A Generalization of the {\L}o\'s-Tarski Preservation Theorem over Classes of Finite Structures

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by  Abhisekh Sankaran

A Generalization of the L o´s-Tarski Preservation
Theorem over Classes of Finite Structures
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Abhisekh Sankaran, Bharat Adsul, Supratik Chakraborty
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Department of Computer Science and Engineering,

n IITBombay, Mumbai, India
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J
January 24, 2014

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] Abstract
O
WeinvestigateageneralizationoftheL o´s-Tarskipreservationtheorem
L viathesemanticnotionofpreservationundersubstructuresmodulok-sized
.
s cores. It was shown earlier that over arbitrary structures, this semantic
c notion for first-order logic corresponds to definability by ∃k∀∗ sentences.
[
Inthispaper,weidentifytwopropertiesofclassesoffinitestructuresthat

1 ensuretheabovecorrespondence. Thefirstisbasedonwell-quasi-ordering
v undertheembedding relation. Thesecond is a logic-based combinatorial
3 property that strictly generalizes the first. We show that starting with
5 classessatisfyinganyoftheseproperties,theclassesobtainedbyapplying
9 operationslikedisjointunion,cartesianandtensorproducts,orbyforming
5 words and trees over the classes, inherit the same property. As a fallout,
.
1 we obtain interesting classes of structures over which an effective version
0 of theL o´s-Tarski theorem holds.
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1 Keywords: finite modeltheory, preservationtheorem,well-quasi-ordering,com-
: position method
v
i
X
1 Introduction
r
a
Preservationtheorems in first-order logic (henceforth called FO) have been ex-
tensively studied in model theory [3]. A FO preservation theorem asserts that
the collection of FO definable classes closed under a model-theoretic operation
correspondstothe collectionofclassesdefinable bya syntacticfragmentofFO.
A classical preservation theorem is the L o´s-Tarski theorem, which states that
overarbitrary structures,aFOsentenceispreservedundersubstructuresiffitis
equivalent to a universal sentence [3]. In [11], it was conjectured that the L o´s-
Tarskitheoremcanbegeneralizedusingasimpleyetdelicatesemanticnotionof
a class of structures being preserved under substructures modulo k-sized cores.
This semantic notion, denoted PSC(k) and explained in detail in Section 2, is
parameterized by a quantitative model-theoretic parameter k. For k = 0, this
reducestotheusualnotionofpreservationundersubstructures. Theconjecture
in [11] was settled in [10], where it was shown that over arbitrary structures, a
FOsentenceispreservedundersubstructuresmodulok-sizedcoresiffitisequiv-
alent to a ∃k∀∗ sentence. This result, which we abbreviate as PSC(k)=∃k∀∗,
provides a non-trivial generalizationof the L o´s-Tarskitheorem.
Since classes of finite structures are the most interesting from a computational
point of view, researchers have studied preservation theorems over finite struc-
tures in the past [1, 5, 2, 9]. Most preservation theorems, including the L o´s-
Tarski theorem, fail over the class of all finite structures1. Recent works [1, 5]
havestudied structuraland algorithmicproperties of classesof finite structures
that allow the L o´s-Tarski theorem to hold over these classes. Unfortunately,
these studies don’t suffice to identify classes over which PSC(k)=∃k∀∗ holds.
In this paper, we try to fill this gap by formulating and studying new abstract
propertiesofclassesoffinite structures. As a falloutofourstudies, wenotonly
identify interesting classes over which PSC(k) = ∃k∀∗ holds for all k ≥ 0, but
also identify classes that lie beyond those studied by [1, 5] and yet satisfy the
L o´s-Tarskitheorem.
The remainder of the paper is organized as follows. In Section 2, we discuss
preliminariesandsetupthenotation. Section3introducesawell-quasi-ordering
based property and a logic-based combinatorial property of classes of finite
structures, where both properties are parameterized by a natural number k.
We show that PSC(k) = ∃k∀∗ holds over classes satisfying these properties.
