Testing first-order properties for subclasses of sparse graphs

0.28 MB·English

by  Zdenek Dvorak

Testing first-order properties for subclasses of
sparse graphs∗
1
1
0 Zdenˇek Dvoˇr´ak† Daniel Kr´al’† Robin Thomas‡
2

p
e
S
Abstract

3
Wepresentalinear-timealgorithmfordecidingfirst-orderlogic(FOL)
2
propertiesinclasses ofgraphswithboundedexpansion,anotion recently

] introduced by Neˇsetˇril and Ossona de Mendez. Many natural classes of
M graphshaveboundedexpansion: graphsofboundedtree-width,allproper
minor-closed classes ofgraphs, graphsofboundeddegree,graphswithno
D
subgraph isomorphic to a subdivision of a fixed graph, and graphs that
.
s can be drawn in a fixed surface in such a way that each edge crosses at
c mostaconstantnumberofotheredges. Wededucethatthereisanalmost
[
linear-timealgorithmfordecidingFOLpropertiesinclassesofgraphswith

1 locally bounded expansion.
v Moregenerally,wedesignadynamicdatastructureforgraphsbelong-
6 ing to a fixed class of graphs of bounded expansion. After a linear-time
3 initialization the data structure allows us to test an FOL property in
0 constant time, and the data structure can be updated in constant time
5 after addition/deletion of an edge, provided the list of possible edges to
.
9 be added is known in advance and their simultaneous addition results in
0 agraphintheclass. Allourresultsalsoholdforrelationalstructuresand
1 are based on theseminal result ofNeˇsetˇril andOssona deMendezon the
1 existence of low tree-depth colorings.
:
v
i
X 1 Introduction
r
a
A celebrated theorem of Courcelle [1] states that for every integer k ≥ 1 and
every property Π definable in monadic second-order logic (MSOL) there is a
linear-timealgorithmtodecidewhetheragraphoftree-widthatmostksatisfies
Π. While the theorem itself is probably not useful in practice because of the
large constants involved, it does provide an easily verifiable condition that a
certain problem is (in theory) efficiently solvable. Courcelle’s result led to the
∗ApreliminaryversionofthispaperappearedinFOCS2010.
†Institute for Theoretical Computer Science (ITI), Faculty of Mathematics and Physics,
CharlesUniversity, Prague, Czech Republic. E-mail: {rakdver,kral}@kam.mff.cuni.cz. In-
stituteforTheoreticalComputerScience(ITI)issupportedbyMinistryofEducationofCzech
Republicasprojects1M0545.
‡SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA.Partiallysupported
byNSFunderNo.DMS-0739366.
1
development of a whole new area of algorithmics, known as algorithmic meta-
theorems;see the survey[19]orthe forthcoming one[18]. Forspecific problems
there is very often a more efficient implementation, for instance following the
axiomatic approach of [29].
While the class of graphs of tree-width at most k is fairly large, it does
not include some important graph classes, such as planar graphs or graphs of
boundeddegree. Courcelle’stheoremcannotbeextendedtotheseclassesunless
P=NP,becausetesting3-colorabilityis NP-hardforplanargraphsofmaximum
degree at most four [17] and yet 3-colorability is expressible in monadic second
order logic.
Thusinanattemptatenlargingtheclassofinputgraphs,wehavetorestrict
the set of properties to be tested. One of the first result in this direction was
a linear-time algorithm of Eppstein [11, 12] for testing the existence of a fixed
subgraph in planar graphs. He then extended his algorithm to minor-closed
classesofgraphswithlocallyboundedtree-width[13]. Sincetestingcontainment
ofafixedsubgraphcanbeexpressedinfirstorderlogic(FOL)byaΣ1-sentence,
this can be regardedas a precursor to FOL formula testing. Prior to our work,
the following were the most general results:
• a linear-time algorithm of Seese [30] to test FOL properties of graphs of
bounded degree,
• alinear-timealgorithmofFrickandGrohe[15]fordecidingFOLproperties
of planar graphs,
• an almostlinear-time algorithmof Frick andGrohe [15] for deciding FOL
properties for classes of graphs with locally bounded tree-width,
• afixedparameteralgorithmofDawar,GroheandKreutzer[2]fordeciding
