variable, such that each is satisfied by exactly one element of 9.1, and each element of 9/satisfies at least one of the formulae ~(v) from S. If a recursive structure 9.1 is such that some expansion (92, 2) of 92 by a sequence g of finitelymany constants from P2 has an r.e. defining family, then it is clear from the following that 92 is recursively stable. For any recursive ~- P2 and any f " ~ ~- 92, we have f ' ( ~ , b) -~ (92, g) for suitable b. Then f is the unique isomorphism from ( ~ , b) to (9.1,g), and its graph is r.e., since it is the set of pairs (d, c) such that, for some qp in the family, ( ~ , b) ~ ~[d] and (P2, g ) ~ qp[c]. The converse statement, that a recursively stable structure 92 has a finite expansion with an r.e. defining family, does not hold without some further assumption. Let us define the existential diagram of a recursive structure 9.1 to be the set of existential sentences true in the structure (P2, (a}~eA). Since P2 is a recursive structure, this set is clearly r.e.. T h e o r e m 2.1 Let 9.1 be a recursive structure whose existential diagram is recursive. Then 9.1 is recursively stable if and only if some finite expansion of ~ by constants has an r.e. defining family. This result is implicit in [12]. The non-trivial direction is obtained by constructing simultaneously a recursive structure ~ and an isomorphism f " ~ -~ P.l which is not recursive, using a finite injury priority argument. This argument is also given in [7]. As with other results to come, Theorem 2.1 needs the assumption that at least one recursive isomorphic copy has some extra recursive properties. In [13], an example is constructed of a recursive structure P2 which is recursively stable but has no r.e. defining family.
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Intrinsically recursive relations
For a recursive structure 92 and an additional relation R on A, a construction similar to that used to prove Theorem 2.1 can often be used to give a recursive structure ~ and an isomorphism f : 91% ~ for which f ( R ) is not recursive. If there are no such ~ and f, then the relation R is said to be intrinsically recursive on 9.1. If a relation R on 9.1 has a definition of the form n
for some sequence ~ from 92 and some sequence ~n(x,y) of existential formulae, then R will be called formally r.e. on 92. Clearly, if both R and its complement are formally r.e., then R is intrinsically recursive. The converse can be shown provided that, again, (9.1,R) satisfies further recursive conditions. T h e o r e m 3.1 (Ash [71) Let (P.I, R) be a recursive structure whose existential diagram is recursive. Then R is intrinsically recursive on 9.1 if and only if both R and its complement are formally r.e. on 91 The construction for this attempts to make both f ( R ) and its complement not r.e.. A similar construction only attempting to make f ( R ) not r.e. gives the more basic result: T h e o r e m 3.2 (Ash [7]) Let (91, R) be a recursive structure whose existential diagram is recursive. Then R is intrinsically r.e. on 9.1 if and only if R is formally r.e. on 9.1. The examples mentioned earlier can all be deduced from Theorem 3.1 without repeating the priority argument. Some assumption is needed for the equivalence in Theorem 3.1. In [16], an example is given of a recursive (92, R) for which R is intrinsically recursive on 92 but is not formally r.e. on 92. If R is not intrinsically recursive on 92, we may define the spectrum of R on 91 to be the set of all possible Turing degrees of the relations f ( R ) for recursive structures ~3 and isomorphisms f : 91 ~ ~ . The possible spectra obtained in this way are considered in detail by V. Harizanov in [15].
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Isomorphic Recurswe Structures
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Recursive categoricity
A recursive structure 91 is said to be recursively categorical if, for every recursive ~ "~ 91, there is at least one recursive isomorphism f 9 ~ -~ 91. For example, the ordering of the rationals (Q, <) has this property, as can be shown using an effective back-and-forth argument. Clearly (Q, <) is not recursively stable, since it has 2 ~~ automorphisms, and so has 2 ~~ isomorphisms with any recursive copy. In fact, it follows fairly briefly from the definitions that a structure 91 is recursively stable if and only if, for some finite sequence ~ from 91, (91, ~) is both rigid and recursively categorical. The example of (Q, <) suggests a sufficient condition for 9.1 to be recursively categorical. D e f i n i t i o n 4.1 An r.e. automorphism family for 91 is an r.e. family of existential formulae r in different numbers of variable, such that every finite sequence from 91 satisfies at least one of these formulae while, if 91 ~ ~n[b] and 91 ~ ~n[~], then (P2, b) "~ (91, ~). -
-
m
Clearly, if PJ, or more generally some (9.1,~), has an r.e. automorphism family, then, by an effective back-and-forth argument, PJ is recursively categorical. The converse can be shown subject to further assumptions on 9./. An 3V sentence means one of the form 3 5 V ~ ~(~, ~), where T(~, ~) is quantifierfree. T h e o r e m 4.1 (Goncharov [12]) Let 91 be a recursive structure whose 3V diagram is recursive. Then 92 is recursively categorical if and only if some finite expansion (9.1,-5) has an r.e. automorphism family. In the case where there is no r.e. automorphism family, one can construct not just two, but infinitely many recursive copies of 91 which are pairwise not recursively isomorphic. In other words: T h e o r e m 4.2 (Goncharov [13]) Let 91 be a recursive structure whose 3V diagram is recursive. Then either there is only one recursive isomorphism type of recursive copies of 91, or there are infinitely many. In contrast, Goncharov in [14] has shown that there are recursive structures having any other number, 2, 3, 4, . . . , of recursive isomorphism types of recursive copies, but of course not having recursive 3V diagrams.
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It follows quickly from the definitions, that if a finite expansion (9.1,g) of 92 is recursively categorical, then so is 9.1. The converse appears to be an open question: Is there an 9.1 which is recursively categorical while some (9.1,g) is not? ~ From Theorem 4.1, there is no such 92 whose 3V diagram is recursive. Millar in [17] shows, more generally, that there is no such 92 whose existential diagram is recursive.
5
A~
While the structure (co, < , S) is recursively stable, (co, <) is not, as can be deduced from Theorem 2.1. On the other hand, for any recursive copy (A, <), the successor relation is II ~ so the unique isomorphism is A ~ We call such a structure A~ and in general: D e f i n i t i o n 5.1 A recursive structure 9.1 is A ~ (a < w~K) if, for every recursive ~ -~ 92, every isomorphism f " ~ ~- 92 is A ~ Thus, each (fl, <) with w <_ /3 < co2 is A~ stable.
but not recursively
For any recursive copy of (co2, <), there is a A ~ procedure which, given m and n, locates the element corresponding to the ordinal w m + n. It follows that (w 2 , <) is A~ It seems less obvious how to show that (co2 <) is not A~ Similarly, if ~o~ <_/3 < aj ~+1 and a >_ 1, then it is not difficult to show that (/3, <) is A~ rem 6.1 below shows that (fl, <) is not A~ for any 7 < 2c~.