We also formulate an ‘effective’ version of the logic-based property that allows
us to compute from a sentence defining a class in PSC(k), an equivalent ∃k∀∗
sentence. InSection4,weundertakeanexhaustivecomparisonofthecollections
of classesthat the aforementionedproperties define. We show in Section 5 that
the classically interesting and well-studied classes of words and trees over a
finite alphabet belong to these collections. Finally, in Section 6, we establish
composition theorems for the above collections of classes under a set of natural
compositionoperatorssuchasdisjointunion,cartesianandtensorproducts. We
further show that the above collections are also closed under the operations of
forming words and trees. The results in Sections 5 and 6 are amongst the most
technically involved results in this paper. Throughout, we provide examples of
interesting classes satisfying the various properties discussed.
2 Notation and Preliminaries
LetNdenote the naturalnumbersincluding zero. We assumethatthe readeris
familiarwithstandardnotationandterminologyoffirst-orderlogic. Weconsider
only finite vocabularies, represented by τ, that are relational (i.e. contain only
predicate and constant symbols). Standard notions of τ-structures, substruc-
tures and extensions (see [3]) are used throughout. All τ-structures considered
in this paper are assumed to be finite. Given a τ-structure A, we use UA to
denote the universe of A and |A| to denote its cardinality or size. If A is a
1Anotableexception isthehomomorphismpreservationtheorem[9].
2
subsetofUA,weuseA(A)todenotethesubstructureofAinducedbyA. Given
τ-structures A and B, we use A ⊆ B to denote that A is a substructure of B.
If A and B are sets, we also use A ⊆ B to denote set containment. We say
that A embeds in Bif A is isomorphicto a substructure of B. Notationally,we
represent this as A֒→B. It is easy to see that ֒→ is a pre-order over any class
ofτ-structures. All classes of τ-structures, and subclasses thereof, considered in
this paper are assumed to be closed under isomorphism.
We denote byFO(τ) the setofallFOformulaeoverτ. Asequence(x ,...,x )
1 k
of variables is written as x¯. For notational convenience, we abbreviate a block
of quantifiers of the form Qx ...Qx by Qkx¯, where Q ∈ {∀,∃}. Given a τ-
1 k
structure A and a FO(τ) sentence ϕ, if A |= ϕ, we say that A is a model of
ϕ. Given a class S of τ-structures of interest, every FO(τ) sentence ϕ defines
a unique subclass of S consisting of all models of ϕ. Therefore, when S is clear
from the context, we interchangeably talk of a set of FO(τ) sentences and the
corresponding collection of subclasses of S.
Thenotionofaclassofτ-structuresbeingpreservedundersubstructuresmodulo
bounded cores was introduced in [11]. This notion is central to our work. The
following is an adapted version of the definition given in [11].
Definition 2.1 Let S be a class of τ-structures and k ∈ N. A subclass C of S
is said to be preservedundersubstructuresmodulok-sizedcoresoverS if every
τ-structure A∈C has a subset Core of UA such that (i) |Core|≤k, and (ii) for
every B ∈ S, if B ⊆ A and Core ⊆ UB, then B ∈ C. The set Core is called a
k-core of A with respect to C over S.
As an example, if S represents the class of all graphs, the subclass C of acyclic
graphs is preserved under substructures module k-sized cores over S, for every
k ≥ 0. Like Definition 2.1, most other definitions, discussions and results in
this paper are stated with respect to an underlying class S of structures. For
notational convenience, when S is clear from the context, we omit mentioning
“overS”. In Definition 2.1, if the subclass C of S and k ∈N are alsoclear from
the context, we call Core simply as a core of A.
Given a class S, let PSC(k) denote the collection of all subclasses of S that are
preserved under substructures modulo k-sized cores. As shown by the example
of acyclic graphs above, PSC(k) may contain subclasses that are not definable
overS by any FO sentence. Since our focus in this paper is onclassesdefinable
by FO sentences, we define PSC(k) to be the collection of classes in PSC(k)
that are definable over S by FO sentences. As before, we interchangeably talk
of PSC(k) as a collection of classes and as a set of the defining FO sentences.
