Selection in the monadic theory of a countable ordinal

SELECTION IN THE MONADIC THEORY OF A COUNTABLE

ORDINAL

ALEXANDER RABINOVICH AND AMIT SHOMRAT

Abstract. A monadic formula (Y ) is a selector for a formula '(Y ) in a structure

M if there exists a unique subset P of M which satis�es and this P also satis�es '.

We show that for every ordinal � � !! there are formulas having no selector in the

structure (�;<). For � � !1, we decide which formulas have a selector in (�;<), and

construct selectors for them. We deduce the impossibility of a full generalization of the

B�uchi-Landweber solvability theorem from (!;<) to (!!; <). We state a partial extension

of that theorem to all countable ordinals. To each formula we assign a selection degree

which measures \how di�cult it is to select". We show that in a countable ordinal all

non-selectable formulas share the same degree.

Contents

1. Introduction 22. Preliminaries and background 32.1. Notations and terminology 32.2. The Monadic Logic of Order (MLO) 42.3. Elements of the composition method 52.4. The monadic theory of countable ordinals 72.5. Uniformization 83. The selection property 83.1. The Periodicity Lemma 93.2. Failure of the selection property in � � !! 104. The selection problem 134.1. Solvability of the selection problem in (�;<) for � 2 [!!; !1] 134.2. Idempotents and selection 155. Comparing di�culty of selection in a countable ordinal 175.1. Selection degrees 175.2. Automata on ordinal words 185.3. An unbounded !-sequence is easier than any non-selectable formula 205.4. Any formula is easier than an unbounded !-sequence 236. Every formula is selectable in some countable ordinal 287. Selection between �rst-order and second-order logic 298. The Church synthesis problem 32

Received by the editors October 12, 2007.

1

2 ALEXANDER RABINOVICH AND AMIT SHOMRAT

x1. Introduction.

Definition 1.1 (Selection). Let '( �Y ), ( �Y ) be formulas in some logic L andM a structure for L. We say that selects (or, is a selector for) ' in M i�:

1. M j= 9�1 �Y ( �Y ),2. M j= 8 �Y ( ( �Y )! '( �Y )), and3. M j= 9 �Y '( �Y )! 9 �Y ( �Y ).

Here �Y is a tuple of distinct variables and \9�1 �Y : : : " stands for \there existsat most one: : : ".We say that M has the L-selection property i� every L-formula ' has a selector in M.

We explore the question \Does there exist a ( �Y ) that selects '( �Y ) in M?"when ' and are formulas in the second-order monadic logic of order (MLO),�Y is a tuple of monadic variables ranging over subsets of the domain, andM :=(�;<) is some ordinal � equipped with its natural order.MLO allows quanti�cation over subsets of the domain (monadic predicates).

The binary relation symbol `<' is its only non-logical constant. Since our struc-tures are ordinals, we usually assume that it is interpreted as a well -order of thedomain.From now on, unless otherwise explicitly stated, \formula" means \MLO-

formula", the \selection property" is the \MLO-selection property", etc.In [LiS98], Lifsches and Shelah show that any � < !! has the selection prop-

erty (in fact, they prove a much stronger property called the uniformizationproperty ; see De�nition 2.20).In section 3, we show that the selection property fails for any � � !!. In

particular, we show that the formula �! ub(Y ) saying \Y is an unbounded !-sequence" has no selector in (!!; <). This naturally leads to the following algo-rithmic problem:

Definition 1.2 (Selection problem). The selection problem in a structureMis:

Input: an MLO formula '( �Y ).Task: check whether there is a selector ( �Y ) for ' inM, and if so, construct

one.

We show the solvability of the selection problem in (�;<) for every � � !1(section 4).In section 5, we try and compare formulas in terms, intuitively, of how di�cult

they are to select. The idea is, roughly, that a formula '0(Y ) is easier to selectthan a formula '1(Y ) in a structure M if, from any given subset P of (thedomain of)M which satis�es '1 inM, we can de�ne a subset Q of (the domainof) M which satis�es '0 in M. This is easily seen to be a partial preorderon the formulas, and its equivalence classes we call selection degrees (De�nition5.3). It turns out, interestingly, that in every countable � � !! there arejust two degrees: one containing all selectable formulas and one containing allnon-selectable formulas. This implies, for instance, that with an unbounded!-sequence as a parameter one can select every formula in (!!; <).

SELECTION IN THE MONADIC THEORY OF A COUNTABLE ORDINAL 3

In section 6, we note that while there is a particular formula lacking a selectorin each and every uncountable ordinal, no formula can be found which is non-selectable in every � 2 [!!; !1).An analysis of our proofs shows that for countable ordinals the above results

concerning the selection property and the selection problem carry through to alllogics between �rst-order and MLO. On the other hand, we show that every�rst-order formula has a selector in (!1; <). In fact, an interesting dichotomyholds:

(a) a �rst-order formula selectable in (!!; <) is also selectable in (!1; <), andwe can choose for it a selector that works in both;

(b) a �rst-order formula which is not selectable in (!!; <) is not even satis�edin (!1; <).

For these results, see section 7.We end by relating our work to the Church synthesis problem (De�nition 8.3).

In their seminal paper [BL69], after stating their main theorem, which proves thecomputability of the synthesis problem in (!;<), B�uchi and Landweber write:

\We hope to present elsewhere a corresponding extension of [our maintheorem] from ! to any countable ordinal."

We will see, however, that the formula �! ub is a counter-example to a full exten-sion of their theorem to (!!; <). We also provide a brief discussion of what isknown and what remains to be studied with respect to its extension to ordinalsabove !.Finally, this paper focuses on matters related to selection in a particular struc-

ture (of the form (�;<)). But, it is also interesting to consider selection overclasses of structures:

Definition 1.3. Let '( �Y ), ( �Y ) be formulas and C a class of structures. Wesay that selects (or, is a selector for) ' over C i� selects ' in every M2 C.C is said to have the selection property i� every formula has a selector over C.

In [RS] we employ the results of the current paper to explore the selectionproperty and the selection problem over classes of countable ordinals. We alsoshow there the solvability of a restricted version of the uniformization problem ina countable ordinal, which is a a natural generalization of the selection problem(see De�nition 2.22).

x2. Preliminaries and background.

2.1. Notations and terminology. We use n; k; l;m; p; q for natural num-bers, �; �; ; �; �; � for ordinals. Our ordinals are von Neumann ordinals: anordinal is identical with the set of all ordinals below it. In particular, 0 = ?,1 = f0g = f?g, 2 = f0; 1g, etc. ! = f0; 1; 2; : : : g is the set of natural num-bers. !1 is the �rst uncountable ordinal. We write � + �, ��, �� for the sum,multiplication and exponentiation, respectively, of ordinals � and �.We denote by On and Lim the classes of all ordinals and of limit ordinals,

respectively.For sets A;B, we denote by BA the set of all functions from B into A. Also,

!>A :=Sn2!

nA.


If f : B ! A and B0 � B, f [B0] = f 00B0 := ff(b0) j b0 2 B0g is the image ofB0 under f .We use the expressions \chain" and \linear order" interchangeably. We use

standard notation for sub-intervals of a chain: if (A;<) is a chain and b < a arein A, we write (b; a) := fc 2 A j b < c < ag, [b; a) := (b; a) [ fbg, etc.We use the symbol `�=' for isomorphism.

2.2. The Monadic Logic of Order (MLO). The presentation of MLOgiven here is similar to the one given in [Th97], and is equivalent to the standardde�nition of MLO which allows both �rst-order and second-order variables overthe signature `<'.

2.2.1. Syntax. The vocabulary of MLO consists ofmonadic second-order vari-ables fXi j i 2 !g, binary relation symbols `<' and `�', and unary relationsymbols `Sing' and `Emp'. Atomic formulas take the form Xi < Xj ; Xi �Xj ; Sing(Xi);Emp(Xi). All other formulas are built up from these by meansof the usual boolean connectives and second-order quanti�ers 9Xi, 8Xi. Thequanti�er depth of a formula ' is de�ned as usual and denoted by qd(').We use upper case letters X, Y , Z,... to denote variables; with an overline, �X,

�Y , etc. to denote �nite tuples of variables (always assumed distinct). We denoteby lg( �X) the length of a tuple �X, that is, the number of variables appearing init.

2.2.2. Semantics. A structure is a tuple M := (A;<M; �PM) where: A is anon-empty set, <M is a binary relation on A, and �PM := hPM0 ; : : : ; PMl�1i is a�nite tuple of subsets of A.If lg( �PM) = l, we call M an l-structure. If <M linearly orders A, we call

M an l-chain. When the speci�c l is unimportant, we simply say that M is alabeled chain.SupposeM is an l-structure and ' a formula with free-variables amongX0; : : : ; Xl�1.

We de�ne the relation M j= ' (read: M satis�es ') as follows: M j= Emp(Xi)i� PMi = ?;M j= Sing(Xi) i� P

Mi is a singleton;M j= Xi � Xj i� P

Mi � PMj ;

and M j= Xi < Xj i� there are b <M a in A with PMi = fbg, PMj = fag. Theboolean connectives are handled as usual and quanti�ers range over subsets ofA.Let M be an l-structure. The monadic theory of M, MTh(M), is the set of

all formulas with free-variables among X0; : : : ; Xl�1 satis�ed by M.From now on, we omit the superscript `M' in `<M' and ` �PM'. We often

write (A;<) j= '( �P ) meaning (A;<; �P ) j= '. Note also the following notationsendemic to this paper:

Notation 2.1. Let M := (A;<; �P ) be a structure, �Q 2 !>P(A). The expan-sion of M by �Q is M_ �Q := (A;<; �P ; �Q), where we write ` �P ; �Q' meaning thetuple obtained by concatenating �P and �Q.

Definition 2.2. (1) Let l1; l2 2 !, M an l1-structure, '( �X) a formula withlg( �X) = l1 + l2. The relation de�ned by ' in M is

D(';M) := f �Q 2 l2P(dom(M)) j M_ �Q j= 'g.

(2) Of every �Q 2 D(';M) we say that it satis�es ' in M.(3) If D(';M) 6= ?, we say that ' is satis�ed in M.


(4) If D(';M) is a singleton f �Qg, we say that ' de�nes �Q in M and that �Qis de�nable in M.

2.2.3. First-order formulas.

Definition 2.3. Let '( �Y ) be a formula. We call a quanti�er appearing in ' a�rst-order quanti�er i� it has the form 9X(Sing(X)^: : : ) or 8X(Sing(X)! : : : ).We say that ' is a �rst-order formula i� all quanti�ers appearing in ' are

�rst-order.

Note that the free variables in ' are still second-order, so that ' is satis�ed(or not) by (tuples of) subsets of the domain. When we want to indicate thata variable is to range only over elements of the domain (equivalently, over sin-gletons), we use lower case letters x, y, etc. For instance, \Let '(x; Y ) be aformula..." means that ' has the form \Sing(X) ^ '0(X;Y )". In this context,we use obvious abbreviations such as \x 2 Y " for \Sing(X) ^X � Y ".

2.2.4. Relativization.

Notation 2.4. Let �P := hP0; : : : ; Pl�1i be a �nite tuple and D any set. The�-intersection of �P and D is �P �\D := hP0 \D; : : : ; Pl�1 \Di.

Notation 2.5. LetM := (A;<; �P ) be a structure and D a non-empty subsetof A. The restriction of M to D is the structure M�D := (D;<; �P �\D).

Lemma 2.6 (Relativization). Let '( �Y ) be a formula, U a variable not appear-ing in '. We can compute a formula '�U ( �Y ; U) such that for every lg( �Y )-structure M and every non-empty subset D of its domain,

M_D j= '�U ( �Y ; U) i� M�D j= '( �Y );

that is, '�U holds in M with U interpreted as D i� ' holds in the restriction ofM to D.When this is the case, we say that ' holds in M relativized to D.

We are mostly interested in the case where M is a labeled chain and D is aninterval [b; a) for some b < a in M.

2.3. Elements of the composition method. Our proofs make use of thetechnique known as the composition method, introduced in [FV59] and adapted(and ingeniously applied) to the case of MLO in [Sh75]. To �x notations and toaid the reader not familiar with this technique, we brie y review those de�nitionsand results that we require. A more detailed presentation can be found in [Th97]or [Gu85], for instance.

2.3.1. Hintikka formulas and n-types.

Notation 2.7. Let n; l 2 !. Denote by Formn;l the set of formulas of quan-ti�er depth � n and with free variables among X0; : : : ; Xl�1.

Definition 2.8. Let n; l 2 ! and M;N be l-structures. We say that M andN are n-equivalent, denoted M �n N , i� for every ' 2 Formn;l, M j= ' i�N j= '.

