Download ON THE COMPUTABILITY-THEORETIC COMPLEXITY OF TRIVIAL

ON THE COMPUTABILITY-THEORETIC
COMPLEXITY OF TRIVIAL, STRONGLY MINIMAL
MODELS
BAKHADYR M. KHOUSSAINOV AND STEFFEN LEMPP
Abstract. We show the existence of a trivial, strongly minimal
(and thus uncountably categorical) theory for which the prime
model is computable and none of the other countable models is
00 -computable.
One of the recurring themes of computable model theory has been
the study of the computability-theoretic complexity of various models
of a fixed first-order theory. Much of the work has focused on the case
of uncountably categorical theories, where the classical model theory
is quite well understood.
We first recall that a model is said to be decidable if it has an isomorphic copy with universe ω for which the full elementary diagram forms
a computable set of formulas; and computable if the open diagram of
some such copy forms a computable set of formulas.
Harrington [Ha74] and Khisamiev [Kh74] showed that all countable
models of a decidable uncountably categorical theory are decidable.
In contrast, Goncharov [Go78] first exhibited an uncountably categorical theory for which one but not all countable models are computable; in fact, for his theory, the prime model is computable while
all other countable models are only 00 -computable. More examples of
uncountably categorical theories for which some but not all countable
models are computable were found by Kudaibergenov [Ku80], Khoussainov/Nies/Shore [KNS97], Herwig/Lempp/Ziegler [HLZ99], and others.
A natural question arises as to how far apart the computabilitytheoretic complexity of the various countable models of an uncountably categorical theory can be. In the special case of a trivial, strongly
2000 Mathematics Subject Classification. Primary: 03C57; Secondary: 03D45.
Key words and phrases. computable model, uncountably categorical, strongly
minimal, trivial geometry.
The first author’s research was partially supported by The Marsden Fund of
New Zealand. The second author’s research was partially supported by NSF grant
DMS-0140120.
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BAKHADYR M. KHOUSSAINOV AND STEFFEN LEMPP
minimal theory, an answer was given in a purely model-theoretic result by Goncharov/Harizanov/Laskowski/Lempp/McCoy [GHLLM03]:
Any such theory is model complete after naming constants in one
model. Thus, by a result of Kueker [Kuta], such a theory is ∃∀∃axiomatizable, and so, if there is a computable model, then all countable models are 000 -decidable. Here, a model is strongly minimal if any
definable subset is finite or cofinite; a strongly minimal model has trivial pregeometry or is trivial if the algebraic closure of any subset A of
the model is the union of the algebraic closures of the elements of A.
(These two notions depend only on the theory of the model. Strong
minimality implies uncountable categoricity. All models discussed in
the previous paragraph turn out to be trivial and strongly minimal. A
good reference for these notions is Buechler [Bu96].)
There is a computability-theoretic gap between the results of Goncharov [Go78] and of Goncharov/Harizanov/Laskowski/Lempp/McCoy
[GHLLM03]: Is it possible for a trivial strongly minimal theory to have
a computable model and another countable model which is only computable in 000 , or at least not computable in 00 ? In this paper, we give
a positive answer to the second part of this question:
Theorem 1. There is a trivial, strongly minimal (and thus uncountably
categorical) first-order theory T such that the prime model T is computable and all other countable models of T are not 00 -computable (but,
by Goncharov/Harizanov/Laskowski/Lempp/McCoy [GHLLM03], all
are 000 -decidable).
The rest of this paper is devoted to the proof of this theorem. To
this end, we first generalize the notion of “cubes” introduced by Khoussainov/Nies/Shore [KNS97] in the following
Definition 2. Fix a language L consisting of binary relation symbols Fi
(for i ∈ ω), which we will all assume to be symmetric and irreflexive
and to have in-degree and out-degree ≤ 1, so we can think of them as
coding permutations fi of order 2 on subsets of the model.
Given a nonempty (finite or infinite) subset S ⊆ ω, we let S = {n0 <
n1 < . . . } be a (not necessarily effective) enumeration of S in order,
and set Sk = {n0 , . . . , nk } (for k < |S| if S is finite). We now define
the notion of an S-cube by induction on k:
An S0 -cube is a 2-element subset {x, y} of an L-model such that
Fn0 (x, y) (coding fn0 (x) = y and fn0 (y) = x) and ¬Fn (x, y) for all
n 6= n0 .
An Sk+1-cube is the disjoint union of two Sk -cubes C0 and C1 such
that Fnk+1 induces an L-isomorphism between C0 and C1 , and such
that ¬Fn (x0 , x1 ) for all n 6= nk+1 , all x0 ∈ C0 and all x1 ∈ C1 .