Since PSC(0) coincides with the property of preservation under substructures,
we abbreviate PSC(0) as PS and PSC(0) as PS in the remainder of the paper.
Givenk,p∈N, let∃k∀p denote the setofallFO(τ) sentences inprenexnormal
formwhosequantifierprefixhask existentialquantifiersfollowedbypuniversal
quantifiers. We use ∃k∀∗ to denote Sp∈N∃k∀p. As before, when the class S of
τ-structures is clear from the context, we use ∃k∀p and ∃k∀∗ to also denote the
corresponding subclasses of S. We refer the reader to [11] for interesting ex-
3
amples from the collections PS,PS,∀∗,PSC(k),PSC(k),∃k∀∗ and for inclusion
relationships among these collections.
Using the above notation, the L o´s-Tarskitheorem can be stated as follows.
Theorem 2.2 Over arbitrary structures, PS =∀∗.
In [10], this was generalized to give the following result.
Theorem 2.3 Over arbitrary structures, for every k ∈N, PSC(k)=∃k∀∗.
It is easy to see that if ϕ is an ∃k∀∗ sentence and A|=ϕ, then every witness of
the existential variables of ϕ forms a k-core of A. However, the converse is not
necessarily true [11]. Specifically, let τ = {E}, where E is a binary predicate.
ConsidertheFO(τ)sentenceϕ ≡ ∃x∀y E(x,y),andtheτ-structureAdefined
byUA ={0,1}andEA ={(0,0),(0,1),(1,1)}. Clearly,A|=ϕandthereisonly
one witness of the existential quantifier, viz. 0. However, both {0} and {1} are
cores of A!
In [11], the notion of relativizing a FO sentence with respect to a finite set of
variables was introduced. We recall this for later use. Let Const be the set of
constantsinarelationalvocabularyτ. Givenasentenceφoverτ andasequence
of variables x¯, let φ| denote the quantifier-free formula with free variables x¯,
x¯
obtainedasfollows. SupposeX istheunderlyingsetofx¯. Wefirstreplaceevery
∀ in φ by ¬∃, and then replace everysubformula of the form ∃xψ(x,y ,...,y )
1 k
by Wz∈X∪Constψ(z,y1,...,yk). The formula φ|x¯ is called φ relativized to X.
Informally, given a τ-structure A, the formula φ| asserts that φ is true in the
x¯
substructure of A induced by the underlying set of x¯. More precisely, for every
(a ,...a )∈Uk, we have (A,a ,...,a )|=φ| iff A({a ,...,a })|=φ.
1 k A 1 k x¯ 1 k
As mentioned in Section 1, recent studies have identified structural and algo-
rithmicpropertiesofclassesoffinitestructuresthatallowPS =∀∗ toholdover
theseclasses[1,5]. Forexample,theclassofstructureswhoseGaifmangraphis
acyclic wasshownto admitPS =∀∗ in [1]. LetS be the classof graphs,where
each graph is a disjoint union of finite undirected paths. Clearly, every such
graph has an acyclic Gaifman graph, and hence PS = ∀∗ over S. However, as
shown in [11], PSC(k) 6= ∃k∀∗, for every k ≥ 2, over S. This motivates us to
ask: Can we identify abstract properties of classes of finite structures that allow
PSC(k) = ∃k∀∗ to hold over these classes? Our primary contribution is the
identification of two properties that answer the above question affirmatively.
3 Two Properties of Classes of Structures
We define two properties of classes of finite structures, each of which entails
PSC(k)=∃k∀∗ over the class.
3.1 A property based on well-quasi-orders
Recall that a pre-order(Π,(cid:22)) is well-quasi-ordered (w.q.o.) if for every infinite
sequenceπ ,π ,...ofelementsofΠ,thereexistsi<jsuchthatπ (cid:22)π (see[4]).