Clearly, �n is an equivalence relation. For any n 2 ! and l > 0, the setFormn;l is in�nite. However, it contains only �nitely many semantically distinct


formulas. So, there are �nitely many �n-classes of l-structures. In fact, we cancompute \representatives" for these classes:

Lemma 2.9 (Hintikka Lemma). For n; l 2 !, we can compute a �nite Hn;l �Formn;l such that:

(a) For every l-structure M, there is a unique � 2 Hn;l such that M j= � .1

(b) If � 2 Hn;l and ' 2 Formn;l, then either � j= ' or � j= :'. Furthermore,there is an algorithm that, given such � and ', decides which of these twopossibilities holds.

Any member of Hn;l we call an (n; l)-Hintikka formula.2

Definition 2.10 (n-type). For n; l 2 ! and M an l-structure, we denote bytypen(M) the unique member of Hn;l satis�ed by M and call it the n-type ofM.

Thus, typen(M) determines (e�ectively) which formulas of quanti�er-depth� n are satis�ed by M.

2.3.2. The ordered sum of labeled chains and of n-types.

Definition 2.11. (1) Let l 2 !, let I := (I;<I) be a linear order and letS := hM� j � 2 Ii be a sequence of l-chains. Write M� := (A�; <

�; �P�) andassume A� \ A� = ? whenever � 6= � are in I. The ordered sum of S w.r.t. Iis the l-chain X

I

S := ([�2I

A�; <I;S; h

[�2I

P�0 ; : : : ;[�2I

P�l�1i)

where:

if �; � 2 I, a 2 A�, b 2 A� , then b <I;S a i� � < � or � = � and b <� a.

If the domains of theM� are not disjoint, replace them with isomorphic l-chainsthat have disjoint domains, and proceed as before.(2) If for all � 2 I, M�

�=M for some �xed M, we denotePI S by MI.

(3) If I = (2; <) and S = hM0;M1i, we denotePI S by M0 +M1.

The next proposition says that taking ordered sums preserves n-equivalence.

Proposition 2.12. Let n; l 2 !. Assume:

1. (I;<) is a linear order,2. hM0

� j � 2 Ii and hM1� j � 2 Ii are sequences of l-chains, and

3. for every � 2 I, M0� �

n M1�.

ThenP�2IM

0� �

nP�2IM

1�.

This allows us to de�ne the sum of formulas in Hn;l with respect to any linearorder.

Definition 2.13. (1) Let n; l 2 !, I := (I;<) a chain, H := h�� j � 2 Iia sequence of (n; l)-Hintikka formulas. The ordered sum of H w.r.t. I, denotedPI H, is an element of Hn;l such that:

1Equivalently,WHn;l is a tautology and for distinct �; � 0 2 Hn;l, � ^ �

0 is not satis�able.2Hintikka formulas made their �rst appearance in [Hi53], in the framework of �rst-order

logic.


if S := hM� j � 2 Ii is a sequence of l-chains and typen(M�) = �� for all� 2 I, then

typen(XI

S) =XI

H:3

(2) If for all � 2 I, �� = � for some �xed � 2 Hn;l, we denotePI H by � I.

(3) If I = (2; <) and H = h�0; �1i, we denotePI H by �0 + �1.

We are usually interested in cases (2) and (3) above. For addition (case 3) wehave:

Lemma 2.14 (Addition Lemma). �n; l 2 !:��0; �1 2 Hn;l:�0 + �1 is recursive.

For multiplication, we have the following fundamental result of Shelah's ([Sh75]):

Theorem 2.15. There is a recursive c : ! � ! ! ! such that for any n; l 2 !and chain I, given � 2 Hn;l and typec(n;l)(I), we can compute � I.In particular, �n; l 2 !:�� 2 Hn;l:� I is recursive in the monadic theory of

I.

2.4. The monadic theory of countable ordinals. B�uchi (for instance[BS73]) has shown that there is a �nite amount of data concerning any ordinal� !1 which determines its monadic theory:

Theorem 2.16. Let � 2 [1; !1]. Write � = !!� + � where � < !! (this canbe done in a unique way). Then the monadic theory of (�;<) is determined by:

1. whether � is countable or � = !1,2. whether � < !!, and3. �.

We can associate with every � � !1 a �nite code which holds the data requiredin the previous theorem. This is clear with respect to (1) and (2). As for (3), if� 6= 0, write

� =Pi�n !

n�i � an�i, where ai 2 ! for i � n and an 6= 0

(this, too, can be done in a unique way), and let the sequence han; : : : ; a0i encode�. The following is then implicit in [BS73]:

Theorem 2.17 (Code Theorem). There is an algorithm that, given a sentence' and the code of an � 2 [1; !1], determines whether (�;<) j= '.

Agreement. From now on, we shall simply say that an algorithm is \givenan ordinal..." or \returns an ordinal...". We always mean the code of the ordinal.

From the last theorem and 2.15, one obtains:

Theorem 2.18 (Multiplication Theorem). There exists an algorithm that, givenn; l 2 !, � 2 Hn;l and � 2 [1; !1], computes � �.

Finally, we often use the following facts concerning de�nability below !!.

3The reader may wonder why we do not say: \PIH is the unique element of Hn;l such

that...". This is because there are in Hn;l formulas that are not satis�ed in any structure. If

one of these apppears in H, then any � 2 Hn;l could serve as the sum.


Lemma 2.19 (De�nability-below-!! Lemma). For any � < !!, we can com-pute �rst-order sentences �def� and �def<� such that for every � 2 On n 1:

(a) (�;<) j= �def� i� � = �.(b) (�;<) j= �def<� i� � < �.

2.5. Uniformization.

Definition 2.20 (Uniformization). Let '( �X; �Y ), ( �X; �Y ) be formulas andM a structure. We say that uniformizes (or, is a uniformizer for) ' in M i�:

1. M j= 8 �X9�1 �Y ( �X; �Y ),2. M j= 8 �X8 �Y ( ( �X; �Y )! '( �X; �Y )), and3. M j= 8 �X

�9 �Y '( �X; �Y )! 9 �Y ( �X; �Y )

�.

M has the uniformization property i� every formula ' has a uniformizer overM.

Note that selection is the special case of uniformization where �X is the emptytuple. In [LiS98], Lifsches and Shelah show that an ordinal � has the uni-formization property i� � < !!. Thus, in particular, any ordinal � < !! hasthe selection property. Moreover, from their proof an algorithm can be extractedas follows:

Proposition 2.21 (Uniformization below !!). There is an algorithm that, givenk 2 ! and '( �X; �Y ), computes a ( �X; �Y ) that uniformizes ' in (�;<) for every� 2 !k n 1.

Now, uniformization, too, naturally leads to a decision problem:

Definition 2.22 (Uniformization problem). The uniformization problem in astructure M is:

Input: a formula '( �X; �Y ).Task: determine whether ' has a uniformizer in M, and if so, construct one

for it.

Using results from this paper, we show in [RS] the solvability of the uniformiza-tion problem in (!!; <) when the �X variables are restricted to range over subsetsof !! of order-type < � for some �xed � < !!. Solving (or showing the insolv-ability of) the full uniformization problem in (!!; <) remains an interesting openproblem.

x3. The selection property. In this section we show that an ordinal � hasthe selection property i� � < !!. Of course, by Proposition 2.21, it su�ces toshow that selection fails in each � � !!. We begin by looking at the ordinals!! and !1. In subsection 3.1, we show that a tuple de�nable in either of theseordinals is periodic (De�nition 3.3). Since a formula ' (which is satis�ed in(�;<)) is selectable in (�;<) i� it satis�ed by a de�nable tuple, we infer thatif ' is selectable in (!!; <) or (!1; <) then it is satis�ed by a periodic tuple.We note that every unbounded periodic subset of !! must have order-type !!,to obtain the non-selectability of an unbounded !-sequence in (!!; <). Usingsimilar considerations, we present a formula �stat(Y ) not selectable in (!1; <).Since !1 is a de�nable ordinal, we deduce that �stat itself is not selectable in


every � � !1. Finally, to show � 2 (!!; !1) lacks the selection property, wenote that any such � is monadically equivalent to a � of the form � = !! + �with � < !!, and that !! is de�nable in !! + �.

3.1. The Periodicity Lemma. The next few de�nitions and lemmas (upto and including 3.5) are repeatedly used in the sequel.

Definition 3.1 (Part and tail). Let �; � be ordinals with � > 0. Write � =�� + � with � < � (this can be done in a unique way). We call �� the �-part of� and � its �-tail. We also write (�mod�) for �.

Definition 3.2 (Periodic functions). Let �0; �1; �0 2 On, t0 : �0 ! T and

t1 : �1 ! T where T is any set.

1. De�ne t0 + t1 : �0 + �1 ! T . For all � < �0 + �1,

(t0 + t1)(�) :=

�t0(�) if � < �0t1(� � �0) if � � �0

:

2. De�ne (t1 �0) : �1 � �0 ! T by: (t1 �0)(�) := t1(�mod�1), for all� < �1 � �0.

3. Suppose � 2 On, t : � ! T and there exist �0 < �, �1 2 � n 1, �0 � �,t0 : �0 ! T and t1 : �1 ! T such that t = t0 + t1 �0. Then we say that tis periodic.

4. We call (�0; �1) a phase and period for t.4

5. If � is an ordinal such that �0; �1 < �, we say that t is �-periodic. 2

In (5), we will usually have � = !!. Next, we de�ne the notion of a periodictuple of subsets of an ordinal.

Definition 3.3 (Periodic tuple). Let l 2 !, � 2 On and �P 2 lP(�). Thecharacteristic function of �P in � is =��P : �! l2 given by:

=��P (�) := h�0(�); : : : ; �l�1(�)i where for all � < � and i < l,

�i(�) :=

�1 if � 2 Pi0 else

:

We shall apply to �P all terms and notations of De�nition 3.2, referring to =��P .

Thus, we speak of �P being periodic, of a phase and period for �P , etc.

The following is a special case of Theorem 3.5(B) of [Sh75] (only our languageis di�erent):

Lemma 3.4 (p-Lemma). For every n 2 !, there is a natural number p(n) <jHn;0j such that the function �� 2 !1: type

n(�;<) is periodic with phase and

period (!p(n); !p(n)). In particular, any two non-0 countable multiples of !p(n)

are n-equivalent.

Now, every multiple of !! is a multiple of !p(n) for each n 2 !. Thus,

Lemma 3.5 (Tail Lemma). If �,� 2 [!!; !1) have the same !!-tail, thenMTh(�;<) = MTh(�;<).

The following is a standard fact about n-types.

4Thus, t may have many phases and periods.


Lemma 3.6. Let n; l 2 ! and let be M a structure. Then typen(M) deter-mines typen(M_h?; : : : ;?| {z }

l times

i). In fact, the latter is computable from the former.

We need the following lemma only for the case � = !! or � = !1, but sincethis involves no signi�cant e�ort, we prove something slightly more general.

Lemma 3.7 (Periodicity Lemma). Let � 2 [!!; !1] and suppose �P 2 lP(�) isde�nable in (�;<) by a formula of quanti�er depth � n. If � is a multiple of

!!, then �P is periodic with phase and period (!p(n+l); !p(n+l)).

Proof. We show that for every � 2 �n1, the (n+l)-type of (�;<) determines=��P (�).

5 Since, by the p-Lemma, �� 2 �: typen+l(�;<) is periodic with phase

and period (!p(n+l); !p(n+l)), this will �nish the proof.Let ( �Y ) be a formula de�ning �P in (�;<) with qd( ) � n. Let � :=

h�0; : : : ; �l�1i be a binary l-tuple (that is, � : l ! 2). Then there is a for-mula ��(x) of quanti�er depth � n+ l such that for every � < �, (�;<) j= ��(�)i� =��P (�) = �. Indeed,

��(x) := 9 �Y� ( �Y ) ^

^i<l;�i=0

x =2 Yi ^^

i<l;�i=1

x 2 Yi�:

Of course, whether or not (�;<) j= ��(�) is determined by the (n + l)-typeof (�;<; f�g). But, for any � 2 � n 1 we have, by de�nition of the sum of(n+ l)-types (see 2.13),

typen+l(�;<; f�g) = typen+l(�;<;?) + typen+l([�; �); <; f�g):

We claim that typen+l([�; �); <; f�g) is independent of �. Indeed, �x � 2 �n1.Assume �rst � is countable. Since � is a multiple of !!, so is the order-type of[�; �). By the Tail Lemma, ([�; �); <) is monadically equivalent to (�;<). For�(X) any formula, let �min� be a sentence saying: \if x is the minimal element of

the domain, then �(fxg) holds." Then ([�; �); <; f�g) j= � i� ([�; �); <) j= �min�

i� (�;<) j= �min� . But, MTh(�;<) is, of course, independent of �. If � = !1, we

even have ([�; �); <; f�g) �= (!1; <; f0g).Thus, =��P (�) is determined by typen+l(�;<;?), which Lemma 3.6 tells us is

in turn determined by typen+l(�;<). This �nishes the proof. a

3.2. Failure of the selection property in � � !!.

Notation 3.8. Let P be a set of ordinals. Then P is well-ordered by theusual ordering of ordinals. The order-type of P , denoted otp(P ), is the uniqueordinal isomorphic to (P;<).