FOL properties for classes of graphs locally excluding a minor, and
• alinear-timealgorithmofNeˇsetˇrilandOssonadeMendez[22]fordeciding
Σ1-properties for classes of graphs with bounded expansion.
Our maintheorem givesa commongeneralizationof these five results. Inorder
to state it we need a couple of definitions. For an integer r ≥ 0, a graph G is
an r-shallow minor of a graph H if G can be obtained from a subgraph of H
by contracting vertex-disjoint subgraphs of radii at most r (and removing the
resulting loops and parallel edges). A class G of graphs has bounded expansion
if there exists a function f : N → R+ such that for every integer r ≥ 0 every
r-shallow minor of a member of G has edge-density at most f(r). (The edge-
density of a graph G is |E(G)|/|V(G)|.) A preliminary version of our main
theorem can be stated as follows.
Theorem 1. Let G be a class of graphs with bounded expansion, and let Π be
a first-order property of graphs. Then there exists a linear-time algorithm that
correctly decides whether a graph from G satisfies Π.
2
In fact, we prove a more general theorem: we prove there exists a linear-
time algorithm for L-structures “guarded” by a member of G, and we design
several data structures that allow the L-structure to be modified and support
FOL property testing in constant time.
Using known techniques we derive the following corollary from Theorem 1.
A class G of graphs has locally bounded expansion if there exists a function
g :N×N→R+ suchthatforeverytwointegersd,r ≥0,foreverygraphG∈G
and for every v ∈V(G), every r-shallow minor of the d-neighborhood of v in G
hasedge-densityatmostg(d,r). We saythatthereexists analmost linear-time
algorithm to solve a problem Π if for every ε > 0 there exists an algorithm to
solve Π with running time O(n1+ε), where n is the size of the input instance.
Corollary 2. Let G be a class of graphs with locally bounded expansion, and let
Π be a first-order property of graphs. Then there exists an almost linear-time
algorithm that correctly decides whether a graph from G satisfies Π.
We announced our results in the survey paper [7]. Dawar and Kreutzer [3]
posted an independent proof of Theorem 1 and a proof of Corollary 2 for the
moregeneralclassesofnowhere-densegraphs(introducedbelow). However,the
proofs in [3] are incorrect. A correct proof of Theorem 1, different from ours,
appears in [18].
Thus it remains an interesting open problem whether Corollary 2 can be
generalizedtothe moregeneralclassesofnowhere-densegraphs. Thisis ofsub-
stantial interest from the point of view of fixed parameter tractability, because
nowhere density of classes of graphs gives a natural limitation (subject to a
widely believed complexity-theoryassumption). Indeed, we provethe following
in Theorem 5 below. Let L be a language consisting of one binary relation
symboland letG be a classof graphsclosedunder taking subgraphsthat is not
nowheredense. WeprovethatiftestingwhetheraninputgraphfromG satisfies
a given Σ1-sentence ϕ is fixed parameter tractable when parametrized by the
size of ϕ, then FPT=W[1].
Inthe restofthis sectionweintroduceterminologyandstate allourresults.
1.1 Logic theory definitions
Ourlogicterminologyisstandard,exceptforthefollowing. Allfunctionsymbols
have arity one, and hence all functions are functions of one variable. If L is a
language, then an L-term is simple if it is a variable or it is of the form f(x)
where f is a function symbol and x is a variable. An L-formula is simple if all
terms appearing in it are simple. The rest of our logic terminology is standard,
and so readers familiar with it may skip the rest of this subsection.
Alanguage LconsistsofadisjointunionofafinitesetLr ofrelation symbols
and a finite set Lf of function symbols. Each relation symbol R∈Lr is associ-
ated with an integer a(R)≥0, called the arity of R. In this paper all function
symbols have arity one.
IfLisalanguage,thenanL-structureAisatriple(V,(RA)R∈Lr,(fA)f∈Lf)