1 2 i j
4
If(Π,(cid:22)) is a w.q.o.,we saythat “Πis aw.q.o. under (cid:22)”. It is a basicfact that
if Π is a w.q.o. under (cid:22), then for every infinite sequence π ,π ,... of elements
1 2
of Π there exists aninfinite subsequence π ,π ,... suchthat i <i <... and
i1 i2 1 2
π (cid:22)π (cid:22)....
i1 i2
Given a vocabulary τ and k ∈ N, let τ denote the vocabulary obtained by
k
adding k new constant symbols to τ. Let S be a class of structures. We use S
k
todenotethe classofallτ -structureswhoseτ-reducts arestructuresinS. Our
k
first property, denoted P (S,k), can now be defined as follows.
wqo
Definition 3.1 If S is a w.q.o under the isomorphic embedding relation ֒→,
k
we say that P (S,k) holds.
wqo
Observe that P (S,0) holds iff S is a w.q.o. under ֒→. Furthermore, if
wqo
P (S,k) holds and S′ is a subclass of S, then P (S′,k) holds as well. If
wqo wqo
S is a finite class of structures, then P (S,k) holds trivially for each k ∈ N.
wqo
The next lemma provides a more interesting example.
Lemma 3.2 LetS betheclass ofallfinitelinearorders. ThenP (S,k)holds
wqo
for all k ∈N.
Proof: Fix k. Let I = (A ,a1,...,ak) be an infinite sequence of elements
i i i i≥1
from S . Since there are only finitely many order-types of a k-tuple from a
k
linear order, there exists an infinite subsequence J = (B ,b1,...,bk) of I
j j j j≥1
such that the order-type of b1,...,bk in B is the same for all j.
j j j
Consider an element (B,b1,...,bk) of J. Let b0 and bk+1 be the minimum and
maximum elements of B. W.l.o.g. assume b0 ≤B b1 ≤B ... ≤B bk ≤B bk+1
where ≤B is the linear order of B. Then (B,b1,...,bk) can be represented
(upto isomorphism) by a (k +1)-tuple tB ∈ Nk+1, where the rth component
of tB is |{b | b ∈ UB, br−1 ≤B b ≤B br}|. Applying Dickson’s lemma to the
sequence (tBj)j≥1 there exist p,q such that p < q and tBp is component-wise
≤ tB . Since a linear order of length m can always be embedded in a linear
q
order of length n for n ≥ m, it follows that (B ,b1,...,bk)֒→ (B ,b1,...,bk).
p p p q q q
Hence, P (S,k) holds.
wqo
Asa“mixed-example”,considerS tobetheclassofallfinite(undirected)paths.
Itis easyto see thatboth P (S,0)andP (S,1) hold. However,P (S,k)
wqo wqo wqo
fails for all k ≥ 2. We illustrate this failure for k = 2. Consider the sequence
(A ) of structures from S , where A =(P ,a ,b ) and P is a path of length
i i≥2 2 i i i i i
i with end-points a and b . It is easy to check that for all i6=j, A 6֒→A .
i i i j
The following lemma provides a “logical” characterization of P (S,0). The
wqo
proof can be found in Appendix A.
Lemma 3.3 Let S be a class of structures. P (S,0) holds iff PS=PSC(0)=
wqo
∀∗ over S.
We are now ready to state the main result of this subsection.
Proposition 3.4 Let k ∈N and S be a class of structures such that P (S,k)
wqo
holds. Then PSC(k)=∃k∀∗ over S.
5
Proof: Consider C ∈ PSC(k) over S. Define D to be the subclass of S con-
k
sisting of all elements (M,a ,...,a ), where M ∈ C and the underlying set of
1 k
(a ,...,a ) is a k-coreof M w.r.t. C overS. It follows fromDefinition 2.1 that
1 k
D ∈ PSC(0) over S . Since P (S,k) holds, by definition, P (S ,0) holds.
k wqo wqo k
By Lemma 3.3, D is definable by a ∀∗ sentence ψ over S . We now replace
k
each constant in τ \τ that appears in ψ by a fresh variable, and existentially
k
quantify these variables to get a ∃k∀∗ sentence defining C over S.