Lemma 3.9. If � is a non-0 multiple of !! and P � � is !!-periodic, thensup(P ) < !! or otp(P ) = �.

Proof. Suppose �0; �1 < !! are a phase and period for P . Since � is amultiple of !!, � � �0 = � and �1� = �. Thus, (�;<; P ) �= (�;<; P )�[0;�0) +(�;<; P )�[�0;�0+�1) �. If P \ [�0; �0 + �1) = ?, then sup(P ) � �0 < !!. If�1 2 P \ [�0; �0 + �1), then f�0 + �1 + �1 j 2 �g has order-type � and is asubset of P , so otp(P ) � �. But, P � �, so otp(P ) = �. a

5Recall that =��P is the characteristic function of �P in �.


From the last lemma and the Periodicity Lemma, we immediately obtain:

Corollary 3.10. The formula �! ub(Y ) which says \Y is an unbounded !-sequence" has no selector in (!!; <).

To handle !1 recall the following de�nitions:

Definition 3.11. (1) Let C � !1. C is called:

closed i� for every limit � < !1, if sup(C \ �) = �, then � 2 C.6

a club i� C is closed and unbounded in !1.

(2) S � !1 is called stationary i� for every club C � !1, S \ C 6= ?.

Note that being a club and being stationary are de�nable properties of a subsetof !1. In de�ning clubs we do not even use second order-quanti�cation, and inde�ning stationary subsets we quantify over clubs.

Corollary 3.12. Let �(Y ) say: \both Y and its complement are stationary".Then � is not selectable in (!1; <).

Proof. It takes an amount of the Axiom of Choice to prove that D(�; !1) 6= ?(that is, that there are stationary co-stationary subsets of !1).

7 But, using ACit can even be shown that there is a family of !1 disjoint stationary subsets of!1. For a proof see, for instance, [HJ99]. By the Periodicity Lemma, it su�cesto show that any periodic P � !1 either contains a club or is non-stationary.Indeed, let �0; �1 < !1 be a phase and a period for P , respectively. Assume�0 2 P . Then, by periodicity, C := f�0 + �1� j � < !1g � P . It is obviousC is unbounded. Also, if � < !1 is a limit point of C then we can pick anincreasing h�i j i 2 !i � � such that � = supf�0 + �1�i j i 2 !g. But, then� = �0 + �1 � supf�i j i 2 !g is a member of C, so C is also closed. Similarly, if�0 =2 P , then f�0 + �1� j � < !1g � !1 n P is a club. a

Now, the reduction from any other ordinal � � !! to the case of !! or !1 isbased on the following simple observation.

Definition 3.13. Let (A;<) be a linear order. Call S � A a segment of(A;<) if (b; a) � S whenever b < a are in S.

Lemma 3.14 (Segment Lemma). Let M be a labeled chain and a formula

which de�nes a tuple �P inM. Let S be a segment ofM. Then typeqd( )((M_ �P )�S)de�nes �P �\S in M�S.

Proof. Set n := qd( ) and let �P 0 satisfy typen((M_ �P )�S) in M�S . Wemust show that �P 0 = �P �\S.Write S� := fb 2 dom(M) j 8a 2 S:b < ag and S+ := dom(M) n (S� [ S).

Both are segments of M. Assume S� and S+ are nonempty. Then

typen(M_ �P ) = typen((M_ �P )�S�) + typen((M_ �P )�S) + typen((M_ �P )�S+):

By assumption, typen((M_ �P )�S) = typen(M�S_ �P 0), so

6That is, C is closed under taking sup.7It is consistent with Zeremelo-Frankel set theory that D(�; !1) = ?, but ZF+\D(�; !1) =

?" is a rather strange set theory. In section 5 of [BS73] the reader will �nd an enlighteningdiscussion of the relation between \D(�; !1) = ?" and de�nability in (!1; <).


(M_ �P )�S� +M�S_ �P 0 + (M_ �P )�S+ j= .

But, this structure equals M_( �P �\S� [ �P 0 [ �P �\S+). Since de�nes �P in M,it follows ( �P �\S� [ �P 0 [ �P �\S+) = �P , so �P 0 = �P �\S.Finally, if S� = ? ignore the leftmost \summand" above, and if S+ = ?,

ignore the rightmost \summand". a

Note that for every � � !1, we can de�ne the segment [0; !1) in (�;<) by saying:\x is in this segment i� for every limit ordinal y smaller than or equal to x, thereis an unbounded !-sequence in [0; y)". Therefore, the formula �stat(Y ) appearingbelow does exist.

Proposition 3.15. Let �stat(Y ) say: \Both Y \!1 and !1 nY are stationaryin !1". Then �stat has no selector in (�;<) for every � � !1.

Proof. Suppose P � � satis�es �stat in (�;<). Then P \ !1 satis�es the �of Corollary 3.12 in (!1; <), hence is not de�nable in (!1; <). By the SegmentLemma (applied to the initial segment !1 = [0; !1)), P is not de�nable in (�;<).

a

To handle � 2 (!!; !1) note �rst that the monadic theory of � \knows" exactlywhich formulas select which others in (�;<):

Definition 3.16 (Selection axiom). For formulas '( �Y ); ( �Y ), the ( ; �)-selectionaxiom, denoted sel-ax( ;'), is the conjunction of the sentences appearing in thede�nition of selection (De�nition 1.1).

Clearly, selects ' in (�;<) i� (�;<) j= sel-ax( ;'). By the Tail Lemma,this means that to show that in � 2 [!!; !1) selection fails, it su�ces we handle� of the form � = !! + � where � < !!.

Proposition 3.17. Let k 2 ! n 1. There is a formula �k(Y ) such that for all� 2 !k n 1, �k is not selectable in (!! + �;<).

Proof. Let �k(Y ) say:

\If x is the least such that the order-type of [x;1) is smaller than !k, thenY \ [0; x) is an unbounded !-sequence in [0,x)."

Suppose � 2 [1; !k). Then !! is the least � < !!+� such that otp([�; !!+�)) <!k. Therefore, if P satis�es �k in (!! + �;<), then P \ !! is an unbounded!-sequence in !!. By Corollary 3.10, P \!! is not de�nable in (!!; <). By theSegment Lemma, neither is P de�nable in (�;<). a

Summing up, we have indeed shown:

Theorem 3.18 (Selection property). An ordinal � has the selection propertyi� � < !!.

In [LiS98], Lifsches and Shelah present a particular formula '(X;Y ) not uni-formizable in every � � !! (whether countable or not). Above, a formula notselectable in every � � !1 was presented. Is there a single formula not selectablein every � 2 [!!; !1)? The answer turns out to be negative:

Proposition 3.19. Let k 2 ! and '( �Y ) a formula with qd(9 �Y ') � k. Then

for every � 2 [!p(k); !!), ' is selectable in (!! + �;<).8

8The selector may depend on �, of course.


Because it requires some preparation, we postpone the proof to section 6.

x4. The selection problem. In this section we show the solvability of theselection problem in (�;<) for � � !1. For � < !! this is handled by Proposition2.21. In the previous section we have seen that any formula which is (satis�edand) selectable in (!!; <) or (!1; <) is satis�ed by a periodic tuple. Here weshow that: (1) conversely, a formula satis�ed by a periodic tuple is selectable;(2) it is decidable whether a formula is satis�ed by a periodic tuple. We handle(2) in a way that allows us to construct a selector for ' when one exists.

4.1. Solvability of the selection problem in (�;<) for � 2 [!!; !1]. Thefollowing is not hard to derive from the p-Lemma.

Lemma 4.1 (Satis�ability Lemma). (a) Let k 2 ! and '( �Y ) a formula suchthat qd(9 �Y ') � k. If ' is satis�able in some countable ordinal, then ' is also

satis�ed in some � < !p(k) � 2.(b) There is an algorithm that, given k and ' as above, decides whether ' is

satis�ed in a countable ordinal, and if so, returns an � < !p(k) � 2 in which ' issatis�ed.

Proposition 4.2. Let n; l 2 !, ' 2 Formn;l and � 2 [!!; !1] a multiple of!!. Suppose ' is satis�ed in (�;<). Then the following are equivalent:

(a) ' is selectable in (�;<).(b) There exists an !!-periodic �P 2 D('; �).(c) There are �0; �1 2 Hn;l such that:

(c1) �0; �1 are satis�able in a countable ordinal,(c2) �0 + �1 � j= '.9

Furthermore, there is an algorithm that, given �0; �1 as in (c), constructs a se-lector (�0;�1) for ' in (�;<).

Proof. (a)) (b) is the Periodicity Lemma.(b) ) (c): Let �P 2 D('; �) be periodic and assume �0; �1 are a phase and

period < !! for �P . Write

�0 := typen((�;<; �P )�[0;�0)); �1 := typen((�;<; �P )�[�0;�0+�1)):

As in the proof of Lemma 3.9,

(�;<; �P ) �= (�;<; �P )�[0;�0) + (�;<; �P )�[�0;�0+�1) �;

so typen(�;<; �P ) = �0+�1�: Since (�;<; �P ) j= ', this means �0+�1� j= '.(c)) (a): Fix �0; �1 as in (c). We construct (�0;�1) as follows. Note that the

construction depends only on �0; �1 and not on �. Fix i 2 2:

1. By (c1), �i is satis�able in a countable ordinal. By the Satis�ability Lemma,we can compute an �i 2 !! n 1 in which �i is satis�able.

2. Apply Proposition 2.21 to compute a selector i( �Y ) for �i in (�i; <). Denoteby �Pi the unique element of D( i; �i).

3. By the De�nability-below-!! Lemma, we can compute a sentence �def�i

which is satis�ed only in (�i; <).

9Multiplication by � is a bit misleading here. It only matters whether � is countable or !1.Indeed, if � is countable, then by the Tail Lemma, � is monadically equivalent to !! . Thus,by Theorem 2.15, �1 � = �1 !! .


Now,

4. M := f�0 + �1� j � < �g is de�nable in (�;<). Indeed, M is the uniqueunbounded subset of � such that:

min(M) = �0,for any � 2M , if �0 is the successor of � in M , then otp([�; �0)) = �1,

andM is closed under limits.

Therefore, M is de�ned by the formula �M (W ) saying:\W is unbounded; �def�0

holds relativized to [0;min(W )); for every x 2 W ,

�def�1holds relativized to [x; y), where y is the successor of x in W ; and W

is closed under limits."5. Finally, let (�0;�1)(

�Y ) say: \for every (i.e. the unique) W that satis�es�M (W ), 0 holds relativized to the segment [0;min(W )), andfor every x 2 W , 1 holds relativized to [x; y), where y is the successor ofx in W ."

Then, by choice of 0; 1, we have that (�0;�1) de�nes in (�;<) the unique �Psuch that

(�;<; �P ) �= (�0; <; �P0) + (�1; <; �P1) �:

But, typen(�0; <; �P0) = �0, typen(�1; <; �P1) = �1 and �0 + �1 � j= ', by (c2).

Hence,

(�0; <; �P0) + (�1; <; �P1) � j= ';

and all is well. a

An immediate corollary is:

Corollary 4.3. For � 2 f!!; !1g, there is an algorithm that, given a for-mula '( �Y ), decides whether ' has a selector in (�;<), and if so, constructssuch a .

Proof. By the Code Theorem, we can decide whether ' is satis�ed in (�;<).If ' is not satis�able, then Y0 6= Y0 (say) selects ' in (�;<). Assume henceforth

' is satis�ed in (�;<). Let n := qd('), l := lg( �Y ) and denote by Sat<!!

n;l

the set of (n; l)-Hintikka formulas satis�able in some countable ordinal. By the

Satis�ability Lemma, Sat<!!

n;l is computable. Let

Sel�' := f(�0; �1) 2 Sat<!!

n;l � Sat<!!

n;l j �0 + �1 � j= 'g.

It is immediate from the \(a) , (c)" equivalence of the previous propositionthat ' is selectable in (�;<) i� Sel�' 6= ?. But, Sel

�' is computable. Indeed, �x

�0; �1 2 Sat<!!

n;l . By the Multiplication Theorem, we can compute �1 �. Bythe Addition Lemma, we can then compute �0+ �1�. Finally, by the HintikkaLemma, we can decide whether �0 + �1 � j= '. It follows that we can decidewhether ' has a selector in (�;<).Now, if Sel�' 6= ?, pick any (�0; �1) 2 Sel�'. By the previous proposition, we

can compute from (�0; �1) a selector for ' in (�;<). a

As in the previous section, there remains the simple task of reducing the selectionproblem in � 2 (!!; !1) to the selection problem in (!!; <).