6
Theo-
Back-and-forth relations
Between structures 91 and ~ of the same type, we define the relations <~ for all ordinals ~/_> 1 as follows. 1Recently, the question has been answered affirmatively by Cholak, Goncharov, Khoussainov and Shore [11].
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Let 9.1 _<1 ~ if every existential sentence true in ~ is also true in 9.1. For 3' > 1, let 92 _<.~ ~ if, for each/3 with 1 _< r < 3' and each finite sequence b from ~ , there is a finite sequence g from P2 (of the same length as b) such that ( ~ , b) _ 2, the subset cl~(g) of A by c E cl~(g) if there exists b from 9.1 and fl < c~ with fl >_ 1 such that, for all c' and b' with -6, c,-b <_;3-6, c',-b', we have c' - c. m
T h e o r e m 6.1 (Ash [11) Suppose that ~ >_ 2, and that 92 is a recursive structure for which the relations <~ on 9.1 are r.e. uniformly in fl < c~. Suppose also that there is a recursive procedure which yields for each -d from 9.1 an element of 9.1 not in cl~(g). Then 91 is not A~Ot To apply this result, one needs to consider these relations _
7
R e c u r s i v e infinitary formulae
The E~ and H~ formulae of L~l~ are defined by transfinite induction" the E0 and II0 formulae are the quantifier-free formulae, the E~ formulae are those of the form V =l~qp~ for which each ~n is a IIz formula for some/3 < c~, and n
the II~ formulae are those of the form A Y y = ~ for which each ~= is a EZ formula for some/3 < c~. n
One may then show, by transfinite induction on a, that 91 <~ ~3 if and only if every II~ sentence true in 91 is also true in ~ , or equivalently, every E~ sentence true in ~ is also true in 92. We define, for c~ < w~K, the recursive E~ and II~ formulae to be those in which all the infinitary disjunctions and conjunctions are recursively enumerable. For this to make sense, we need to define simultaneously indices for
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these formulae, in the style of Kleene's system of notations for the ordinals below w 1 . We can then define a 2 o defining family for a structure 9.1 to be a 2 ~ set S of (indices for) recursive P~ formulae, qp(v), each satisfied by exactly one element, such that each element of 9.1 is satisfied by at least one ~(v) from S. Then it is clear that if a recursive structure 9.1 has a finite expansion (91, ~) with a Eo defining family, then it is A~ For a converse, we need several assumptions. CK
c~
Assume that the existential diagram of 92 is recursive, and that the relations
It follows that, under the same assumptions, if c E cl~(~), then there is a recursive E~ formula ~(v, ~) satisfied in 9.1 by c alone, and such a formula can be found recursively from c (and ~). Thus, if A - cl~(~), then (91, ~) has a Eo defining family. So Theorem 6.1 gives" T h e o r e m 7.1 (Ash [1]) Suppose that 91 is a recursive structure for which no finite expansion (91, ~) has a E ~ defining family. Suppose also that the existential diagram of 91 is recursive, that the relations <_~ are r.e. uniformly in ~ < c~, and that the relation c ~ cl~(~) is r.e.. Then 91 is not A~
8
A~
In a similar way, we may define a recursive structure 91 to be A~ if, for every recursive ~ ~- 91, there exists a A~ f 9 ~ ~- 91. Again, for a sufficiently well-behaved 91, either Theorem 8.1 below can be used to show that 91 is not A~ or some (91, ~) has a Eo automorphism f a m i l y - the obvious generalization of an r.e. automorphism family, above and so 91 is A~-categorical. 0 Theorem 8.1 below gives the most categoricity. For a recursive structure define C~(~) to be the set of sequences 91 and some /3 < a, for all ~' and b' a,c,b _<~ a ~', then (9.1,~') -~ (9.1,~).
straightforward result for non-A ~ 91, a > 2, and elements ~ of 92, we ~ from 91 such that, for some b from of the same lengths as ~ and b, if
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T h e o r e m 8.1 (Ash [2]) Suppose that a >_ 2, and that 92 is a recursive structure on which the relations <_~ are r.e. uniformly in/3 < a, and the relation (9.1,-d') ~ (9.1,-d) is r.e.. Suppose also that there is a recursive procedure which gives, from each finite sequence -g, a sequence -d ~ C~(-d). Then 92 is not A~ Under similar recursive assumptions on 9.1, including that its existential diagram is recursive, if every sequence is in Co(g), then we can obtain a 2 ~ automorphism family for (9.1,g), so that a sufficiently well-behaved structure 92 will be A~ if and only if there is some g such that (9.1,g) has such a family. A slight refinement of Theorem 8.1 is shown in [2], in which the definition of Co(g) replaces (92, ~') -~ (92, ~) by g, ~ _>0 g, ~'. This is an oversight, and should read g, ~ >_0 a, c' and -d,-d <_0 -d,-d'. By way of example, we consider in [2] certain Boolean algebras. We let B(/3) denote the Boolean algebra generated by a well-ordered set of type/3. Then, for infinite/3, B(/3) has 2 ~~ automorphisms, and so is not A~ for any a. For ,J~ <_/3 < (aJ c~-t-1 and a >_ 1, B(/3)is A~ and not A~ for any 3' < 2a.
9
Intrinsically
C~
relations
Say that a relation R on a recursive structure 9.1 is intrinsically 2 ~ if, for every recursive ~3 ~ 92 and every isomorphism f " 9./~ ~ , f ( R ) is No. If R is defined in some (9.1,g) by a recursive 2o formula, then it is clear that R is intrinsically 2 ~ Otherwise, provided that (92, R) has sufficiently many recursive properties, Theorem 9.1 below shows that R is not intrinsically No. Assume that a _> 2. For ~ from 92, define the relation Rcl~(~) on A by g E Rcl~(g) if there exist b and/3 < a such that, for all ~' and b' with a, c, b _<~ g, Y, b, we have R(-d'). T h e o r e m 9.1 (Barker [S]). Suppose that a >_ 2, and that the relations <~ are r.e. uniformly in /3 < a. Suppose further that there is a recursive procedure for finding, given a, some ~ ~ Rcl~(~). Then R is not intrinsically o n 9a.