3.2 A logic-based combinatorial property
For every m ∈ N, τ-structures A and B are said to be m-equivalent, denoted
A≡ B, iff A and B agree on the truth of every FO(τ) sentence of quantifier
m
rank ≤ m. We assume the reader is familiar with Ehrenfeucht-Fra¨ıss´e games
(henceforth called EF games) [7, 6]. The classical Ehrenfeuct-Fra¨ıss´e theorem
(henceforth called EF theorem) states that A and B are m-equivalent iff the
duplicator has a winning strategy in the m-round EF game between A and B.
Let k ∈ N and S be a class of structures. Our second property, namely
P (S,k), can now be stated as follows.
logic
Definition 3.5 Suppose there exists a function f : N → N such that for each
m∈N, for each structure A of S and for each subset W of UA of size at most
k, there exists B⊆A such that (i) B∈S, (ii) W ⊆UB, (iii) |B|≤f(m) and
(iv) B ≡ A. Then, we say that P (S,k) holds, and call f(m) a witness
m logic
function of P (S,k).
logic
Clearly, if P (S,k) holds and S′ is a subclass of S that is preserved under
logic
substructures over S, then P (S′,k) also holds.
logic
Revisiting the examples of the previous subsection, we see that if S is a finite
class,thenP (S,k)holdswithf(m)beingtheconstantfunctionthatreturns
logic
the size of the largest structure in S. For linear orders, we have the following
result.
Lemma 3.6 Let S be the class of all finite linear orders. Then P (S,k)
logic
holds for all k ∈N.
Proof: Fix m and k. Let A∈S and let W ={a ,...,a }, r≤k be a subset of
1 r
UA. Define f(m)=max{2m,k}. We now show that there is a linear sub-order
B of A such that B contains W, |B|≤f(m) and B≡ A.
m
If |A| ≤ f(m), choose B = A, and we are done. Otherwise, let B be any
substructure of A that contains W and is of size f(m). Observe that, in this
case,both A and Bareof size at least2m. It is well-knownthat any two linear
orders of length ≥2m are m-equivalent. Hence, B≡ A, and all conditions in
m
Definition 3.5 are satisfied by B.
Finally, if S is the class of all finite paths, it is easy to see that P (S,0) and
logic
P (S,1) hold. However, P (S,k) fails for all k ≥ 2. This is because if
logic logic
A is a path containing two distinct end-points, and if W contains both these
end-points,thenAisitsonly substructurein S thatcontainsW. Thisprecludes
6
the existenceofauniform(i.e. independentof|A|)functionf(m)bounding the
size of a substructure of A containing W, as required in Definition 3.5.
The next theorem is one of the main results of this paper. Before stating the
theorem, we note that given a class S of structures and n ∈ N, the subclass of
allstructuresinS ofsize ≤n is definable overS bya FO sentence in∃n∀∗. We
call this sentence ξ in the following theorem.
S,n
Theorem 3.7 Let S be a class of structures and k ∈ N such that P (S,k)
logic
holds. Then PSC(k) = ∃k∀∗ over S. More precisely, for every defining FO
sentence φ in PSC(k) over S, there exists p ∈ N such that φ is semantically
equivalent to ∃kx¯∀py¯ ψ| over S, where ψ ≡(ξ →φ).
x¯y¯ S,k+p
Proof: It is obvious that ∃k∀∗ ⊆PSC(k) over S. To prove containment in the
other direction, consider φ in PSC(k) over S, and let φ have quantifier rank
m. Let f(m) be the witness function of P (S,k). Consider the sentence
logic
ϕ ≡ ∃kx¯∀py¯ ψ| , where p = f(m). Since φ is in PSC(k) over S, every model
x¯y¯
A of φ in S also satisfies ϕ. To see why this is so,note that the elements of any
k-core of A can serve as witnesses of the existential variables in ϕ. Therefore,
φ→ϕ over S. To show ϕ→φ over S, suppose A is a model of ϕ in S. Let W
bethesetofwitnessesinAforthekexistentialvariablesinϕ. Clearly,|W|≤k.