consisting of a finite set V and for each m-ary relation symbol R ∈ Lr a set
3
RA ⊆ Vm, the interpretation of R in A, and for each function symbol f ∈ Lf
a function fA :V →V of one variable, the interpretation of f in A. We define
V(A) := V. For example, graphs may be regarded as L-structures, where L is
the language consisting of a single binary relation. We define the size |A| of A
to be |V(A)|+ |RA|+|Lf||V(A)|. A language L′ extends a language L
R∈Lr
if every function symbol of L is a function symbol of L′ and the same holds for
P
relation symbols. If a language L′ extends a language L, A is an L-structure
andA′ is anL′-structure suchthat V(A)=V(A′) andA andA′ havethe same
interpretations of symbols of L, then we say that A′ is an expansion of A.
Assume that we have an infinite set of variables. An L-term is defined as
follows:
1. each variable is an L-term, and
2. if f ∈Lf and t is an L-term, then f(t) is an L-term.
Each L-term is obtained by a finite number of applications of these two rules.
We say that an L-term is simple if it is a variable or is of the from f(x) where
f ∈ Lf and x is a variable. A term t appears in a term t′ if either t = t′ or
t′ =f(t′′) for some f ∈Lf and t appears in t′′.
An atomic L-formula ϕ is either the symbol ⊤ (which represents a tautol-
ogy); or its negation ⊥; or R(t1,...,tm), where R is an m-ary relation symbol
of L and t1,...,tm are L-terms; or t1 = t2, where t1 and t2 are L-terms. A
term t appears in ϕ if it appears in one of the terms t1,...,tm. An L-formula
is definedrecursivelyasfollows: everyatomic L-formulais anL-formula,and if
ϕ1 andϕ2 areL-formulasandxis avariable,then¬ϕ1, ϕ1∨ϕ2, ϕ1∧ϕ2, ∃xϕ1
and ∀x ϕ1 are L-formulas. Every L-formula is obtained by a finite application
of these rules. We write t1 6=t2 as a shortcut for ¬(t1 =t2).
A term t appears in an L-formula ϕ1∨ϕ2 if it appears in ϕ1 or ϕ2, and we
define appearance for the other cases analogously. An L-formula is simple if all
terms appearing in it are simple. A variable x appears freely in an L-formula ϕ
if either ϕ is atomic and x appears in ϕ; or ϕ = ϕ1 ∨ϕ2 or ϕ = ϕ1∧ϕ2 and
x appears freely in at least one of the formulas ϕ1 and ϕ2; or ϕ = ∃y ϕ′ or
ϕ = ∀y ϕ′, x is distinct from y and x appears freely in ϕ′. Occurences of x in
the formula ϕ not inside the scope of a quantifier bounding x, i.e., those that
witness that x appears freely in ϕ, are called free and the variables that appear
freely in ϕ are also referred to as free variables. If ϕ is a formula, then the
notation ϕ(x1,...,xn) indicates that all variables that appear freely in ϕ are
amongx1,...,xn. AnL-sentenceisanL-formulasuchthatnovariableappears
freely in it. If ϕ(x1,...,xn) is an L-formula and A is an L-structure, then for
v1,...,vn ∈ V(A), we define A |= ϕ(v1,...,vn) in the usual way. We denote
the length of a formula ϕ by |ϕ|.
1.2 Classes of sparse graphs
Allgraphsconsideredinthispaperaresimpleandfinite. Thenotionofaclassof
graphsofboundedexpansionwasintroducedbyNeˇsetˇrilandOssonadeMendez
4
in [20] and in the series of journal papers [21, 22, 23]. Examples of classes of
graphs with bounded expansion include proper minor-closed classes of graphs,
classes of graphs with bounded maximum degree, classes of graphs excluding
a subdivision of a fixed graph, classes of graphs that can be embedded on a
fixedsurfacewithboundednumberofcrossingspereachedgeandmanyothers,
see [25]. Many structural and algorithmic properties generalize from proper
minor-closed classes of graphs to classes of graphs with bounded expansion,
see [7, 26].
A class G of graphs is nowhere-dense if
log|E(G)|
lim lim sup ≤1,
r→∞k→∞ G log|V(G)|
where the supremum is taken over all graphs G on at least k vertices such that
G is an r-shallow minor of a member of G. It can be shown that every class of
graphswith(locally)boundedexpansionisnowhere-dense[24],buttheconverse
is false. One can also define a “locally nowhere-dense” class of graphs, but it
turns out that such classes are nowhere-dense [24].