Again, assuming also that the existential diagram of 92 is recursive, if for some ~ every sequence C of the relevant length is in Rcl~(~), then we can
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obtain a recursive E~ definition of R in (9.1,g). So, for a sufficiently recursive (9.1, R), R is intrinsically E ~ if and only if it has such a recursive E~ definition. By way of example, Barker shows in [8] that, if/3 >_ r and a _> 1, then the set of limit ordinals of the a - t h kind in (/3, <) is intrinsically II~ but not intrinsically E~ In [9], he also considers intrinsically E ~ subsets of reduced abelian p-groups.
10
Non-recursive structures
By a structure here (in this section) we mean one which has a recursive similarity type, and has as universe a recursive set of natural numbers, but is not necessarily recursive. Instead, as a sequence of sequences of relations and operations, it has a well-defined Turing degree. Let 9.1 be a recursive structure, and let R be a relation on A. Define R to be relatively intrinsically E ~ on 9.1 if, for all ffl, not necessarily recursive, and all f " 91 ~- ~ , f ( R ) is N ~ relative to if3. We have seen that, under the assumption of recursive properties of (9.1, R), if R has no recursive N~ definition in any (9.l,g), then R is not intrinsically recursive on 92. By contrast, without any provisos, one can show the following, using forcing. T h e o r e m 10.1 R is relatively intrinsically E ~ on 92 if and only if R has a recursive E~ definition in some (9.1,~). Suitable definitions of relative A~ and A~ also have exact characterizations in terms of recursive infinitary formulae. These results are obtained in [10] and, independently, in [6]. In many ways, such results are more pleasing that our earlier ones. In this paper, however, we take the view that what matters is the construction of recursive structures.
11
c
-systems
Theorems 6.1, 8.1 and 9.1 were obtained using the author's "metatheorems" from [1]. While relying on the same proof, these results have since then been improved and streamlined in [3] and [4]. We take the opportunity to state the present version from [4], and show how Theorem 6.1 can be obtained from it. This is a significant improvement on the argument in [1], since complete
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recursive m e t r i c spaces are no longer mentioned, and a single a r g u m e n t covers the cases where a is and is not a limit ordinal. Let T be an r.e. tree having no t e r m i n a l nodes. Let U and V denote the sets of nodes of T at even levels and at odd levels, respectively, so t h a t the root of T is in U. Let E assign to each v C V an r.e. set E(v), uniformly in v. Consider the game, ~ , in which players I and II choose, alternately, nodes of T to form an infinite sequence u0, v0, u l , v l , . . . , where I chooses the ui and II chooses the vi in such a way t h a t u0 is the root of T, each vi is a successor of ui in T, and each ui+l is a successor of vi. (By a successor of a node in a tree, we m e a n an immediate successor.) We say t h a t such a sequence is a play of the g a m e ~5, and t h a t this play is winning for II if UiE(vi) is r.e., and otherwise is winning for I. A I - s t r a t e g y is a function s which assigns to each v E V a successor, s(v), of v in T. T h e play u0 v0 u l vl . . . is said to follow the I - s t r a t e g y s if for each i, ui+l = s(vi). A winning strategy for player I is an I - s t r a t e g y s such t h a t every play which follows s is winning for player I. P r o p o s i t i o n G Suppose t h a t there exist uniformly r.e. relations (C~)~<~ on V satisfying the conditions below. T h e n there is no A ~ winning s t r a t e g y for player I in the g a m e ~5.
(i) (ii)
If v Co v', then E(v) C_ E(v'). If ~ / < / 3 < c~ and v _Co b', then v C_~ v'.
(iii)
Each C_~ is reflexive and transitive.
(iv)
If a > 71 > 72 > "'" > 7k for k = 1 , 2 , . . . , and Vl C_.),1 v2 ~-')'2 " ' " ~--'Yk--I vk with u any successor of vl, then there is a successor v of u such t h a t vi C_~, v for each i = 1 , 2 , . . . , k. rn
Thus, for any A ~ I - s t r a t e g y , there is a play which follows it, and is winning for II. F u r t h e r information about such a play is given by: P r o p o s i t i o n U Under the supposition of Proposition G, from a recursive index for T, an r.e. index for E and a A ~ index for a I - s t r a t e g y s, we m a y recursively c o m p u t e both a A ~ index for a play u0, v0, u l , Vl, . . . of ,D such t h a t I.JiE(vi) is r.e., and also an r.e. index for UiE(vi). [-1
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P r o o f of T h e o r e m 6.1. We may assume that A is infinite, since otherwise the second supposition cannot hold. Let B be an infinite recursive set. We wish to define a recursive structure ~ having universe B, and an isomorphism f " if3 ~ 91 which is not A ~ Let c* E A with c* ~ cl~(0), and let d* C B be arbitrary. We let the tree T have root ((0, 0, c*), d*), and have as other nodes all longer sequences of the form ((f0, a0, Co), do, (fa, or1, Cl), d~, ... ), satisfying the conditions: (1) fo - r a o -
O, Co - c*, do - d*,
and, for each i, the conditions:
(2)
fi is a finite partial one-to-one function from B to A, fi+a D fi and ran(fi+a) and dom(fi+~) contain at least the first i elements of A and B, respectively. Also, ci e ran(fi+~) and f&~a(ci) 7/= di. m
(3) ai is a finite set of sentences ~(b) for which ~(V) is a quantifier-free formula of the language, b C dom(fi) and r true in 9.1. Also, eri+l 2 cri, and ai+x contains each of the first i sentences ~(b) which are induced in this sense by fi+l. (4) ci C A and ci ~ cl~(ran(fi)). (5) d~ 6 B. Thus, a play in ~b consists of the initial segments of an infinite sequence
((f0,
co), do, (f,,
c,), d,, ... )
for which [.Jioi determines a structure ~3 with universe B, and f - Uifi is an isomorphism from ~ to 9.1 such that for each i, f-1 (ci) # di. We consider a A~ strategy for player I in which each
(--', (fi+l, tYi+l, Ci+l)) is extended to a sequence
(..., where di+l is different from the i-th partial A ~ function applied to ci+x, if this is defined, and is arbitrary otherwise. Then, for a play which follows this strategy, f - Uifi is different from each A ~ function.