Since Plogic(S,k) holds, there exists B⊆A such that (i) B∈S, (ii) W ⊆UB,
(iii) |B| ≤ f(m) = p, and (iii) B ≡ A. Since A |= ϕ, by instantiating the
m
universal variables in ϕ with the elements of UB, we have B |= φ. Since the
quantifier rank of φ is m and B ≡ A, it follows that A |= φ. Therefore, φ is
m
semantically equivalent to ϕ over S. This proves the theorem.
Remark: Suppose A ∈ S is a model of φ, and a¯ = (a ,a ,...,a ) ∈ Uk.
1 2 k A
Define φ′(x¯)≡ ∀py¯ ψ| . It is easy to see that A|= φ′(a¯) iff the underlying-set
x¯y¯
{a ,a ,...,a } is a k-core of A w.r.t. φ. Thus, the k-cores of φ are defined by
1 2 k
the FO formula φ′(x¯) if P (S,k) holds.
logic
The witness function f(m) in the definition of P (S,k) may not be com-
logic
putable, in general. By requiring f to be computable in Definition 3.5, we ob-
tainaneffectiveversionofP (S,k),whichwecallPcomp(S,k). Notethatfor
logic logic
all examples considered so far where P (S,k) holds, we actually gave closed
logic
form expressions for f(m). Hence Pcomp(S,k) also holds for these classes. We
logic
will soon see a class S that is in P (S,k), but not in Pcomp(S,k).
logic logic
The following is an important corollary of Theorem 3.7.
Corollary 3.8 Let S be a class of structures and k ∈N such that Pcomp(S,k)
logic
holds. For every φ in PSC(k) over S, the translation to a semantically equiva-
lent (over S) ∃k∀∗ sentence ϕ is effective.
4 Relations between properties
We begin by comparing the classes for which P (S,k) hold with those for
wqo
whichP (S,k)hold. Surprisingly,itturnsoutthatP (S,k)impliesP (S,k)!
logic wqo logic
7
Proposition 4.1 For each class S of structures and each k ∈N, if P (S,k)
wqo
holds, then P (S,k) holds as well.
logic
Proof: We give a proofby contradiction. Suppose, if possible, P (S,k) holds
wqo
but P (S,k) fails. By Definition 3.5, there exists m ∈ N such that for all
logic
p∈N,thereexistsastructureA inS andasetW ofatmostk elementsofA
p p p
suchthatforanysubstructureBofAp,wehave(cid:0)(B∈S)∧(Wp ⊆UB)∧(|B|≤
p)(cid:1)→(B6≡m Ap).
Foreachp≥1,fixthestructureA andthesetW thatsatisfytheseproperties.
p p
Now, let a¯ be any k-tuple such that the components of a¯ are exactly the
p p
elementsofW . LetA′ betheτ -structure(A ,a¯ )inS . Considerthesequence
p p k p p k
(A′) . Since P (S,k) holds, S is a w.q.o. under ֒→. Therefore, there
i i≥1 wqo k
exists an infinite sequence of indices I =(i ,i ,...) such that i <i <... and
1 2 1 2
A′ ֒→A′ ֒→.... Let ∆ be the set of all equivalence classesof the ≡ relation
i1 i2 m
over the class of all τ-structures. Given m and τ, ∆ is clearly a finite set.
Therefore,there exists an infinite subsequence J of I with indices j <j <...
1 2
suchthat(i)A′ ֒→A′ ֒→...,and(ii)thecorrespondingτ-reductsA ,A ,...
arem-equivalejn1t. Letjr2 =|A′ | (whichisthe sameas|A |)andletnj>1 1jb2ean
index such that j ≥ r. Thenj1A′ ֒→ A′ and A ≡ Aj1 . Fix an embedding
ı : A′ ֒→A′ in Sn . We abuse nj1otationjnand denjo1te mthe ijnnduced embedding of
ı onjt1he τ-rejnducts kalso by ı:A ֒→A in S.