Theorem 4.4 (Solvability of selection). There exists an algorithm that, given� 2 [!!; !1] and a formula '( �Y ), decides whether ' has a selector in (�;<),and if so, constructs one for it.

Proof. If � is a multiple of !!, then either � = !1 or � is monadicallyequivalent to !!. In each case, the previous corollary does the job. Otherwise,by the Tail Lemma, we may assume � = !! + � for � 2 !! n 1. Again, wecan decide if ' is satis�ed in (�;<) and may assume this is the case. Set n :=

qd('), l := lg( �Y ). Let Sel!!

n;l be the set of (n; l)-Hintikka formulas that are bothsatis�able and selectable in (!!; <). By the previous corollary, we can compute

Sel!!

n;l . Let Sat�n;l be the set of (n; l)-Hintikka formulas satis�ed in (�;<). By the

Code Theorem, Sat�n;l is computable. We claim that ' has a selector in (�;<)i�

(Sel): there are � 2 Sel!!

n;l and �0 2 Sat�n;l such that � + � 0 j= '.

Furthermore, given such �; � 0, we can compute a selector for ' in (�;<).Suppose �rst �P 2 D('; �) is de�nable in (�;<). If we let

� := typen((�;<; �P )�[0;!!)) and �0 := typen((�;<; �P )�[!!;�)),

then � + � 0 j= ' and � 0 2 Sat�n;l. By the Segment Lemma (3.14), �P \ !! is

de�nable in (!!; <). But, �P \ !! satis�es � in (!!; <), so � 2 Sel!!

n;l .Assume conversely that (Sel) holds. By the previous corollary, we can compute

a that selects � in (!!; <). By Proposition 2.21, we can compute a 0 thatselects � 0 in (�;<). As in the proof of 3.17, we can de�ne !! in (�;<). Thus,the ( �Y ) that says: \ holds relativized to [0; !!) and 0 holds relativized to[!!;1)" selects ' in (�;<).Finally, by the Addition Lemma and the Hintikka Lemma, it is decidable

whether (Sel) holds. a

4.2. Idempotents and selection. In this subsection we reexamine the con-struction of the selector (�0;�1) in the \(c)) (a)" direction of Proposition 4.2,to extract data that will turn out useful in section 7 and is also interesting initself.Recall that (�0;�1) says: \for the unique W that satis�es �M (W ), �0 holds

relativized to [0;min(W )), and �1 holds relativized to [x; y) whenever x; y aresuccessive elements of W ." As noted in the proof of 4.2, the construction of (�0;�1) did not depend on the � given in that proposition. In showing that (�0;�1) selects ' in (�;<), we did use the assumption that we can pick �0; �1 <minf�; !!g such that:

1. for i 2 2, �i is satis�able in (�i; <).2. �� 0 is a multiple of �1, and3. �0 + �1

��0�1

j= '.

Note that from assumptions (1) and (2) alone it follows that (�0;�1) de�nes in

(�;<) a tuple �P such that typen(�;<; �P ) = �0 + �1��0�1

. Thus,

Lemma 4.5. Let n; l 2 ! and �0; �1 2 Hn;l both satis�able in some countableordinal. Then we can compute a formula (�0;�1) such that for every non-0

multiple � of !p(n+l)+1, (�0;�1) selects �0 + �1��0�1

in (�;<).


Proof. By the Satis�ability Lemma we can pick �0; �1 < !p(n+l)+1 such thatfor i 2 2, �i is satis�ed in (�i; <). Now, let (�0;�1) be constructed as in the proof

of 4.2. Fix � 2 On n 1 and let � := !p(n+l)+1�. Then � � !p(n+l) � 2 implies�0; �1 < �. Since � is a multiple of !p(n+l)+1, so is � � �0 = �. Thus, � � �0is divisible by �1. a

We needed assumption (3) to show that the �P de�ned by (�0;�1) satis�es ' in(�;<). We now show that we can pick �1 to ensure that (�0;�1) will select ' in

every countable multiple of !p(n+l)+1.

Definition 4.6. Let n; l 2 !, � 2 Hn;l and � 2 On n 1. We call � an �-idempotent i� � � = � .

Lemma 4.7. An !-idempotent is an �-idempotent for every � 2 !1 n 1.

Proof. Let n; l 2 ! and � 2 Hn;l an !-idempotent. We proceed by inductionon � 2 !1 n 1. For � = 1, there is nothing to prove. Assume � 2 !1 n 2 andour claim holds for all �0 < �. If � = �0 + 1 is a successor, then � (�0 + 1) =

� �0 + �induction

= � + �assumption

= � + � ! = � (1 + !) = � ! = �.Assume � is limit. Pick a strictly increasing h�i j i 2 !i � � with supf�i j i 2!g = � and �0 := 0. For i 2 !, let �i := �i+1 � �i. Then � =

Pi2! �i, so

� � =Pi2! � �i

induction=

Pi2! � = � ! = �. a

It turns out that when ' is selectable, we can pick an !-idempotent �1.

Proposition 4.8. Let n; l 2 !, ' 2 Formn;l and � a non-0 countable multipleof !!. Assume ' is satis�ed in (�;<). Then the following are equivalent:

(a) ' is selectable in (�;<).(b) There exist �0; �1 2 Hn;l such that:

(b1) �0; �1 are satis�able in a countable ordinal,(b2) �0 + �1 j= ', and(b3) �1 is an !-idempotent.

Furthermore, when this is the case, we can compute �0; �1 as in (b).

Proof. We show that (b) here holds i� clause (c) of Proposition 4.2 holds.Assume (b) here holds. Then (c1) is (b1). By the previous lemma, �0 + �1 =�0 + �1 � and (c2) follows. Assume (c) there holds, i.e. there are � 00; �

01 2 Hn;l

both satis�able in a countable ordinal such that � 00 + � 01 � j= '. Let �0 := � 00and �1 := � 01 �. Then (b2) is immediate. Also, if �01 < !! and � 01 is satis�edin (�01; <), then �1 is satis�ed in (�01�;<). Hence, (b1) holds. Finally, � and�! have the same monadic theory. By Theorem 2.15, this means �1 = � 01 � =� 01 (�!) = (� 01 �) ! = �1 !.By the proof of Corollary 4.3, when (c) of 4.2 holds, we can compute � 00; �

01 as

above. Since � 01 � is computable from � 01, the \furthermore" holds. a

Corollary 4.9. There is an algorithm that, given a formula '( �Y ) selectable

in (!!; <), constructs a which selects ' in !p(qd(9�Y '))+1� for every � 2 !1n1.

Proof. Let n := qd(') and l := lg( �Y ). Compute �0; �1 2 Hn;l as in (b) ofthe last proposition. The desired formula is the (�0;�1) of Lemma 4.5. a


x5. Comparing di�culty of selection in a countable ordinal.

5.1. Selection degrees. We know that the formula �! ub(Y ) saying \Y isan unbounded !-sequence" has no selector in (!!; <). Now, let us look at theformula �!2 ub saying \Y is unbounded and of order type !2". It is immediatefrom Lemma 3.9, that �!2 ub has no selector in (!!; <). But are there any otherinteresting relations between these two formulas? Can we say, for instance, that�!2 ub is even \harder" to select than �! ub (whatever that might mean)? Or,perhaps the other way round? Do we feel that the example of �!2 ub \contains anew idea" when it comes to our discussion of selection?To turn these admittedly vague questions into mathematical ones, we require

a notion of comparing formulas and perhaps an equivalence relation on them.But, as our example shows, semantical equivalence seems not to be the rightnotion. Note, however, the following. For any unbounded !2-sequence S2 � !!,the set of limit points of S2 (i.e., those � < !! such that sup(S2 \�) = �) is anunbounded !-sequence. Also, this set is de�nable from S2. On the other hand,given an unbounded !-sequence S1 � !!, the set f� + n j � 2 S1; n 2 !g isan unbounded !2-sequence. And, again the latter set is de�nable from S1. Theexample suggests the following de�nition:

Definition 5.1 (Reduction). Let '0( �Y ), '1( �X) be formulas and M a struc-ture. We say that '0 is easier to select than '1 in M (in symbols: '0 �M '1)i� either :

1. '0 is selectable, or2. '1 is satis�ed in M and there exists a formula ( �X; �Y ) such that for all

�P 2 D('1;M), ( �P ; �Y ) selects '0 in M.

We call a reduction of '0 to '1 in M.'0; '1 are equally easy to select over M (in symbols: '0 �M '1) i� both

'0 �M '1 and '1 �M '0.

We leave the proof of the following easy lemma to the reader.

Lemma 5.2. Let M be a structure. Then:

(a) �M is a re exive, transitive relation on the formulas.(b) �M is an equivalence relation on the formulas.

Definition 5.3 (Degrees). LetM be a structure. We call every�M-equivalenceclass an M-degree.

The main result of this section is:

Theorem 5.4. For every � 2 [!!; !1), there are two degrees in (�;<): oneconsisting of all formulas selectable in (�;<) and the other consisting of all non-selectable formulas.

Note �rst that,

Lemma 5.5. All formulas selectable in a structure M form one M-degree.

Proof. It is immediate from (1) of De�nition 5.1 that all selectable formulasshare a degree. It remains to be seen that any formula easier than a selectable


formula is itself selectable. Suppose then that ( �X; �Y ) reduces some formula'0( �Y ) to '1( �X), where '1 is selectable in M. Let ( �Y ) be any selector for '1in M. Then 9 �X(( �X) ^ ( �X; �Y )) selects '0 in M. a

It remains to be seen that all non-selectable formulas are equally easy to selectin (�;<). To prove this, we show in subsection 5.3, that the formula ��! ub(Y )saying \Y is an unbounded !-sequence in the !!-part of �" is easier to selectthan every non-selectable formula. Then, in subsection 5.4, we show that everyformula is easier than ��! ub.Note that MTh(�;<) holds all information concerning the degrees in (�;<).

Lemma 5.6. Let '0( �Y ); '1( �X) be formula, andM;M0 monadically equivalentstructures. Then '0 is easier than '1 to select in M i� it is easier than '1 toselect in M0. When this is the case, the same formulas are reductions of '0 to'1 in M as are in M0.

Proof. Because the monadic theory of a structure determines which formulasare selectable in it, '0 is selectable in M i� it is selectable in M0. Similarly, '1is satis�able in M i� it is satis�able in M0. Finally, for any formula ( �X; �Y ),

M j= 8 �X('1( �X)! sel-ax( ( �X; �Y ); '0( �Y )))

i�M0 satis�es this sentence. Looking again at De�nition 5.1, it is clear that ourclaim is proved. a

Thus, it su�ces we handle � of the form !!+ � for � < !!. In practice, we shallhandle the case of !! and leave the easy generalization to other !! + � to thereader.Note �nally that in proving Theorem 5.4, we shall be solving a restricted case

of the uniformization problem in a countable ordinal (recall De�nition 2.22). In-deed, if '0( �Y ) and '1( �X) are both satis�able in a structureM, then '0 is easierthan '1 to select in M i� '1( �X) ^ '0( �Y ) has a uniformizer in M.

We have seen that tuples de�nable in (!!; <) are \combinatorially simple" -they are periodic. Tuples satisfying non-selectable formulas, however, are neverperiodic (by Proposition 4.2). To handle such tuples, we will use a somewhatmore re�ned tool of \combinatorial analysis", namely, B�uchi's notion of an au-tomaton acting on ordinal words and his translation of formulas to automata.We begin, therefore, by recalling the relevant de�nitions.

5.2. Automata on ordinal words.

Definition 5.7. Let � 2 On n 1, � : �! Q with Q any set. The limit of � is

lim(�) := fq 2 Q j ��1(q) is unbounded in �g:

We call the elements of lim(�) limit states of �.

Definition 5.8 (Automata). 1. An automaton is a tuple A := hQ;�; �suc; �limiwhere:� Q and � are �nite non-empty sets,� �suc : Q� �! P(Q) n 1, and


� �lim : P(Q)! P(Q) n 1.10

2. We call Q the states of A, � its input alphabet, �suc the successor transitionfunction and �lim the limit transition function.

3. We call A deterministic i� for all q 2 Q and � 2 �, j�suc(q; �)j = 1 and forall K � Q, j�lim(K)j = 1. In that case, we assume that �suc : Q � � ! Qand �lim : P(Q)! Q.

4. An initialized automaton is a pair A := hA; qIi with A as above and qI 2 Q.

Definition 5.9 (Runs). Let A := hQ;�; �suc; �lim; qIi be an initialized au-tomaton. Let � 2 On, = : �! �.