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To use Proposition G, we arrange that, for a play which is winning for II, the diagram UiO'i of 0~3 is r.e., so that ~B is a recursive structure. This follows if we define E ( ( . . . , (f, o., c)) to be o". Of course, for c~ in Proposition G, we now read c~ + 1. To complete the argument, we need to define suitable relations C_~ for _< c~. We first define relations _<.y for 7 >- 1 between finite partial oneto-one functions f and g from B to A by f _<~ g if dom(g) _D d o m ( f ) and -5 <_~ g(f-l(-d)), where f f - r a n ( f ) . Then for v - (...,
(f, a, c))
v'-
and
(... , (f',a',c')),
we define v Co v' if o" C o"', we define v Ca v' if f C f ' and o" C_ o"', and, for 1 < -y < a, we define v C~ v' if f <~ f ' and o" C_ o"'. Conditions (i), (ii) and (iii) of Proposition G are immediate, so to prove the theorem it remains only to verify condition (iv). Suppose, then, that a + 1>
"/1 >
"'"
>
"Yk,
and that v c_~, v2 c_~ . . . c_~_, vk.
For simplicity we may assume, without loss of generality, that 3'1 - c~, "/k - 0 and, since c~ > 2, that k > 2, and so 72 > 0. Let v i - ( . . . , ( f i, o'i, ci)), and let u - ( . . . , (]'1,o.l,Cl), d). Now we have fl C_ ]'2 -<',2 "'" <'Yk-, fk and O"0 C 0" 1 C
...
C
o" k.
We appeal at this stage to the principal result about the relations <~ that, since f2 _<~ "'" _<~,_, fk, there is a g _D f2 with dom(fk) C_ dora(g) and fi <_.y, g for each i - 3 , . . . , k - 1. This fact is fairly easy to show by induction on k; it appears as L e m m a 4 in [1], and is proved as Proposition 1.3 in [5]. By extending this g, we may assume that cx E ran(g). If g-l(cl) :/: d, then take h - g. If g - l ( c l ) - d, then, since Cl q~ cl~(ran(fl)), there exists h ___Dfx w i t h y <~2 h and h(d) 7~ c1. In either case, we may c h o o s e v to be ( . . . , ( f l, ax, Cl), d, (f, a, c, )), where f D___h, o. D O'k and c are chosen to satisfy the conditions (2), (3) and (4). Condition ( i v ) i s thus verified, and the theorem proved. Q
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References [1] C. J. Ash, Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees, Trans. Amer. Math. Soc., 298 (1986) 497514; errata ibid., 310 (1988) 851. [2] C. J. Ash, Categoricity in hyperarithmetical degrees, Ann. Pure Appl. Logic, 34 (1987) 1-14. [3] C. J. Ash, Labelling systems and r.e. structures, Ann. Pure Appl. Logic, 47 (1990)99-119. [4] C. J. Ash, A construction for recursive linear orderings, J. Symbolic Logic, 56 (1991)673-683. [5] C. J. Ash and J. F. Knight, Pairs of recursive structures, Ann. Pure Appl. Logic, 46 (1990) 211-234. [6] C. J. Ash, J. F. Knight, M. Manasse and T. A. Slaman, Generic copies of countable structures, Ann. Pure Appl. Logic, 42 (1989) 195-205. [7] C. J. Ash and A. Nerode, Intrinsically recursive relations, in: Aspects of Effective Algebra, (Proc. Conf. Monash Univ., Clayton, Australia, Aug. 1-4, 1979), J. N. Crossley (ed.), (Upside Down A Book Co., Yarra Glen, Victoria, Australia, 1981), 26-41. [8] E. J. Barker, Intrinsically E ~ relations, Ann. Pure Appl. Logic, 39 (1988) 105-130.
[9] E.
J. Barker, Hyperarithmetical Properties of Relations on Abelian pGroups and Orderings, Ph.D. Thesis, School of Cognitive Science, Univ. Edinburgh, Edinburgh, Scotland, (1991).
[io] J. Chisholm,
Effective model theory vs. recursive model theory, J. Symbolic Logic, 55 (1990) 1168-1191.
[11] P.
A. Cholak, S. S. Goncharov, B. Khoussainov and R. A. Shore, On recursively categorical models, (in preparation). S. S. Goncharov, Autostability and computable families of constructivizations (Russian), Algebra i Logika, 14 (1975) 647-680; [translated in: Algebra and Logic, 14 (1975) 392-409].
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[13] S. S. Goncharov, The quantity of nonautoequivalent constructivizations (Russian), Algebra i Logika, 16 (1977) 257-282,377; [translated in: Algebra and Logic, 16 (1977) 169-185]. [14] S. S. Goncharov, The problem of the number of nonautoequivalent constructivizations (Russian), Dokl. Akad. Nauk SSSR, 251 (1980) 271-274; [translated in: Soviet M a t h . - Dokl., 21 (1980) 411-414]. [15] V. S. Harizanov, Some effects of Ash-Nerode and other decidability conditions on degree spectra, Ann. Pure Appl. Logic, 55 (1991) 51-65. [16] M. S. Manasse, Notes on a theorem of Ash and Nerode, in: Recursion Theory, (Proc. AMS-ASL Summer Inst., Ithaca, NY, June 2 8 - July 16, 1982), A. Nerode and R. A. Shore, (eds.), Proc. Sympos. Pure Math., 42 (1985). [17] T. S. Millar, Recursive categoricity and persistence, J. Symbolic Logic, 51 (1986) 430-434.
Recursive structures Recursive stability Intrinsically recursive relations Recursive categoricity A~ Back-and-forth relations Recursive infinitary formulae A~ Intrinsically E ~ relations Non-recursive structures c~-systems References
An early result about recursive structures is the following, attributed to S. Tennenbaum. There is a recursive linear ordering of order type co + co* having no infinite recursive ascending or descending sequence. There are also, of course, more obvious recursive orderings of this type, for example 0 , 2 , 4, ... , 5, 3, 1. An ordering of the first sort, which can be constructed by a priority argument, is classically isomorphic to the second, but cannot be recursively isomorphic since it has different recursive properties. Some other examples are mentioned in [7]: a recursive linear ordering of type w on which the successor relation is not recursive; a recursive vector space over any recursive infinite field having no recursive basis; a recursive algebraically closed field of any characteristic having no recursive transcendence basis. *Due to the untimely death of the author, the final version of this paper was prepared by Professor J. Knight. The editors of the volume express their gratitude to Professor Knight for this valuable assistance. HANDBOOK OF RECURSIVE MATHEMATICS Edited by Yu. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. R e m m e l ~) 1998 Elsevier Science B.V. All rights reserved.
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In each of these cases it is clear that within the isomorphism type of a single structure we have recursive structures which are not recursively isomorphic. In this article we discuss general results which have been obtained concerning this and related matters.