j1 jn
Let B be the image of A under ı. Then B has the following properties: (i)
j1
B∈S, sinceAj1 ∈S andS is closedunder isomorphisms,(ii) Wjn ⊆UB,since
ı : (A ,a¯ ) ֒→ (A ,a¯ ) and the components of a¯ are exactly the elements
j1 j1 jn jn jn
of W , (iii) |B| = r ≤ j , and (iv) B ≡ A . This contradicts the property
jn n m jn
of A stated at the outset, completing the proof.
jn
Giventheaboveresult,itisnaturaltoaskwhetherP (S,k)impliesPcomp(S,k)
wqo logic
as well. Proposition 4.2 provides a strong negative answer to this question.
Proposition 4.2 There exists a class S of structures for which P (S,k),
wqo
and hence P (S,k), holds for every k ∈ N, but Pcomp(S,0) fails, and hence
logic logic
Pcomp(S,k) fails for every k ∈N.
logic
Proof: Recallthatthe setofallcomputablefunctions fromNto Nis countable.
Fix an enumeration f ,f ,f ,f ,··· of the computable functions. Now define
0 1 2 3
a class S of words over the alphabet Σ = {a,b,c} as follows: S = {w =
i
abiacfi(i+3) |i∈N}. We show in Section 5 that the class Σ∗ of all finite words
satisfies P (Σ∗,k) for every k ∈ N. It follows that P (S,k) also holds for
wqo wqo
every k ∈N. Now we prove by contradictionthat Pcomp(S,0) fails. If possible,
logic
suppose Pcomp(S,0) holds, and let f (m) be a computable witness function for
logic r
Plcoogmicp(S,0). Considerthewordwr =abracfr(r+3). BydefinitionofPlcoogmicp(S,0),
there exists a subwordw′ ∈S suchthat |w′|≤f (r+3)andw′ ≡ w . Since
r r+3 r
w′ ≡ w ,itiseasytoseethatw′ andw mustagreeonthefirstr+2letters.
r+3 r r
However, since w′ must also be in S, by the definition of S, we must choose
w′ =w . This contradicts |w′|≤f (r+3).
r r
Remark: If S is recursively enumerable, the witness function in Definition 3.5
lies in the second level of the Turing hierarchy. See Appendix B for a proof.
8
The following proposition, along with Proposition 4.2 shows that Pcomp(S,k)
logic
and P (S,k) are incomparable.
wqo
Proposition 4.3 There exists a class S of structures for which Pcomp(S,k)
logic
holds for all k ∈ N, but P (S,0) fails, and hence P (S,k) fails for all
wqo wqo
k ∈N.
Proofsketch: LetC (respectively,P )denoteanundirectedcycle(respectively,
n n
path) of length n. Let mP denote the disjoint union of m copies of P . Let
n n
Hn = Fii==30nnPi and Gn = C3n ⊔Hn, where ⊔ denotes disjoint union. Now
consider the class S of undirected graphs given by S = S ∪S , where S =
1 2 1
{H | n ≥ 1} and S = {G | n ≥ 1}. That P (S,0) fails is easily seen
n 2 n wqo
by considering the sequence (G ) , and noting that C cannot embed in
n n≥1 3n
C unless m = n. The proof that Pcomp(S,k) holds for all k is deferred to
3m logic
Appendix C.
Towardsacomparisonofclassessatisfyingthe variouspropertiesdefinedabove,
we considerthe followingeightnaturalcollectionsof classesandstudy the rela-
tions between them.