ON THE COMPLEXITY OF TRIVIAL, STRONGLY MINIMAL MODELS
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For infinite S, we define an S-cube to be the “limit” of the Sk -cubes,
i.e., an infinite subset A of an L-model such that for all k ∈ ω, every
element x ∈ A is an element of a unique Sk -cube ⊂ A and such that any
two elements x0 , x1 ∈ A are connected by a finite chain of Fi -relations.
We will now construct the computable prime model of our trivial,
strongly minimal theory, which will essentially, modulo some finite perturbations, be the disjoint union of (S n)-cubes, one for each n > 0,
where S is a Π02 -complete set. We will then argue that each model
of our theory consists of the prime models plus κ many copies of the
S-cube for some cardinal κ. Finally, S will be existentially definable
in any nonprime model with one parameter from the nonprime part of
the model, so S is Σ01 relative to the open diagram of any nonprime
model. Thus the open diagram of any nonprime model cannot be ∆02 ,
i.e., 00 -computable, finishing the argument.
So we first fix a Π02 -complete set S and a computable function g such
that m ∈ S iff the c.e. set Wg(m) is infinite. Without loss of generality,
we may furthermore assume that
(1) each Wg(m) is either of the form [m, p] for some p ≥ m or of the
form [m, ∞),
(2) each Wg(m) is enumerated in order, and
(3) for each stage s > 0, there is a unique pair hm, ni such that n
enters Wg(m) at stage s.
We now define the prime model M in stages s, starting by defining
M0 = E, the set of even integers, and the relations Fk not to hold for
any k ∈ ω and any pair from M0 . At subsequent stages, we will add
odd numbers to the model and build a finite cube (denoted by Cn )
around each element 2n ∈ M0 disjoint from all other such cubes as
follows: If n enters Wg(m) at stage s then we use unused odd numbers
to build a copy of the current cube connected to 2n, and connect the
two by defining Fm between the two cubes to form a new cube.
We now finish our formal proof in a sequence of lemmas.
Lemma 3. The model M is computable and consists of the disjoint
union of finite cubes Cn , each containing the corresponding even number 2n. The cube Cn is an Sn -cube where Sn = {m ≤ n | n ∈ Wg(m) }.
Proof. The first sentence of the lemma is immediate by the construction. The second sentence follows since we add Fm -relations to the
cube Cn containing 2n iff n enters Wg(m) at some stage.
Lemma 4. For any n > 0, all but finitely many elements x ∈ M are
contained in an (S n)-cube but not in a T -cube for any finite set T
with S n ⊂ T ⊆ [0, n).
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BAKHADYR M. KHOUSSAINOV AND STEFFEN LEMPP
Proof. This follows by the properties (1)–(3) of the sequence of c.e. sets
{Wg(m) }m∈ω : Fix n > 0, and fix m ≥ n sufficiently large such that for
all i < n, m ∈ Wg(i) iff Wg(i) is infinite. Then by Lemma 3, x ∈ M can
fail to satisfy the statement above only if x is in a cube Ci for some
i < m.
Lemma 5. All models of the first-order theory T of M consist of the
disjoint union of M and κ many copies of the S-cube. Thus M is the
prime model of T , and T is trivial and strongly minimal (and thus in
particular uncountably categorical).
Proof. By Lemma 3 and since S is infinite, each element of M satisfies
a principal type. By Lemma 4, there is a unique nonprincipal 1-type
over M, namely, that of being a member of an S-cube. This easily
implies triviality and strong minimality.
Lemma 6. No nonprime model M0 of T has a 00 -computable copy.
Proof. Fix any element x ∈ M 0 − M where M is the copy of the prime
model inside M0 . By Lemmas 4 and 5, n ∈ S iff there is some y ∈ M 0
such that M0 |= Fn (x, y). Thus S is Σ01 relative to the open diagram
of M0 . But if (the open diagram of) M0 were 00 -computable then this
would make S a Σ02 -set, contradicting the Π02 -completeness of S.
This completes the proof of Theorem 1.
We end with the following
Conjecture 7. For any n > 0, there is a trivial, strongly minimal
(and thus uncountably categorical) first-order theory such that its prime
model is computable and all other countable models of it are not acomputable for any degree a which is lown over 00 .
We suspect that the techniques of Marker [Ma89], jacking up the
Morley rank of the model and obfuscating the binary relations by relations of higher arity, can be used to prove this conjecture.
References
[Bu96]
[Go78]
[GK04]
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ON THE COMPLEXITY OF TRIVIAL, STRONGLY MINIMAL MODELS
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[GHLLM03] Goncharov, Sergey S.; Harizanov, Valentina S.; Laskowski, Michael
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Department of Computer Science, University of Auckland, Private
Bag 92019, Auckland, New Zealand
E-mail address: [email protected]
Department of Mathematics, University of Wisconsin, Madison, WI
53706-1388, USA
E-mail address: [email protected]