1. A run of A on = is a function � such that:� rng(�) � Q,� if � is zero or successor, dom(�) = �+ 1,� if � is limit, dom(�) 2 f�; �+ 1g,

and, for each � 2 dom(�):� �(0) = qI ,� if � + 1 2 dom(�), �(� + 1) 2 �suc(�(�);=(�)),11 and� if � is limit, �(�) 2 �lim(lim(�j�)).12

2. We call � a strict run in all cases but when � is limit and dom(�) = �+ 1.3. If A is deterministic there exists a unique strict run of A on =, denoted by

runA(=).4. We call � a dom(�)-run to indicate its length.

So far we have automata that can run on ordinal words. To allow them tode�ne classes of such words, we must provide them with acceptance conditions.

Definition 5.10. (Muller automata)

1. AMuller automaton is a tuple A := hQ;�; �suc; �lim;Fi where hQ;�; �suc; �limiis an automaton and F � P(Q).

2. We call F the acceptance condition of A.3. An initialized Muller automaton is a pair A := hA; qIi where A is a Muller

automaton and qI 2 Q.4. Let � 2 On, = : � ! �. We say that = is accepted by A i� there exists a

strict run � of A on = with lim(�) 2 F . We call such a run accepting.

Definition 5.11. Let ' be a formula with free variables Y0; : : : ; Yl�1, A aninitialized Muller automaton with input alphabet l2 and C a class of ordinals.We say that A and ' are equivalent over C i� for every � 2 C and �P 2 lP(�),

�P 2 D('; �) i� =��P is accepted by A:

The following is proved in [BS73], section 4. The proof is far from trivial.

Theorem 5.12 (B�uchi, 1973). There exists an algorithm that, given a formula' with free-variables Y0; : : : ; Yl�1, returns an initialized deterministic Mullerautomaton A with input l2 which is equivalent to ' over the countable ordinals.

10Our de�nition is a bit more restrictive than the de�nition of non-deterministic automataoften found in the literature, in that we require that for all q 2 Q and � 2 �, �suc(q; �) 6= ?,and for all K � Q, �lim(K) 6= ?.

11Or, when A is deterministic, �(� + 1) = �suc(�(�);=(�)).12Or, when A is deterministic, �(�) = �lim(lim(�j�)).


Remark 5.13. The fact that A is deterministic will be important when weprove that �! ub represents the minimal non-selectable degree in (!!; <). It willnot be used when proving that it represents the maximal degree.Note also that the equivalence between MLO formulas and Muller automata,

as de�ned here, breaks in (!1; <). Consult section 6 of [BS73] for more details.13

Let � 2 Onn1 and A an initialized automaton. The following lemma essentiallysays that MTh(�;<) knows \everything there is to know" concerning the runsof A on inputs of length �: are there runs taking the automaton into a givenstate? are there runs with a particular limit? are there runs in which certainstates appear while others do not?, etc.While we only state the existence of a particular formula, it is an easy exercise

in formalization to use it (and variants thereof) to \answer" every such question.

Lemma 5.14. Let l 2 !, A an initialized automaton with input l2 and statesQ := fq0; : : : ; qk�1g. Let q 2 Q. Then there is a formula

�run-onA!q (Y0; : : : ; Yl�1;W0; : : : ;Wk�1; )

such that for � 2 On n 1, �P 2 lP(�) and �Q 2 kP(�):

(�;<) j= �run-onA!q ( �P ; �Q) i� there exists an (� + 1)-run � of A on =��P suchthat8i < k:Qi = ��1(qi) \ � and last(�) = q.

Thus, �run-onA!q de�nes (a natural encoding of) the set of (� + 1)-runs of Aon =��P that take it into state q. If A is deterministic, there exists a unique�Q 2 D(�run-onA!q ( �P ; �W ); �).

Remark 5.15. In section 7 it will be important that �run-onA!q can be taken a

�rst-order formula (in the sense of De�nition 2.3).

5.3. An unbounded !-sequence is easier than any non-selectable for-mula. Suppose '( �Y ) is a formula not selectable in (!!; <) and A an automatonequivalent to it over the countable ordinals. Let � be an accepting !!-run of Awith limit F . De�ne an !-sequence �0; �1; : : : as follows: �0 is the �rst ordinalafter which only limit states of � appear in �, �1 is the �rst � above �0 such that� visits all states of F between �0 and �, �2 is the �rst �

0 after �1 such that �visits all states of F between �1 and �0, etc. If we let �! := supf�n j n 2 !g,then lim(�j�! ) = F . Using the fact that ' is not selectable, we will argue that�! = !!, i.e. that �0; �1; : : : is an unbounded !-sequence. Since the sequence�0; �1; : : : is de�nable from �, this will show that given any tuple satisfying ',we can de�ne an unbounded !-sequence, as desired. Below are the details.We know that ' is selectable in (!!; <) i� it is satis�ed by a periodic tuple.

Let us therefore begin by relating the existence of such a tuple with the existenceof periodic accepting runs of A.

Notation 5.16. Let S be a set of ordinals and f a function with S � dom(f).Then f � S := f � �S where �S : otp(S) ! S is the unique order-preservingisomorphism.

13Neeman ([Ne]) de�nes a generalized notion of �nite automaton for which the equivalenceholds in every ordinal.


Lemma 5.17. Let '(X0; : : : ; Xl�1) be a formula, A := hA; qIi an initializedMuller automaton equivalent to ' over the countable ordinals and � a countablemultiple of !!.

If A has an accepting !!-periodic �-run, then there is an !!-periodic �Psatisfying ' in (�;<).14

Proof. Let � be an accepting !!-periodic �-run of A. Suppose � is a runon �P 0 for some �P 0 2 lP(�).15 Let �0; �1 < !! be a phase and period of �,respectively. Let �P0 := �P 0�\�0, �P1 := �P 0�\[�0; �0 + �1) and �P := �P0 + �P1 �.16

Then � is a run of A on �P . Since � is accepting, �P 2 D('; �). a

Our second step is to present a simple \automata-theoretic" criterion for theexistence of periodic runs (Lemma 5.23). To do this we need two de�nitions andtwo lemmas.

Definition 5.18 (Pumpability). LetA be an initialized automaton with statesQ and limit transition function �lim. Let K � Q, � 2 On. We say that K is:

1. stable in A i� K \ �lim(K) 6= ?;2. an �-limit in A i� there exist some limit � < � and a �-run � of A such

that lim(�) = K;3. �-pumpable in A i� it is both stable and an �-limit in A.

The following easy lemma explains why stable sets are good for us.

Lemma 5.19. Let A be an automaton with states Q and limit transition func-tion �lim. Let K � Q. Suppose �1 2 Lim and �1 is an �1-run of A with�1(0) 2 K \ �lim(K) and rng(�1) = lim(�1) = K. Then for every � 2 On n 1,�1 � is a run of A with rng(�1 �) = lim(�1 �) = K.

Proof. We proceed by induction on �. For � = 1, there is nothing to show.Let � = �0 + 1 be a successor. Then �1 � = �1 �0 + �1. By the inductiveassumption, lim(�1 �0) = K. Since �1(0) 2 �lim(K), �1 �0 + �1 is a run ofA. Also, lim(�1 �) = lim(�1) and rng(�1 �) = rng(�1 �0) [ rng(�1) = K.Assume � is limit. It is easy to see that the union of an increasing (by inclusion)sequence of runs of A is also a run. Thus, �1 � =

S 2� �1 is a run. Also,

rng(�1 �) =S 2� rng(�1 ) = K. Finally, lim(�1 �) = rng(�1) = K. a

Definition 5.20 (Fiber). Let � 2 Lim and � : � ! Q with jQj < !. Byinduction on n 2 ! de�ne:

�0(�) := minf� < � j �[[�; �)] = lim(�)g. That is, �0(�) is the least ordinalafter which all states appearing in � are limit states of �.�n+1(�) := minf� 2 (�n(�); �) j �[[�n(�); �)] = lim(�)g. That is, �n+1(�)is the �rst � after �n(�) such that � visits all states of lim(�) in the interval[�n(�); �).

14We also have that if A is deterministic and �P 2 lP(�) is !!-periodic, then runA(=��P) is

!!-periodic. However, there exists a non-deterministic automaton A with input alphabet 2such that the only = : !! ! 2 accepted by A is �� 2 !! :0 (say), and yet A has no periodicaccepting !!-runs.

15We shall speak of runs on tuples of subsets of an ordinal, always meaning runs on theircharacteristic functions.

16By which we mean �P is the unique such that (�;<; �P ) �= (�0; <; �P0) + (�1; <; �P1) �.We shall continue to use such abbreviations when no confusion is likely to occur.


The �ber of � is �b(�) := f�n j n 2 !g.

Remark 5.21. �b(�) is an !-seqeunce and lim(�jsup(�b(�))) = lim(�).

Lemma 5.22. Let A be an initialized automaton with states Q and K � Q.Suppose � 2 On and K is an �-limit in A. Then K is also an !!-limit in A.

Proof. We only need the case � = !1. Using Lemma 5.14, write a formula�limK ( �W ) saying: \The domain is a limit ordinal and �W encodes a run of A (oflength the domain) whose limit is K." If K is an !1-limit in A, then by (2)of De�nition 5.18, there is some countable limit � where �limK is satis�ed. Bythe Satis�ability Lemma (4.1), this means that there is also some limit �0 < !!

where �limK is satis�ed. Thus, K is an !!-limit in A, as claimed.For the general case, note that �limK can be taken �rst-order and apply Theorem

7.7. a

Lemma 5.23 (Fiber Lemma). Let A be an initialized automaton with statesQ, K � Q and � a multiple of !!. Then the following are equivalent:

(a) K is �-pumpable in A.(b) There exists an !!-periodic �-run of A whose limit is K.(c) There exists an �-run � of A with bounded �ber and with limit K.

In fact, the �ber of every !!-periodic run is bounded in !!.

Proof. (a) ) (b): Since K is an �-limit in A, there are � 2 Lim \ � anda �-run % of A with lim(%) = K. By the previous lemma, we may assume� < !!. Since K is stable, we can pick a q 2 K \�lim(K) (where �lim is the limittransition function of A). Then there is a 0 2 [�0(%); �) such that %( 0) = q (infact, there are co�naly many such 0 below �). Let �0 := %j 0 , �1 := % � [ 0; �)and � := �0 + �1 �. Since � is a multiple of !! and 0; � � 0 < !!, � = 0 + (� � 0)�. Thus, dom(�) = �, indeed. By de�nition of 0, �1(0) = q. Bychoice of �1, lim(�1) = K, which implies K � rng(�1). By de�nition of �0(%),%[[�0(%); �)] = K, so, in particular, rng(�1) = K. By Lemma 5.19, �1 � is arun of A from state q with limit K. Hence, � is a run of A with limit K.(b) ) (c): Let � be an !!-periodic �-run of A with lim(�) = K. Let �0,

�1 < !! be a phase and period for � respectively, and �0, �1 the correspondingpartial runs. Then K = lim(�) = rng(�1). Therefore, �0(�) � �0. Since alsolim(�0 + �1 !) = K, it follows sup(�b(�)) � �0 + �1! < !!.(c)) (a): Suppose � is an accepting run of A with limitK and �b(�) bounded

in �. Let � := sup(�b(�)). Then � 2 Lim\ � and lim(�j�) = lim(�) = K, whichshows that K is an �-limit in A. Furthermore, since � > �0(�), �(�) 2 K. But,�(�) 2 �lim(lim(�j�)), so �(�) 2 K \ �lim(K). a

Now, if we add to the last result the simple observation that the �ber of a runis de�nable from the run, we obtain the desired result:

Proposition 5.24. There is an algorithm that, given � 2 [!!; !1) and a for-mula '( �X), constructs a formula ( �X;Y ) such that the following are equivalent:

(a) ' is not selectable in (�;<).(b) is a reduction in (�;<) of ��! ub to ',


where ��! ub(Y ) says: \Y is an unbounded !-sequence in the !!-part of �."In particular, for any formula ' not selectable in (�;<), ��! ub is easier to

select in (�;<) than '.

Proof. We handle the case � = !! (equivalently, a countable multiple of !!)and leave the general case to the reader. By Theorem 5.12, we can compute aninitialized deterministic Muller automaton A equivalent to ' over the countableordinals. Let Q = fq0; : : : ; qk�1g be its states. It is easy to write a formula��b(W0; : : : ;Wk�1; Y ) as follows. For every � : !