Recursive structures We consider structures of some fixed recursive similarity type (I, J, K, #, i.,) where I, J and K are recursive sets and # : I -+ w and I., : J --+ w are recursive functions. A structure of this type is a quadruple of the form 9/ -
(A, (R~)iei , ( f j ) / e j , (ak)keK)
where each Ri is a # ( i ) - a r y relation on A, each fi is a u ( j ) - a r y operation on A, and each ak C A. Such a structure 91 is a recursive structure if A is a recursive set, each Ri is a recursive relation on A, uniformly in i, each fj is a recursive operation on A, uniformly in j, and the function k ~-~ ak is recursive. An alternative definition is used by the Russian workers in this area. Define a constructivization a of an abstract structure 92, to be an onto function a : N --+ A, such that for atomic formulae ~ the relations {~]P.l ~ ~[a(6)]} are recursive, uniformly in ~. Then such a pair (9.1,a) is essentially a recursive structure isomorphic to 91 (together with a particular isomorphism). Indeed, this approach is closer to mathematical nature, where one thinks of a classical isomorphism type as being an abstract structure. But, in practice, it seems to reduce the number of symbols significantly to deal only with the recursive structures themselves, and we shall follow this line.
2
R e c u r s i v e stability
A structure with which the least can be done by way of different recursive copies is (w, < , S) where S is the successor relation on ~o. This structure is rigid, that is, has no automorphisms other than the identity function, and it is easily seen that for any two recursive structures, 91 and if3, which are isomorphic to (w, < , S), the unique isomorphism from 9,1 to if3 is recursive. This rather strong property, that all isomorphisms between recursive copies are recursive, has been called "recursive stability".
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D e f i n i t i o n 2.1 A recursive structure 92 is said to be recursively stable if for every recursive structure ~ -~ P.I, every f 9 ~ -~ 9.1 is recursive. It seems plausible to expect that this phenomenon will occur only when the elements of the structure are suitably definable. D e f i n i t i o n 2.2 An r.e. defining family for a structure 92 is an r.e. set S of existential formulae qp(v) of one variable, such that each is satisfied by exactly one element of 9.1, and each element of 9/satisfies at least one of the formulae ~(v) from S. If a recursive structure 9.1 is such that some expansion (92, 2) of 92 by a sequence g of finitelymany constants from P2 has an r.e. defining family, then it is clear from the following that 92 is recursively stable. For any recursive ~- P2 and any f " ~ ~- 92, we have f ' ( ~ , b) -~ (92, g) for suitable b. Then f is the unique isomorphism from ( ~ , b) to (9.1,g), and its graph is r.e., since it is the set of pairs (d, c) such that, for some qp in the family, ( ~ , b) ~ ~[d] and (P2, g ) ~ qp[c]. The converse statement, that a recursively stable structure 92 has a finite expansion with an r.e. defining family, does not hold without some further assumption. Let us define the existential diagram of a recursive structure 9.1 to be the set of existential sentences true in the structure (P2, (a}~eA). Since P2 is a recursive structure, this set is clearly r.e.. T h e o r e m 2.1 Let 9.1 be a recursive structure whose existential diagram is recursive. Then 9.1 is recursively stable if and only if some finite expansion of ~ by constants has an r.e. defining family. This result is implicit in [12]. The non-trivial direction is obtained by constructing simultaneously a recursive structure ~ and an isomorphism f " ~ -~ P.l which is not recursive, using a finite injury priority argument. This argument is also given in [7]. As with other results to come, Theorem 2.1 needs the assumption that at least one recursive isomorphic copy has some extra recursive properties. In [13], an example is constructed of a recursive structure P2 which is recursively stable but has no r.e. defining family.
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Intrinsically recursive relations
For a recursive structure 92 and an additional relation R on A, a construction similar to that used to prove Theorem 2.1 can often be used to give a recursive structure ~ and an isomorphism f : 91% ~ for which f ( R ) is not recursive. If there are no such ~ and f, then the relation R is said to be intrinsically recursive on 9.1. If a relation R on 9.1 has a definition of the form n
for some sequence ~ from 92 and some sequence ~n(x,y) of existential formulae, then R will be called formally r.e. on 92. Clearly, if both R and its complement are formally r.e., then R is intrinsically recursive. The converse can be shown provided that, again, (9.1,R) satisfies further recursive conditions. T h e o r e m 3.1 (Ash [71) Let (P.I, R) be a recursive structure whose existential diagram is recursive. Then R is intrinsically recursive on 9.1 if and only if both R and its complement are formally r.e. on 91 The construction for this attempts to make both f ( R ) and its complement not r.e.. A similar construction only attempting to make f ( R ) not r.e. gives the more basic result: T h e o r e m 3.2 (Ash [7]) Let (91, R) be a recursive structure whose existential diagram is recursive. Then R is intrinsically r.e. on 9.1 if and only if R is formally r.e. on 9.1. The examples mentioned earlier can all be deduced from Theorem 3.1 without repeating the priority argument. Some assumption is needed for the equivalence in Theorem 3.1. In [16], an example is given of a recursive (92, R) for which R is intrinsically recursive on 92 but is not formally r.e. on 92. If R is not intrinsically recursive on 92, we may define the spectrum of R on 91 to be the set of all possible Turing degrees of the relations f ( R ) for recursive structures ~3 and isomorphisms f : 91 ~ ~ . The possible spectra obtained in this way are considered in detail by V. Harizanov in [15].
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4
Isomorphic Recurswe Structures
171
Recursive categoricity
A recursive structure 91 is said to be recursively categorical if, for every recursive ~ "~ 91, there is at least one recursive isomorphism f 9 ~ -~ 91. For example, the ordering of the rationals (Q, <) has this property, as can be shown using an effective back-and-forth argument. Clearly (Q, <) is not recursively stable, since it has 2 ~~ automorphisms, and so has 2 ~~ isomorphisms with any recursive copy. In fact, it follows fairly briefly from the definitions that a structure 91 is recursively stable if and only if, for some finite sequence ~ from 91, (91, ~) is both rigid and recursively categorical. The example of (Q, <) suggests a sufficient condition for 9.1 to be recursively categorical. D e f i n i t i o n 4.1 An r.e. automorphism family for 91 is an r.e. family of existential formulae r in different numbers of variable, such that every finite sequence from 91 satisfies at least one of these formulae while, if 91 ~ ~n[b] and 91 ~ ~n[~], then (P2, b) "~ (91, ~). -
-
m
Clearly, if PJ, or more generally some (9.1,~), has an r.e. automorphism family, then, by an effective back-and-forth argument, PJ is recursively categorical. The converse can be shown subject to further assumptions on 9./. An 3V sentence means one of the form 3 5 V ~ ~(~, ~), where T(~, ~) is quantifierfree. T h e o r e m 4.1 (Goncharov [12]) Let 91 be a recursive structure whose 3V diagram is recursive. Then 92 is recursively categorical if and only if some finite expansion (9.1,-5) has an r.e. automorphism family. In the case where there is no r.e. automorphism family, one can construct not just two, but infinitely many recursive copies of 91 which are pairwise not recursively isomorphic. In other words: T h e o r e m 4.2 (Goncharov [13]) Let 91 be a recursive structure whose 3V diagram is recursive. Then either there is only one recursive isomorphism type of recursive copies of 91, or there are infinitely many. In contrast, Goncharov in [14] has shown that there are recursive structures having any other number, 2, 3, 4, . . . , of recursive isomorphism types of recursive copies, but of course not having recursive 3V diagrams.