1) Γ0 = {S |P (S,0) holds }
wqo wqo
2) Γ∗ = {S |∀k P (S,k) holds }
wqo wqo
3) Γ0 = {S |P (S,0) holds }
logic logic
4) Γ∗ = {S |∀k P (S,k) holds }
logic logic
5) Γ0 = {S |Pcomp(S,0) holds }
comp logic
6) Γ∗ = {S |∀k Pcomp(S,k) holds }
comp logic
7) Γ0 = {S |PS =PSC(0)=∀∗ over S}
8) Γ∗ = {S |∀k PSC(k)=∃k∀∗ over S}
Note that Γ0 (repectively, Γ∗) is the collection of all classes of finite structures
overwhichtheL o´s-Tarskitheorem(respectively, PSC(k)=∃k∀∗)holds. Using
⊆todenotecontainmentforcollectionsofclasses,itistrivialtoseethatΓ∗ ⊆Γ0,
Γ∗ ⊆Γ0 ,Γ∗ ⊆Γ0 ,Γ∗ ⊆Γ0 ,Γ∗ ⊆Γ∗ andΓ0 ⊆Γ0 .
wqo wqo logic logic comp comp comp logic comp logic
Γ0
ttttttSt3ttttttttt:: OOS2
Γ∗ Γ0
OO logic
S2 ttttttSt1ttttt✎t✎t✎✎::✎✎S✎✎4GG WW✴✴✴✴✴✴S✴✴5
Γ∗ Γ0 Γ0
logic wqo comp
S✎✎4✎✎t✎✎t✎✎t✎GGtttWW✴✴t✴✴St✴✴5t✴✴t✴ttSt1tt::tttttttSt1ttttt::
Γ∗ Γ∗
wqo comp
9
The “Hasse” diagram D above depicts all containment relations between the
above eight collections. Every directed arrow represents a “primary” contain-
ment. Some of these containments have already been discussed above. The
remainingfollowfromTheorem3.7andfromProposition4.1. Everypairofun-
orderedcollectionsinthe diagramrepresentsincomparablecollections. We also
annotate every directed arrow with a label that refers to an example demon-
strating the strictness of the containment. The list of examples used to show
strict containments is as follows.
• S is the class of all undirected paths.
1
• S is the class of all undirected cycles.
2
• S is the class of all undirected graphs that are disjoint unions of paths.
3
• S is the class of structures constructed in the proof of Proposition 4.3.
4
• S is the class of words constructed in the proof of Proposition 4.2.
5
For each example in the above list, we indicate below all the lowest/minimal
collections in D that contains it, and all the highest/maximal collections in D
that does not contain it. Due to lack of space, proofs of these inclusions/non-
inclusions are deferred to Appendix D. We discuss only the inclusions/non-
inclusions of S in detail below.
2
• S ∈Γ0 , S ∈Γ0 , S ∈Γ∗, S ∈/ Γ∗ .
1 wqo 1 comp 1 1 logic
• S ∈Γ∗, S ∈/ Γ0 .
2 2 logic
• S ∈Γ0 , S ∈Γ0 , S ∈/ Γ∗.
3 wqo 3 comp 3
• S ∈Γ∗ , S ∈/ Γ0 .
4 comp 4 wqo
• S ∈Γ∗ , S ∈/ Γ0 .
5 wqo 5 comp
Lemma 4.4 The class S belongs to Γ∗\Γ0 .
2 logic
Proof: For any cycle A ∈ S , the only substructure of A in S is A itself. It
2 2
follows that S ∈/ Γ0 . We now show that S ∈ Γ∗. Firstly, observe that any
2 logic 2
subclass of S is in PS over S . So, it suffices to show that S belongs to Γ0.
2 2 2
Towardsthis,letφbe inPS overS , andsupposethequantifierrankofφism.
2
It is well-known that any two cycles of sizes ≥p =2m are m-equivalent. Then
either (i) all cycles of size ≥p are models of φ or (ii) all models of φ in S have
2
sizes <p.
In case (i), we define φ′ = ∀py¯ ψ| where ψ ≡ ξ → φ and ξ is as in
y¯ S2,p S2,p
Theorem3.7. It is easy to verify that (a) allcycles of size >p are models of φ′,
and (b) a cycle of length ≤ p is a model of φ iff it is a model of φ′. Therefore,
φ is equivalent to φ′ over S .
2
In case (ii), let X be the finite set of the sizes of all models of φ in S . It is
2
trivial to see that there exists a quantifier-free formula ξ′ (y ,y ,...,y ) which
X 1 2 p
10