! ! Q and �Q 2 kP(!!), if �Q\encodes" �, i.e. for all i < k, Qi := ��1(qi), then ��b( �Q;Y ) de�nes �b(�) in(!!; <). Let �run-onA ( �X; �Y ) be as in Lemma 5.14.17 Set

( �X;Y ) := 9 �W (�run-onA ( �X; �W ) ^ ��b( �W;Y ));

Let �P 2 lP(!!). Since A is deterministic, there is a unique (strict) run of A on=��P , and the unique �Q that satis�es �runA ( �P ; �Q) encodes this run. Then ( �P ; Y )de�nes in (�;<) the �ber of this run. Let us show this is as claimed.If ' is selectable in (!!; <), then by Lemma 5.5, there can be no reduction of

the non-selectable �! ub to ' in (!!; <). In particular, is not such a reduction.Assume ' is not selectable in (!!; <). By Proposition 4.2, there exists no

!!-periodic �P 2 D('; !!). Thus, by Lemma 5.17, there exists no accepting !!-periodic !!-run of A. The Fiber Lemma now tells us that there is no accepting!!-run of A with bounded �ber. Since, given any �P 2 D('; !!), ( �P ; Y ) de�nesthe �ber of runA(=!

!

�P), it follows that for any �P 2 D('; !!), ( �P ; Y ) de�nes an

unbounded !-sequence in !!, as was to be shown. a

The next corollary extracts from the proof data that are needed in section 7.

Corollary 5.25. Let '( �X) be a formula not selectable in (!!; <). Thenthere is a �rst-order formula ��( �X; �W;Y ) such that:

(a) (!!; <) j= 8 �X8 �W8Y�('( �X) ^ ��( �X; �W;Y ))! �! ub(Y )

�, and

(b) for every � 2 Lim, (�;<) j= 8 �X9 �W9Y (��(X; �W;Y )).

Proof. Using the notation of the above proof, let ��( �X; �W;Y ) := �run-onA ( �X; �W )^��b( �W;Y ). It is easy to see that ��b is �rst-order and, by Remark 5.15, so is�run-onA . By the proof above, (a) holds, while (b) simply says that A has a runon any input of whatever limit length and that this run has a �ber. a

5.4. Any formula is easier than an unbounded !-sequence. Let againA be an automaton equivalent to a non-selectable '. Denote by F the limit of anaccepting !!-run of A. By the Fiber Lemma, F is not !!-pumpable. We shallprove, however, that there is a K ( F which is !!-pumpable (Proposition 5.29).Suppose q 2 K \ �lim(K). Then, given an unbounded !-sequence S, we will beable to de�ne an !!-run of A with limit F roughly as follows (Proposition 5.31):

Use some run �0 of length < !! to get A into state q. From q there is a run�1 of limit length < !! whose limit is K and which mentions only statesin K. Start pumping �1 (that is, repeat it over and over again). This canbe done since q 2 K \ �lim(K). Whenever you encounter an element of S,stop pumping �1 and use (once) a run �2 of length < !! that mentions all

17Only here we do not care what state the run takes A into.


states of F and only them. Then go back to pumping �1, until you againencounter an element of S.

Having de�ned this run, we will be able to de�ne a tuple �P on which it is arun. Since F is an accepting limit, �P will satisfy '. Below are the details.

Notation 5.26. Let A := hQ;�; �suc; �limi be an automaton and F � Q. Therestriction of A to F is the automaton

AjF := hF;�; �sucjF��; �limjP(F )i:

That is, AjF is the automaton gotten from A by removing all states not in F .

Definition 5.27. Let A := hA; qIi be an initialized Muller automaton withstates Q, limit transition function �lim and acceptance condition F . Let K;F �Q, q 2 Q and � 2 On. Call (q;K; F ) an �-pumping triple of A i�:

1. K is an �-limit in A,2. q 2 �lim(K) \K,18

3. K � F ,4. there exist a successor � < � and a �-run � of hAjF ; qi such that last(�) = q

and rng(�) = F , and5. F 2 F .

Denote the set of �-pumping triples by P�A.

We leave the proof of the following combinatorial fact to the reader.

Lemma 5.28 (Finite Partition Lemma). Let � 2 On, � : !� ! D where D is�nite. Then there is d 2 D with otp(��1(d)) = !�.

Lemma 5.29 (Existence of pumping triples). Let A be an initialized Mullerautomaton. Suppose A has an accepting !!-run. Then the set P!!

A of !!-pumping triples is not empty.

Proof. Let � be an accepting !!-run of A, F := lim(�), k := jF j, and letC0 := [�0(�); !

!), where �0(�) is the least ordinal above which only limit statesof � appear.For any B � !!, let @B denote the set of limit points of B (in !!). Let �lim

be the limit transition function of A. We de�ne a sequence hCi j i � k+1i suchthat for all i � k + 1:

(a) otp(Ci) = !!,

and, if i � k:

(b) Ci+1 � @Ci,(c) for ; 0 2 Ci+1, lim(�j ) = lim(�j 0) and �( ) = �( 0).

C0 was de�ned above. Let i � k and assume Ci has already been de�ned. By(a) of the inductive assumption, otp(Ci) = !!, so otp(@Ci) = !!, too. For 2 @Ci, write �( ) := (lim(�j ); �( )). Clearly, rng(�) � P(Q) � Q is �nite.We may therefore apply the Finite Partition Lemma to obtain a Ci+1 � @Ciwith otp(Ci+1) = !! such that j�[Ci+1]j = 1, which is exactly what is neededin (a)-(c). This completes the construction.For i 2 [1; k + 1], let Ki; qi such that for any 2 Ci, �( ) = (Ki; qi). Then,

18Note that (1) and (2) imply that K is �-pumpable in A.


(d) Ki � F . Indeed, Ci � C0, so for each 2 Ci, > �0(�).(e) qi 2 �lim(Ki). This is immediate from the de�nition of a run.

Also, if i � k:

(f) Ki [ fqig � Ki+1. Indeed, let 2 Ci+1. Then 2 @Ci, so we can pick anincreasing h j j j 2 !i � Ci with supf j j j 2 !g = . For each j 2 !,lim(�j j ) = Ki. Therefore, all states of Ki are unbounded in . On theother hand, �( j) = qi, so also qi 2 lim(�j ) = Ki+1.

Thus, hKi j i 2 [1; k + 1]i is an increasing sequence of subsets of F . SincejF j = k, there is i 2 [1; k] with Ki = Ki+1 (note that K1 6= ?). We claim that(qi;Ki; F ) 2 P!!

A (numbers are as in De�nition 5.27):

1. Every 2 Ci shows that Ki is an !!-limit in A.

2. By (e) and (f), qi 2 �lim(Ki) \Ki+1. But, Ki+1 = Ki.3. is (d).4. Pick any 2 Ci. Since F = lim(�), there exists a � 2 ( ; !!) such that�[[ ; �)] = F . Pick 0 2 Ci with 0 > �. Then � � [ ; 0] is a run as in (4)of the de�nition.

5. Since F = lim(�) and � is an accepting run.

a

Comment. A similar proof would show that P�A 6= ? under the weaker as-

sumption that � is a multiple of !jF j+2 (with F the limit of an accepting run).

The next lemma follows easily from Lemma 5.14.

Lemma 5.30. For all � 2 On n 1 and A an initialized Muller automaton, theset of �-pumping triples of A is determined by MTh(�;<) and computable fromit.

Finally, we have:

Proposition 5.31. There exists an algorithm that, given � 2 [!!; !1) and aformula '( �Y ), builds a formula (X; �Y ) which is a reduction in (�;<) of ' to��! ub.In particular, for any formula ' we have that ' is easier to select in (�;<)

than ��! ub.

Proof. We handle the case � = !! (equivalently, a multiple of !!). ByTheorem 5.12, we can compute an initialized Muller automaton A := hA; qIiwhich is equivalent to ' over the countable ordinals. By Lemma 5.30, we cancompute P!!

A . If P!!

A = ?, then by Lemma 5.29, A has no accepting !!-runs. By choice of A, this means ' is not satis�ed in (!!; <) and we cantake (X; �Y ) := X 6= X (say). Assume this is not the case and pick any(q;K; F ) 2 P!!

A . Use Lemma 5.14 to write formulas as follows:

'0( �Y ) says: \There is run of A on �Y (of length the successor of the domain)that takes A into state q".19

'1( �Y ) says: \The domain is a limit ordinal and there is a (strict) run ofhAjK ; qi on �Y whose limit is K".

19Thus, in the notation of Lemma 5.14, '0 is 9 �W�run-onA!q ( �Y ; �W ).


'2( �Y ) says: \There is a run of hAjF ; qi on �Y (of length the successor of thedomain) that takes it into q and whose range is F".

We claim that each of these formulas is satis�ed in some ordinal below !!.Indeed, by (1) of De�nition 5.27, K is an !!-limit in A, i.e. there are � 2 Lim\!!

and a �-run � of A such that lim(�) = �. Let l := lg( �Y ) and pick �P 2 lP(�)such that � is a (strict) run of A on �P . Since q 2 K and lim(�) = K, thereis a 0 2 (�0(�); �) such that �( 0) = q. Then � � [0; 0] is a run of A on�P �\ 0, so ( 0; <; �P �\ 0) j= '0. Next, �[[ 0; �)] � K, since 0 � �0(�). Thus,� � [ 0; �) is a run of hAjK ; qi on �P �\[ 0; �) with limit K. In other words,([ 0; �); <; �P �\[ 0; �)) j= '1. Similarly, satis�ability of '2 below !! follows from(4) of De�nition 5.27.Fix i 2 3. Use the Satis�ability Lemma to compute an �i 2 !! n 1 in which

'i is satis�ed. By Proposition 2.21, compute a i that selects 'i in (�i; <). Let�Pi be the unique element of D( i; �i).Pick k 2 ! such that !k > �0; �1; �2. We de�ne f : D(�! ub; !!) ! D('; !!)

as follows. Fix an unbounded !-sequence S � !!. De�ne Sk := f� + !k j� 2 Sg. Then Sk is also an unbounded !-sequence in !! and every elementin Sk is a multiple of !k. Note further that Sk is de�nable from S (because,by the De�nability-below-!! Lemma, !k is de�nable). Let f�i j i 2 !g be theorder-preserving enumeration of Sk. Then f(S) is the unique �P 2 lP(!!) whichsatis�es:

(a) (!!; <; f(S))�[0;�0)�= (�0; <; �P0) + (�1; <; �P1)

�0��0�1

,

and, for every i 2 !,

(b) (!!; <; f(S))�[�i;�i+1)�= (�2; <; �P2) + (�1; <; �P1)

�i+1�(�i+�2)�1

.

Since !k > �0; �1; �2 and �0 and �i+1 � �i are multiples of !k, �0 = �0 � �0 is a

multiple of �1 and so is �i+1� �i = �i+1� (�i+�2). Therefore, both (a) and (b)make sense.We must show that (!!; <) j= '(f(S)). To this end we present a run � of A on

f(S) which is accepting. Let %0 be an (�0+1)-run of A on �P0 with last(%0) = q.Write �0 := %0j�0 . Let �1 be an �1-run of hAjK ; qi on �P1 with limit K. Finally,let %2 be an (�2+1)-run of hAjF ; qi on �P2 with last(%2) = q and rng(%2) = F . Set�2 := %2j�2 (%0; �1; %2 exist since, for i 2 3, (�i; <) j= 'i( �Pi)). Let � : !

! ! Qsatisfy:

(a0) �j�0 = �0 + �1 �0��0�1

,

and, for every i 2 !,

(b0) � � [�i; �i+1) = �2 + �1 �i+1�(�i+�2)

�1.

It is easy to see that � is a run of A on f(S). Brie y:

1. � begins in qI , since �0 does.2. By Lemma 5.19, for any � 2 On n 1, �1 � is a run of hAjK ; qi with limit

and range K. In particular, this holds for � = �0��0�1

or � = �i+1�(�i+�2)�1

for some i 2 !.3. Since �0 takes us into q, where �1

�0��0�1

begins, the initial segment �j�0is a �0-run with limit K.


4. Since both �2 and �1�i+1�(�i+�2)

�1begin in q and take us back into q, � is

an !!-run as desired.

Now, clearly lim(�) � rng(�1) [ rng(�2) = K [ F = F . But, for every i 2 !,�[[�i; �i + �2)] = rng(�2) = F . Since Sk = f�i j i 2 !g is unbounded in !!,F � lim(�). Thus, � is accepting.We conclude that, indeed, f takes unbounded !-sequences and returns tuples

satisfying '. If we write a (X; �Y ) that de�nes f in (!!; <), we would be done.This being very much like the proof of \(c)) (a)" in Proposition 4.2, we allowourself to be brief.First, write �M (X;W ) such that for any unbounded !-sequence S, if Sk =

f�i j i 2 !g is as above, then �M (S;W ) de�nes in (!!; <) the set

f�0 + �1� j � <�0 � �0�1

g [ Sk [[i2!

f�i + �2 + �1� j � <�i+1 � (�i + �2)

�1g:

Here, all we use is the de�nability of �0; �1; �2 and the de�nability of Sk fromS.Then, the desired (X; �Y ) says: \for the unique W that satis�es �M (X;W ): 0( �Y ) holds relativized to [0;min(W )), and if x; y are successive elements of

W , then:if x 2W , then 2( �Y ) holds relativized to [x; y), andif x =2W , then 1( �Y ) holds relativized to [x; y)." a

A nice corollary of the last proof is:

Corollary 5.32. Let � 2 !1 n1 and '( �Y ) a formula not selectable in (�;<).Then jD('; �)j = 2@0 .