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It follows quickly from the definitions, that if a finite expansion (9.1,g) of 92 is recursively categorical, then so is 9.1. The converse appears to be an open question: Is there an 9.1 which is recursively categorical while some (9.1,g) is not? ~ From Theorem 4.1, there is no such 92 whose 3V diagram is recursive. Millar in [17] shows, more generally, that there is no such 92 whose existential diagram is recursive.
5
A~
While the structure (co, < , S) is recursively stable, (co, <) is not, as can be deduced from Theorem 2.1. On the other hand, for any recursive copy (A, <), the successor relation is II ~ so the unique isomorphism is A ~ We call such a structure A~ and in general: D e f i n i t i o n 5.1 A recursive structure 9.1 is A ~ (a < w~K) if, for every recursive ~ -~ 92, every isomorphism f " ~ ~- 92 is A ~ Thus, each (fl, <) with w <_ /3 < co2 is A~ stable.
but not recursively
For any recursive copy of (co2, <), there is a A ~ procedure which, given m and n, locates the element corresponding to the ordinal w m + n. It follows that (w 2 , <) is A~ It seems less obvious how to show that (co2 <) is not A~ Similarly, if ~o~ <_/3 < aj ~+1 and a >_ 1, then it is not difficult to show that (/3, <) is A~ rem 6.1 below shows that (fl, <) is not A~ for any 7 < 2c~.
6
Theo-
Back-and-forth relations
Between structures 91 and ~ of the same type, we define the relations <~ for all ordinals ~/_> 1 as follows. 1Recently, the question has been answered affirmatively by Cholak, Goncharov, Khoussainov and Shore [11].
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Let 9.1 _<1 ~ if every existential sentence true in ~ is also true in 9.1. For 3' > 1, let 92 _<.~ ~ if, for each/3 with 1 _< r < 3' and each finite sequence b from ~ , there is a finite sequence g from P2 (of the same length as b) such that ( ~ , b) _
T h e o r e m 6.1 (Ash [11) Suppose that ~ >_ 2, and that 92 is a recursive structure for which the relations <~ on 9.1 are r.e. uniformly in fl < c~. Suppose also that there is a recursive procedure which yields for each -d from 9.1 an element of 9.1 not in cl~(g). Then 91 is not A~Ot To apply this result, one needs to consider these relations _
7
R e c u r s i v e infinitary formulae
The E~ and H~ formulae of L~l~ are defined by transfinite induction" the E0 and II0 formulae are the quantifier-free formulae, the E~ formulae are those of the form V =l~qp~ for which each ~n is a IIz formula for some/3 < c~, and n
the II~ formulae are those of the form A Y y = ~ for which each ~= is a EZ formula for some/3 < c~. n
One may then show, by transfinite induction on a, that 91 <~ ~3 if and only if every II~ sentence true in 91 is also true in ~ , or equivalently, every E~ sentence true in ~ is also true in 92. We define, for c~ < w~K, the recursive E~ and II~ formulae to be those in which all the infinitary disjunctions and conjunctions are recursively enumerable. For this to make sense, we need to define simultaneously indices for
C. J. Ash
174
these formulae, in the style of Kleene's system of notations for the ordinals below w 1 . We can then define a 2 o defining family for a structure 9.1 to be a 2 ~ set S of (indices for) recursive P~ formulae, qp(v), each satisfied by exactly one element, such that each element of 9.1 is satisfied by at least one ~(v) from S. Then it is clear that if a recursive structure 9.1 has a finite expansion (91, ~) with a Eo defining family, then it is A~ For a converse, we need several assumptions. CK
c~
Assume that the existential diagram of 92 is recursive, and that the relations
It follows that, under the same assumptions, if c E cl~(~), then there is a recursive E~ formula ~(v, ~) satisfied in 9.1 by c alone, and such a formula can be found recursively from c (and ~). Thus, if A - cl~(~), then (91, ~) has a Eo defining family. So Theorem 6.1 gives" T h e o r e m 7.1 (Ash [1]) Suppose that 91 is a recursive structure for which no finite expansion (91, ~) has a E ~ defining family. Suppose also that the existential diagram of 91 is recursive, that the relations <_~ are r.e. uniformly in ~ < c~, and that the relation c ~ cl~(~) is r.e.. Then 91 is not A~
8
A~
In a similar way, we may define a recursive structure 91 to be A~ if, for every recursive ~ ~- 91, there exists a A~ f 9 ~ ~- 91. Again, for a sufficiently well-behaved 91, either Theorem 8.1 below can be used to show that 91 is not A~ or some (91, ~) has a Eo automorphism f a m i l y - the obvious generalization of an r.e. automorphism family, above and so 91 is A~-categorical. 0 Theorem 8.1 below gives the most categoricity. For a recursive structure define C~(~) to be the set of sequences 91 and some /3 < a, for all ~' and b' a,c,b _<~ a ~', then (9.1,~') -~ (9.1,~).
straightforward result for non-A ~ 91, a > 2, and elements ~ of 92, we ~ from 91 such that, for some b from of the same lengths as ~ and b, if
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T h e o r e m 8.1 (Ash [2]) Suppose that a >_ 2, and that 92 is a recursive structure on which the relations <_~ are r.e. uniformly in/3 < a, and the relation (9.1,-d') ~ (9.1,-d) is r.e.. Suppose also that there is a recursive procedure which gives, from each finite sequence -g, a sequence -d ~ C~(-d). Then 92 is not A~ Under similar recursive assumptions on 9.1, including that its existential diagram is recursive, if every sequence is in Co(g), then we can obtain a 2 ~ automorphism family for (9.1,g), so that a sufficiently well-behaved structure 92 will be A~ if and only if there is some g such that (9.1,g) has such a family. A slight refinement of Theorem 8.1 is shown in [2], in which the definition of Co(g) replaces (92, ~') -~ (92, ~) by g, ~ _>0 g, ~'. This is an oversight, and should read g, ~ >_0 a, c' and -d,-d <_0 -d,-d'. By way of example, we consider in [2] certain Boolean algebras. We let B(/3) denote the Boolean algebra generated by a well-ordered set of type/3. Then, for infinite/3, B(/3) has 2 ~~ automorphisms, and so is not A~ for any a. For ,J~ <_/3 < (aJ c~-t-1 and a >_ 1, B(/3)is A~ and not A~ for any 3' < 2a.