Proof. We handle the case � = !!. Let A be an initialized deterministicMuller automaton equivalent to ' over the countable ordinals.20 Since ' is notselectable in (!!; <), ' is satis�ed in (!!; <). Thus, A has an accepting !!-run,so there are !!-pumping triples of A. Let (q;K; F ) be one. Let �1; �2, �P1,�P2, k 2 ! and f : D(�! ub; !!) ! D('; !!) be as in the proof of the previousproposition. Let Seqk denote the class of all unbounded !-sequences in !

! all ofwhose members are multiples of !k. Clearly, j Seqk j = 2@0 . We show that fjSeqkis 1-1, which will �nish the proof.Let S; S0 2 Seqk, S * S0. Let Sk; S

0k be obtained from S; S0 as in the previous

proof. Pick � 2 SnS0 and write � := �+!k. Then � 2 SknS0k (use the fact that allelements of S; S0 are multiples of !k). Let �, �0 be the unique runs of A on f(S),f(S0), respectively. For i 2 f1; 2g, let �i be the unique run of hA; qi on =�i�Pi . By

de�nition of f, �[[�; �+�2)] = rng(�2) = F . In particular, �[[�; �+!k)] = F (recall

!k > �2). But, if �0 is the least member of S0 above �, then �0 � [�; � 0) = �1

�0��1

.

Since rng(�1) = K, �[[�; � 0)] = K. Since � 0, too, is a multiple of !k, we have,in particular, �[[�; � + !k)] = K. Since ' is not selectable, K 6= F (or else Fis an accepting set which is !!-pumpable in A). Since A is deterministic and�(�) = �0(�) = q, this must mean f(S) \ [�; � + !k) 6= f(S0) \ [�; � + !k), i.e.f(S) 6= f(S0), as was to be shown. a

20Recall that in the previous proof we did not need the assumption A is deterministic.


Note that Proposition 5.31 leaves open an interesting question. It tells us thatwith an unbounded !-sequence S � !! as a parameter, we can select all formulasin (!!; <). But, the formulas we select using S do not themselves \mention" S.In other words, the proposition does not tell us that (!!; <; S) has the selectionproperty. In fact, let '(X;Y ) says: \If x < x0 are successive elements of X, thenY \ [x; x0) is an !-sequence unbounded in [x; x0)" and S := f!k j k 2 !g. Thenit is easy to show ' has no selector in (!!; <; S).21 It is therefore natural to ask:

Can (!!; <) be expanded by �nitely many subsets of !! to have the selec-tion property?

In [RS] we provide an a�rmative answer to this question:

Proposition 5.33 ([RS]). There is an unbounded !-sequence S � !! suchthat:

(a) the monadic theory of (!!; <; S) is decidable,(b) (!!; <; S) has the selection property, and(c) for any formula '(X; �Y ), a selector for ' in (!!; <; S) is computable.

An essential ingredient of the proof given there is Corollary 6.1 below.

x6. Every formula is selectable in some countable ordinal. This shortsection is devoted to the proof of Proposition 3.19. On our way there, we statea simple combinatorial corollary of (the proof of) Proposition 5.31, which seemsof independent interest.

Corollary 6.1. Let n; l 2 ! and ' 2 Formn;l which is satis�ed in (!!; <).Then we can compute �sel; �suf 2 Hn;l such that:

(a) �sel is satis�able and selectable in (!!; <),(b) �suf is satis�able in (!!; <), and(c) �sel + �suf j= '.

Furthermore, �sel and �suf can be computed from '.

Proof. Fix � 2 Hn;l. By Theorem 2.15, there is c = c(n; l) 2 ! such that forevery chain I, � I depends only on the c-type of I. Let p := p(c). Then for�; �0 2 !1 n 1, � !p� = � !p�0. In particular, � !p = � !p+1, whichmeans � !p is an !-idempotent (De�nition 4.6).Now, go back to the proof of 5.31 (we use the notation used there). From what

was just said, it follows typen((�1; <; �P )) !c is an !-idempotent. Of course,typen((�1; <; �P ))!c = typen((�1; <; �P1)!c). But, the l-chain (�1; <; �P1)!c

satis�es every requirement placed on (�1; <; �P1) in 5.31: its domain �1!c is

smaller than !! and �1 !c is a run of A on this l-chain beginning in stateq, mentioning only states in K and with limit K (use Lemma 5.19). We couldtherefore use (�1; <; �P1) !c in our construction of f in 5.31. In other words,we may assume �1 := typen(�1; <; �P1) is an !-idempotent.

21Recall that in our semantics some of the free variables play the role of monadic predicatesymbols. When interpreting '(X;Y ) in (!! ; <; S), we intend for X to be interpreted as S,while Y serves as a free variable in the usual sense. So, in selecting '(X;Y ) we need to de�nea subset of !! and not a pair of subsets (cf. De�nition 2.2).


Let �0 := typen(�0; <; �P0) and �sel := �0 + �1. Proposition 4.8 tells us�sel is selectable in (!!; <). Pick any unbounded !-sequence S � !! and letf(S) 2 D('; !!) be constructed from it as in 5.31. Then �suf := typen((!!; <; f(S))�[�0+�1;!!)) is certainly satis�ed in (!

!; <). However, typen(!!; <; f(S)) =typen((!!; <; f(S))�[0;�0+�1))+type

n((!!; <; f(S))�[�0+�1);!! ) = �sel+�suf . Sincef(S) satis�es ' in (!!; <), it follows �sel + �suf j= ', as desired.We leave it to the reader to show the computability of �sel; �suf . a

The following is Proposition 3.19.

Proposition 6.2. Let n; l 2 ! and ' 2 Formn;l. Then for every � 2 [!p(n+l); !!),' is selectable in (!! + �;<).

Proof. Write p := p(n+ l). Pick � 2 [!! + !p; !! � 2). We must show ' hasa selector ' in (�;<) and may assume ' is satis�ed in (�;<). Denote by �p the!p-part of �. We treat the case �p 6= �, and leave the (simpler) case �p = � tothe reader.There must exist �p; �

0 2 Hn;l with �p satis�ed in �p and �0 in ��p such that

�p + � 0 j= '. Since �p is a multiple of !p, it is (n+ l)-equivalent to !!, so �p issatis�ed in (!!; <). Pick �sel; �suf as in the previous corollary, setting ' := �p.Then �sel is selectable in (!!; <), say by a selector sel. Note that �p � !! isa non-0 multiple of !p (use that � � !! + !p). Thus, �suf is also satis�ed in(�p � !!; <). Since �p � !! < !!, we can compute a selector suf for �suf in(�p � !!; <). Similarly, a selector 0 for � 0 in (� � �p; <) exists. Then the which says \sel;suf and 0 hold relativized to [0; !!), [!!; �p) and [�p; �),respectively" selects ' in (�;<). a

x7. Selection between �rst-order and second-order logic. In this sec-tion we often write \FO" instead of \�rst-order". Recall that in our de�nition ofa FO-formula �( �Y ) quanti�ers range only over elements of the domain, but thefree variables are allowed to range over subsets.22 Let us call the set of �rst-orderMLO formulas the �rst-order fragment of MLO. We show that all results con-cerning the selection property and selection problem in countable ordinals carrythrough from MLO to its FO-fragment.23 On the other hand, (!1; <) has the�rst-order selection property, i.e. every FO formula is selectable in (!1; <) (by aFO formula). To prove the latter statement, we show: (1) every FO formula se-lectable in (!!; <) is also selectable in (!1; <); (2) any FO formula not selectablein (!!; <) is not even satis�ed in (!1; <) (so, trivially, it is selectable). In (2)we use the fact that �! ub represents the least non-selectable degree in (!!; <).First, we need to slightly re�ne the Satis�ability Lemma.

Definition 7.1 (L-chain). Let L0 be the set of one element labeled chains.For i 2 !, let Li+1 be de�ned as follows: a labeled chain C is in Li+1 i� it iseither in Li or there are C0; C1 2 Li such that C = C0 + C1 or C = C1 !.

22This is signi�cant. For instance, there is a �rst-order �(Y ) such that the only subset of! satisfying � in (!;<) is the set of even numbers. On the other hand, there is no �rst-orderformula �(y), with y an individual variable, such that for any n 2 !, (!;<) j= �(n) i� n iseven.

23It is easy to do the same for the results of section 5.


A labeled chain C is an L-chain i� C 2 Li for some i 2 !. The least such i isthen called the rank of C.

Lemma 7.2. Any formula '( �Y ) satis�able in a countable ordinal is satis�edby an L-chain. When this is the case, we may compute a FO formula � de�ningan L-chain satisfying '.

Proof. [Sh75] proves the �rst claim. Shelah further shows that we can com-pute a bound r on the rank of an L-chain satisfying '. Now, every L-chain canbe described by a �nite \list of instructions": we need to know which chainswith domain 1 to begin with, and how to apply addition and multiplication by!. Let n := qd('), l := lg( �Y ). Given any instruction list, we may computethe n-type of the chain constructed from the list. Indeed, we �rst compute then-types of the chains of length 1 with which the construction begins; we thenuse the Addition and Multiplication Lemmas to compute n-types for the chainsobtained at the various stages of the construction, up until the �nal one. Finally,it is clear that there are only �nitely many possible instruction lists for l-chainsof rank at most r. Thus, we may search through these instruction lists, until we�nd one yielding an L-chain whose n-type implies '.To complete the proof the reader will show (by an easy induction on rank)

that, given any instruction list, we may compute a FO formula de�ning thecorresponding L-chain. a

Lemma 7.3. If � 2 [1; !1] and '( �Y ) is an MLO formula selectable in (�;<),then there is a �rst-order �( �Y ) that selects ' in (�;<). Furthermore, � iscomputable from � and '.

Proof. If ' is not satis�ed in (�;<) take � := Y 6= Y (say). AssumeD('; �) 6= ?. If � 2 !! n 1, let �� be a FO-sentence de�ning � in the classof ordinals. Thus, if � is an ordinal, then (�;<) j= �� i� � = �. Then take � tobe a FO-formula de�ning an L-chain satisfying ��^', as in the previous lemma.For � = !! or � = !1, we again look at the proof of Proposition 4.2. We use

the notation used there. By what was just said, we can take the 0, 1 whichselect �0; �1 in �0; �1 to be FO formulas. This leaves one place where second-order quanti�cation is used in (�0;�1). Recall that we had a formula �M (W )de�ning the set M := f�0 + �1� j � < �g. Then (�0;�1) said: \For every (thatis, the unique) W which satis�es �M (W ): : : ". This, of course, is a second-orderquanti�er \8W (�M (W )! : : : )".24

Now, assume we could take �1 = !k > �0 for some k 2 !. Then M wouldconsist of �0 and all multiples of !k below �. But, there is a FO formula �k(x)that is satis�ed (in any ordinal) only by multiples of !k: �1(x) says \x is a limitordinal", �2(x) says \x is the limit of limit ordinals", etc. By the De�niability-below-!! Lemma, there also exists a FO formula ��0(x), which in every ordinal> �0 is satis�ed only by �0. Using ��0(x) _ �k(x) instead of �M , we eliminatethe set quanti�er.So, it remains to be seen we can have �1 = !k. By Proposition 4.8, we may

assume �1 ! = �1. Pick any k such that !k > �0; �1. Then !k is a multiple of

�1. By Lemma 4.7, �1 = �1 !k

�1is also satis�ed in (!k; <), and we are done.

24The fact that �M itself is FO does not help us in eliminating this quanti�er.


Finally, for � = !! + � where � 2 !! n 1, we proceed as in the proof ofProposition 4.4, using the fact that there are FO formulas de�ning [0; !!) and[!!; �) in (�;<). a

Though the following corollary is phrased somewhat vaguely, we hope ourintention is clear.

Corollary 7.4. Let L be a logic such that:

1. For every �rst-order �, we can compute a formula � of L equivalent to it.2. For every formula � of L, we can compute an MLO formula ' equivalent

to it.

Then:

(a) If � 2 !1 n 1, then (�;<) has the selection property for L-formulas i�� < !!, and

(b) the selection problem for L-formulas in any � 2 [1; !1] is solvable.