9
Intrinsically
C~
relations
Say that a relation R on a recursive structure 9.1 is intrinsically 2 ~ if, for every recursive ~3 ~ 92 and every isomorphism f " 9./~ ~ , f ( R ) is No. If R is defined in some (9.1,g) by a recursive 2o formula, then it is clear that R is intrinsically 2 ~ Otherwise, provided that (92, R) has sufficiently many recursive properties, Theorem 9.1 below shows that R is not intrinsically No. Assume that a _> 2. For ~ from 92, define the relation Rcl~(~) on A by g E Rcl~(g) if there exist b and/3 < a such that, for all ~' and b' with a, c, b _<~ g, Y, b, we have R(-d'). T h e o r e m 9.1 (Barker [S]). Suppose that a >_ 2, and that the relations <~ are r.e. uniformly in /3 < a. Suppose further that there is a recursive procedure for finding, given a, some ~ ~ Rcl~(~). Then R is not intrinsically o n 9a.
Again, assuming also that the existential diagram of 92 is recursive, if for some ~ every sequence C of the relevant length is in Rcl~(~), then we can
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obtain a recursive E~ definition of R in (9.1,g). So, for a sufficiently recursive (9.1, R), R is intrinsically E ~ if and only if it has such a recursive E~ definition. By way of example, Barker shows in [8] that, if/3 >_ r and a _> 1, then the set of limit ordinals of the a - t h kind in (/3, <) is intrinsically II~ but not intrinsically E~ In [9], he also considers intrinsically E ~ subsets of reduced abelian p-groups.
10
Non-recursive structures
By a structure here (in this section) we mean one which has a recursive similarity type, and has as universe a recursive set of natural numbers, but is not necessarily recursive. Instead, as a sequence of sequences of relations and operations, it has a well-defined Turing degree. Let 9.1 be a recursive structure, and let R be a relation on A. Define R to be relatively intrinsically E ~ on 9.1 if, for all ffl, not necessarily recursive, and all f " 91 ~- ~ , f ( R ) is N ~ relative to if3. We have seen that, under the assumption of recursive properties of (9.1, R), if R has no recursive N~ definition in any (9.l,g), then R is not intrinsically recursive on 92. By contrast, without any provisos, one can show the following, using forcing. T h e o r e m 10.1 R is relatively intrinsically E ~ on 92 if and only if R has a recursive E~ definition in some (9.1,~). Suitable definitions of relative A~ and A~ also have exact characterizations in terms of recursive infinitary formulae. These results are obtained in [10] and, independently, in [6]. In many ways, such results are more pleasing that our earlier ones. In this paper, however, we take the view that what matters is the construction of recursive structures.
11
c
-systems
Theorems 6.1, 8.1 and 9.1 were obtained using the author's "metatheorems" from [1]. While relying on the same proof, these results have since then been improved and streamlined in [3] and [4]. We take the opportunity to state the present version from [4], and show how Theorem 6.1 can be obtained from it. This is a significant improvement on the argument in [1], since complete
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recursive m e t r i c spaces are no longer mentioned, and a single a r g u m e n t covers the cases where a is and is not a limit ordinal. Let T be an r.e. tree having no t e r m i n a l nodes. Let U and V denote the sets of nodes of T at even levels and at odd levels, respectively, so t h a t the root of T is in U. Let E assign to each v C V an r.e. set E(v), uniformly in v. Consider the game, ~ , in which players I and II choose, alternately, nodes of T to form an infinite sequence u0, v0, u l , v l , . . . , where I chooses the ui and II chooses the vi in such a way t h a t u0 is the root of T, each vi is a successor of ui in T, and each ui+l is a successor of vi. (By a successor of a node in a tree, we m e a n an immediate successor.) We say t h a t such a sequence is a play of the g a m e ~5, and t h a t this play is winning for II if UiE(vi) is r.e., and otherwise is winning for I. A I - s t r a t e g y is a function s which assigns to each v E V a successor, s(v), of v in T. T h e play u0 v0 u l vl . . . is said to follow the I - s t r a t e g y s if for each i, ui+l = s(vi). A winning strategy for player I is an I - s t r a t e g y s such t h a t every play which follows s is winning for player I. P r o p o s i t i o n G Suppose t h a t there exist uniformly r.e. relations (C~)~<~ on V satisfying the conditions below. T h e n there is no A ~ winning s t r a t e g y for player I in the g a m e ~5.
(i) (ii)
If v Co v', then E(v) C_ E(v'). If ~ / < / 3 < c~ and v _Co b', then v C_~ v'.
(iii)
Each C_~ is reflexive and transitive.
(iv)
If a > 71 > 72 > "'" > 7k for k = 1 , 2 , . . . , and Vl C_.),1 v2 ~-')'2 " ' " ~--'Yk--I vk with u any successor of vl, then there is a successor v of u such t h a t vi C_~, v for each i = 1 , 2 , . . . , k. rn
Thus, for any A ~ I - s t r a t e g y , there is a play which follows it, and is winning for II. F u r t h e r information about such a play is given by: P r o p o s i t i o n U Under the supposition of Proposition G, from a recursive index for T, an r.e. index for E and a A ~ index for a I - s t r a t e g y s, we m a y recursively c o m p u t e both a A ~ index for a play u0, v0, u l , Vl, . . . of ,D such t h a t I.JiE(vi) is r.e., and also an r.e. index for UiE(vi). [-1
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P r o o f of T h e o r e m 6.1. We may assume that A is infinite, since otherwise the second supposition cannot hold. Let B be an infinite recursive set. We wish to define a recursive structure ~ having universe B, and an isomorphism f " if3 ~ 91 which is not A ~ Let c* E A with c* ~ cl~(0), and let d* C B be arbitrary. We let the tree T have root ((0, 0, c*), d*), and have as other nodes all longer sequences of the form ((f0, a0, Co), do, (fa, or1, Cl), d~, ... ), satisfying the conditions: (1) fo - r a o -
O, Co - c*, do - d*,
and, for each i, the conditions:
(2)
fi is a finite partial one-to-one function from B to A, fi+a D fi and ran(fi+a) and dom(fi+~) contain at least the first i elements of A and B, respectively. Also, ci e ran(fi+~) and f&~a(ci) 7/= di. m
(3) ai is a finite set of sentences ~(b) for which ~(V) is a quantifier-free formula of the language, b C dom(fi) and r true in 9.1. Also, eri+l 2 cri, and ai+x contains each of the first i sentences ~(b) which are induced in this sense by fi+l. (4) ci C A and ci ~ cl~(ran(fi)). (5) d~ 6 B. Thus, a play in ~b consists of the initial segments of an infinite sequence
((f0,
co), do, (f,,
c,), d,, ... )
for which [.Jioi determines a structure ~3 with universe B, and f - Uifi is an isomorphism from ~ to 9.1 such that for each i, f-1 (ci) # di. We consider a A~ strategy for player I in which each
(--', (fi+l, tYi+l, Ci+l)) is extended to a sequence
(..., where di+l is different from the i-th partial A ~ function applied to ci+x, if this is defined, and is arbitrary otherwise. Then, for a play which follows this strategy, f - Uifi is different from each A ~ function.