Proof. We do (b) and leave (a) to the reader. Let � be an L-formula. Com-pute an MLO ' as in (2). Decide whether ' has a selector in (�;<). If no, weare done. If yes, use the previous lemma, to compute a FO selector for ' in(�;<), and using (1), translate it back into L. a

There are interesting logics as in the corollary. A famous example is weak MLO,where quanti�ers range over �nite subsets of the domain.We now show that (a) of the corollary cannot be extended to the case (!1; <)

even when L = FO. First, we must know that, if in the presentation of thecomposition method in subsection 2.3, we replace everywhere the word \formula"with the words \FO formula", then all results remain valid. Thus, FOFormn;lis the set of FO formulas in Formn;l; inside which we �nd a �nite set FOHn;l

of FO Hintikka formulas, exactly one of which is satis�ed in every l-structureM; we call it the FO n-type of M. The taking of ordered sums preserves FOn-equivalence, so that the sum of FO n-types can be de�ned. Finally, we haveFO analogues of Theorem 2.15 and the Multiplication Theorem. So far, nothingnew. What is new is:

Theorem 7.5 ([GR02]). Let k 2 !, I any chain. Then !k and !k + !k Icannot be distinguished by any FO sentence of quanti�er-depth k. In particular,any two multiples of !! have the same �rst-order theory.

Now, let n; l 2 ! and � 2 Hn;l. If � is satis�able in some structure, then thereis a unique � 2 FOHn;l with � j= �. We call � the FO restriction of � .

Proposition 7.6. If a FO formula is selectable in (!!; <), then it has a FOselector over the class of all multiples of !!.

Proof. Let n; l 2 ! and let � 2 FOFormn;l be selectable in (!!; <). Let�0; �1 2 Hn;l both satis�able below !! such that �0+ �1!! j= �. Let �0, �1 bethe FO restrictions of �0, �1, respectively. Then �0 +FO (�1 FO !

!) j= �. Let� 2 On n 1. By the last theorem, !! and !!� have the same FO-theory. Sincemultiplication of FO-types is recursive in this theory, �0 +FO (�1 FO !

!�) =�0 +FO (�1 FO !

!). By Lemma 4.5, there is formula = (�0;�1) which selects�0 +FO (�1 FO !!) in every multiple of !!. By the proof of Lemma 7.3, weknow it can be taken FO. a


Next, we need to know that for FO formulas, the Satis�ability Lemma can beextended to all ordinals. This is due to L�auchli and Leonard ([LL66], [Ro82]).

Theorem 7.7. Let �( �Y ) be a FO formula satis�ed in some ordinal. Thenthere is � < !! in which � is satis�ed.

Corollary 7.8. Let � be a multiple of !! and �( �Y ) a FO formula which issatis�ed in (�;<). Then � is also satis�ed in (!!; <).

Proof. Suppose (!!; <) j= 8 �Y :�( �Y ). Let n be the quanti�er-depth of thissentence and p := p(n) as in the p-Lemma. Then the MLO sentence 8 �Y :� alsoholds in every countable multiple of !p. Now, there is a FO sentence �p which issatis�ed by an ordinal � i� � is a multiple of !p. Then �( �Y )^ �p is not satis�edin any countable ordinal, so by the previous theorem, it is not satis�able in anyordinal whatsoever. This means that �( �Y ) is not satis�able in any multiple of!p, in particular, in any multiple of !!. a

Lemma 7.9. Let �( �X) be a �rst-order formula not selectable in (!!; <). Then� is not satis�ed in (!1; <).

Proof. Since �( �X) is not selectable in (!!; <), there is a FO formula ��( �X; �W;Y )as in Corollary 5.25. Recall that

1. (!!; <) j= 8 �X8 �W8Y��( �X) ^ ��( �X; �W;Y )! �! ub(Y )

�.

Since �! ub and �( �Y ) are also FO, the formula in brackets here is FO. By theprevious Corollary,

2. (!1; <) j= 8 �X8 �W8Y��( �X) ^ ��( �X; �W;Y )! �! ub(Y )

�.

Since there is no unbounded !-sequence in !1, we must have

3. (!1; <) j= 8 �X8 �W8Y (:�( �X) _ :��X; �W;Y )

�.

But by 5.25,

4. (!1; <) j= 8 �X9 �W9Y ��( �X; �W;Y ).

Therefore, we must have (!1; <) j= 8 �X:�( �X), as desired. a

Summing up, we have

Corollary 7.10. (!1; <) has the FO order selection property, but not theMLO selection property.

x8. The Church synthesis problem. What is known as the \Church syn-thesis problem" was �rst posed by A. Church in [Ch63] for the case of (!;<).Church uses the language of automata theory. It was McNaughton (see [Mc66])who �rst observed that it can be equivalently phrased in game-theoretic lan-guage.

Definition 8.1. For an ordinal � and a formula '(X;Y ), the McNaughtongame G�' is a game of perfect information of length � between two players, Xand Y .At stage � < �, X either accepts or rejects �; then, Y also decides whether

to accept or to reject �.For a play �, we denote by X� (resp. Y�) the set of ordinals < � accepted by

X (resp. Y ) during the play. Then,


Y wins � i� (�;<) j= '(X�; Y�).

What we want to know is: Does either one of X and Y have a winning strategyin G�'? If so, which of them? That is, can X choose his moves so that, whateverway Y responds we have :'(P�; Q�)? Or can Y respond to X's moves in a waythat ensures the opposite?Since at stage � < �, X has access only to Q� \ [0; �) and Y has access only

to P� \ [0; �], it seems that the following formalizes the notion of a strategy inthis game well:

Definition 8.2 (Causal operator). Let � be an ordinal, f : P(�) ! P(�).We call f causal (resp. strongly causal) i� for all P; P 0 � � and � < �, if

P \ [0; �] = P 0 \ [0; �] (resp. P \ [0; �) = P 0 \ [0; �)),

then

f(P ) \ [0; �] = f(P 0) \ [0; �].

That is, if P and P 0 agree up to and including (resp. up to) �, then so do f(P )and f(P 0).

So a winning strategy for Y is a causal f : P(�) ! P(�) such that for everyP � �, (�;<) j= '(P; f(P )). It is easy to come up with examples of formulaswhere Y has no winning strategy. For example, if 'X(X;Y ) says \if X = ?,then Y = All; otherwise, Y = ?" and � � 2, then X has a winning strategy inG�'X . This leads to

Definition 8.3 (Game version of the Church synthesis problem). Let � be anordinal. Given a formula '(X;Y ), decide whether Y has a winning strategy inG�' .

Note the similarities and dissimilarities between this problem and the uni-formization problem. In uniformization we are also given a formula '(X;Y ) andto every P we try and \respond" with a Q such that '(P;Q) holds. Only we donot restrict ourselves to causal responses. Thus, the formula 'X described aboveis its own uniformizer in every �. On the other hand, we do restrict ourselves tode�nable (in (�;<)) responses. In the de�nition just given, we did not requirethat the strategy (=causal operator) be de�nable.In [BL69], B�uchi and Landweber prove the decidability of the Church synthesis

problem in (!;<). What is even more important, they show that in the case of(!;<) we can restrict ourselves to de�nable causal (or strongly causal) operators.

Theorem 8.4 (B�uchi-Landweber, 1969). Let '(X;Y ) be a formula. Then:

� Determinacy: One of the players has a winning strategy in the game G!' .� Decidability: It is decidable which of the players has a winning strategy.� De�nable strategy: The player who has a winning strategy, also has a de-�nable winning strategy.

� Synthesis algorithm: We can compute a formula (X;Y ) that de�nes (in(!;<)) a winning strategy for the winning player in G!' .

25

25If is to de�ne a strategy for X, then we should take Y as the domain variable, so tospeak.


As mentioned in the introduction, it seems that B�uchi and Landweber be-lieved their theorem would generalize to all countable ordinals. However, if welet '(X;Y ) say \Y is an unbounded !-sequence" (so, we ignore X), then byProposition 3.17, Y cannot have a de�nable strategy in G!

!

' . Indeed, if (X;Y )de�ned such a strategy, then 9X(X = ? ^ (X;Y )) (say) would select an un-bounded !-sequence in !!. On the other hand, Y does win this game: shesimply plays a �xed unbounded !-sequence S � !!, ignoring X's moves.What we know at present concerning the extension of B�uchi and Landweber's

result to countable ordinals is summarized in the following theorem.

Theorem 8.5. Let � be a countable ordinal, '(X;Y ) a formula.

� Determinacy: One of the players has a winning strategy in the game G�' .� Decidability: It is decidable which of the players has a winning strategy.� De�nable strategy: If � < !!, then the player who has a winning strategy,also has a de�nable (in (�;<)) winning strategy. For every � � !!, thereis a formula for which this fails.

� Synthesis algorithm: If � < !!, we can compute a formula (X;Y ) thatde�nes a winning strategy for the winning player in G�' .

A proof of this theorem can be found in [Ra]. It uses the composition methodto reduce games of every countable length to games of length !.Note that Theorem 8.5 leaves an interesting question open, namely: can we

decide those McNaughton games for which there exists a de�nable winning strat-egy? It seems reasonable to expect this is the case.

Conjecture 8.6. There is an algorithm that, given � 2 [!!; !1) and a for-mula '(X;Y ), decides whether there is a de�nable winning strategy in G�' , andif so, returns a de�ning one.

Rabinovich ([Ra]) shows that if the conjecture holds for � = !!, then it istrue.Next, we have seen that the only stumbling block for selection in (!!; <)

was selecting an unbounded !-sequence (recall Proposition 5.31). We believe ananalogous statement may be true concerning de�nability of a winning strategyfor games of length !!:

Conjecture 8.7. For every formula '(X;Y ), there is a formula (W;X; Y )such that for every unbounded !-sequence S � !!, (S;X; Y ) de�nes in (!!; <)a winning strategy for the winner of G!

!

' . Moreover, can be computed from '.

Finally, we mention that for uncountable ordinals the situation changes radi-cally. Let 'spl(X;Y ) say: \X is stationary, Y � X and both Y and X n Y arestationary" (recall De�nition 3.11). Then it follows immediately from [LaS] thateach of the following statements is consistent with ZFC:

1. None of the players has a winning strategy in G!1'spl .2. Y has a winning strategy in G!1'spl .3. X has a winning strategy in G!1'spl .

In other words, ZFC can hardly tell us anything concerning this game. On theother hand, S. Shelah (private communication) tells us he believes it should bepossible to prove:


Conjecture 8.8. It is consistent with ZFC that G!1' is determined for everyformula '.

Acknowledgments The second author would like to thank Chen Meiri formany useful discussions on the contents of this paper, particularly concerningsection 5.

REFERENCES

[BL69]J. R. B�uchi, L. H. Landweber, Solving sequential conditions by �nite-state strategies,Trans. Amer. Math. Soc.,Vol. 138 (Apr. 1969), pp. 295-311.

[BS73]J. R. B�uchi, D. Siefkes, The Monadic Second-Order Theory of all Countable Ordinals,Springer Lecture Notes 328, 1973, pp. 1-126.

[Ch63]A. Church, Logic, arithmetic and automata, Proc. Intrnat. Cong. Math. 1963,Almquist and Wilksells, Uppsala, 1963.

[FV59]S. Feferman, R.L. Vaught, The �rst-order properties of products of algebraic systems,Fundamenta Mathematicae 47 (1959), pp. 57-103.

[GR02]Y. Gurevich, A. Rabinovich, De�nability in the rationals with the real order in thebackground, Jou. of Logic and Computation, 12:1 (2002), pp. 1-11.

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[LaS]P. B. Larson, S. Shelah, The stationary set splitting game, to appear in MathematicalLogic Quarterly. Also available on: http://www.users.muohio.edu/larsonpb/preprints.html.

[LiS98]S. Lifsches, S. Shelah, Uniformization and skolem functions in the class of trees, Jou.of Symbolic Logic, Vol. 63(1) (Mar. 1998), pp. 103-127.

[Mc66]R. McNaughton, Testing and generating in�nite sequences by a �nite automaton,Information and Control 9 (1966), pp. 521-530.

[Ne]I. Neeman, Finite state automata and monadic de�nability of ordinals, preprint. Avail-able on: http://www.math.ucla.edu/�ineeman

[Ra]A. Rabinovich, The Church synthesis problem over countable ordinals, submitted.[Rab69]M. O. Rabin, Decidability of second-order theories and automata on in�nite trees,

Trans. Amer. Math. Soc. 141 (1969), pp. 1-35.[Ro82]J. G. Rosenstein, Linear orderings, New York: Academic Press, 1982.[RS]A. Rabinovich, A. Shomrat, Selection over classes of countable ordinals expanded by

monadic predicates, submitted.[Sh75]S. Shelah, The monadic theory of order, Annals of Mathematics, Ser. 2, Vol. 102

(1975), pp. 379-419.[Th97]W. Thomas, Ehrenfeucht games, the composition method, and the monadic theory

of ordinal words, in: J. Mycielski et al. (eds.), Structures in Logic and Computer Science, A

Selection of Essays in Honor of A. Ehrenfeucht, volume 1261 of Lecture Notes in ComputerScience, Springer, 1997, pp. 118-143.

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Selection in the monadic theory of a countable ordinal