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179
To use Proposition G, we arrange that, for a play which is winning for II, the diagram UiO'i of 0~3 is r.e., so that ~B is a recursive structure. This follows if we define E ( ( . . . , (f, o., c)) to be o". Of course, for c~ in Proposition G, we now read c~ + 1. To complete the argument, we need to define suitable relations C_~ for _< c~. We first define relations _<.y for 7 >- 1 between finite partial oneto-one functions f and g from B to A by f _<~ g if dom(g) _D d o m ( f ) and -5 <_~ g(f-l(-d)), where f f - r a n ( f ) . Then for v - (...,
(f, a, c))
v'-
and
(... , (f',a',c')),
we define v Co v' if o" C o"', we define v Ca v' if f C f ' and o" C_ o"', and, for 1 < -y < a, we define v C~ v' if f <~ f ' and o" C_ o"'. Conditions (i), (ii) and (iii) of Proposition G are immediate, so to prove the theorem it remains only to verify condition (iv). Suppose, then, that a + 1>
"/1 >
"'"
>
"Yk,
and that v c_~, v2 c_~ . . . c_~_, vk.
For simplicity we may assume, without loss of generality, that 3'1 - c~, "/k - 0 and, since c~ > 2, that k > 2, and so 72 > 0. Let v i - ( . . . , ( f i, o'i, ci)), and let u - ( . . . , (]'1,o.l,Cl), d). Now we have fl C_ ]'2 -<',2 "'" <'Yk-, fk and O"0 C 0" 1 C
...
C
o" k.
We appeal at this stage to the principal result about the relations <~ that, since f2 _<~ "'" _<~,_, fk, there is a g _D f2 with dom(fk) C_ dora(g) and fi <_.y, g for each i - 3 , . . . , k - 1. This fact is fairly easy to show by induction on k; it appears as L e m m a 4 in [1], and is proved as Proposition 1.3 in [5]. By extending this g, we may assume that cx E ran(g). If g-l(cl) :/: d, then take h - g. If g - l ( c l ) - d, then, since Cl q~ cl~(ran(fl)), there exists h ___Dfx w i t h y <~2 h and h(d) 7~ c1. In either case, we may c h o o s e v to be ( . . . , ( f l, ax, Cl), d, (f, a, c, )), where f D___h, o. D O'k and c are chosen to satisfy the conditions (2), (3) and (4). Condition ( i v ) i s thus verified, and the theorem proved. Q
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References [1] C. J. Ash, Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees, Trans. Amer. Math. Soc., 298 (1986) 497514; errata ibid., 310 (1988) 851. [2] C. J. Ash, Categoricity in hyperarithmetical degrees, Ann. Pure Appl. Logic, 34 (1987) 1-14. [3] C. J. Ash, Labelling systems and r.e. structures, Ann. Pure Appl. Logic, 47 (1990)99-119. [4] C. J. Ash, A construction for recursive linear orderings, J. Symbolic Logic, 56 (1991)673-683. [5] C. J. Ash and J. F. Knight, Pairs of recursive structures, Ann. Pure Appl. Logic, 46 (1990) 211-234. [6] C. J. Ash, J. F. Knight, M. Manasse and T. A. Slaman, Generic copies of countable structures, Ann. Pure Appl. Logic, 42 (1989) 195-205. [7] C. J. Ash and A. Nerode, Intrinsically recursive relations, in: Aspects of Effective Algebra, (Proc. Conf. Monash Univ., Clayton, Australia, Aug. 1-4, 1979), J. N. Crossley (ed.), (Upside Down A Book Co., Yarra Glen, Victoria, Australia, 1981), 26-41. [8] E. J. Barker, Intrinsically E ~ relations, Ann. Pure Appl. Logic, 39 (1988) 105-130.
[9] E.
J. Barker, Hyperarithmetical Properties of Relations on Abelian pGroups and Orderings, Ph.D. Thesis, School of Cognitive Science, Univ. Edinburgh, Edinburgh, Scotland, (1991).
[io] J. Chisholm,
Effective model theory vs. recursive model theory, J. Symbolic Logic, 55 (1990) 1168-1191.
[11] P.
A. Cholak, S. S. Goncharov, B. Khoussainov and R. A. Shore, On recursively categorical models, (in preparation). S. S. Goncharov, Autostability and computable families of constructivizations (Russian), Algebra i Logika, 14 (1975) 647-680; [translated in: Algebra and Logic, 14 (1975) 392-409].
Chapter 3
Isomorphic Recursive Structures
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[13] S. S. Goncharov, The quantity of nonautoequivalent constructivizations (Russian), Algebra i Logika, 16 (1977) 257-282,377; [translated in: Algebra and Logic, 16 (1977) 169-185]. [14] S. S. Goncharov, The problem of the number of nonautoequivalent constructivizations (Russian), Dokl. Akad. Nauk SSSR, 251 (1980) 271-274; [translated in: Soviet M a t h . - Dokl., 21 (1980) 411-414]. [15] V. S. Harizanov, Some effects of Ash-Nerode and other decidability conditions on degree spectra, Ann. Pure Appl. Logic, 55 (1991) 51-65. [16] M. S. Manasse, Notes on a theorem of Ash and Nerode, in: Recursion Theory, (Proc. AMS-ASL Summer Inst., Ithaca, NY, June 2 8 - July 16, 1982), A. Nerode and R. A. Shore, (eds.), Proc. Sympos. Pure Math., 42 (1985). [17] T. S. Millar, Recursive categoricity and persistence, J. Symbolic Logic, 51 (1986) 430-434.