If L is a language,then the Gaifman graph of an L-structure A is the undi-
rected graph G with vertex set V(G ) = V(A) and an edge between two
A A
vertices a,b ∈ V(A) (a 6= b) if and only if there exist R ∈ Lr and a tuple
(a1,...,ar)∈RA suchthat a,b∈{a1,...,ar} orthere existsa functionf ∈Lf
such that b = f(a) or a = f(b). We say that the relational structure A is
guarded by a graph G with vertex set V(G) = V(A) if G ⊆ G. Observe that
A
if G belongs to a class of graphs with bounded expansion, then every subgraph
of G has a vertex of bounded degree, and hence the number of complete sub-
graphs of G is linear in |V(G)| by a result of [31]. It follows that the size |A| of
any L-structureA guardedby a graphbelonging to a fixed class ofgraphs with
bounded expansion is O(|V(A)|).
1.3 Our results
We first state versions of Theorem 1 and Corollary 2 for L-structures.
Theorem 3. Let G be a class of graphs with bounded expansion, L a language
and ϕ an L-sentence. There exists a linear-time algorithm that decides whether
an L-structure guarded by a graph G∈G satisfies ϕ.
Corollary 4. Let G be a class of graphs with locally bounded expansion, L a
language and ϕ an L-sentence. There exists an almost linear-time algorithm
that decides whether an L-structure guarded by a graph G∈G satisfies ϕ.
Ourapproachdiffersfromthemethodsusedtoprovetheresultsfrom[2,15,
30] mentioned above and is based on a seminal result of Neˇsetˇril and Ossona
de Mendez [21] on the existence of low tree-depth colorings for graphs with
bounded expansion, stated below as Theorem 7.
We also consider dynamic setting and design the following data structures,
where the first can be viewed as a dynamic version of Theorem 3. In the
descriptions below we use n to denote the number of vertices of the graph G.
5
• for every class G of graphs with bounded expansion, a language L and an
L-sentence ϕ, a data structure that is initialized with a graph G∈G and
an L-structure A guarded by G in time O(n) and supports:
– adding a tuple to a relation of A in time O(1) provided A stays
guarded by G,
– removing a tuple from a relation of A in time O(1),
– answering whether A|=ϕ in time O(1),
• for every class G of graphs with bounded expansion, an integer d0 and a
language L, a data structure that is initialized with a graph G ∈ G and
an L-structure A guarded by G in time O(n) and supports:
– adding a tuple to a relation of A in time O(1) provided A stays
guarded by G,
– removing a tuple from a relation of A in time O(1),
– answering whether A |= ϕ for any Σ1-L-sentence ϕ with at most d0
variables in time O(|ϕ|) and outputting one of the satisfying assign-
ments, and
if G is only a class of nowhere-densegraphs,then for everyε>0 the data
structure is initialized in time O(n1+ǫ) and supports:
– adding a tuple to a relation of A in time O(nε) provided A stays
guarded by G,
– removing a tuple from a relation of A in time O(nε), and
– answering whether A |= ϕ for any Σ1-L-sentence ϕ with at most d0
variables in time O(|ϕ|) and if so, outputting one of the satisfying
assignments.
The second of these data structures is needed in our linear-time algorithm for
3-coloring triangle-free graphs on surfaces [10], also see [8].
1.4 A hardness result
Theorem1 andCorollary2 fall within the realmoffixed parametertractability
(FPT). We say that a decision problem Π parametrized by a parameter t is
fixed parameter tractable if there exists an algorithm for Π with running time
O(f(t)nc), where n is the size of the input, f is an arbitrary function and c is
a constant independent of t. Analogously to the polynomial hiearchy starting
with the classes P and NP, there exists a hiearchy of classess FPT ⊆ W[1] ⊆
W[2] ⊆··· of parametrized problems, where FPT is the class of problems that
are fixed parameter tractable. See e.g. [5, 14, 27] for more details.
Theorem 5. Let G be a class of graphs closed under taking subgraphs. If G
is not nowhere-dense and the problem of deciding Σ1-properties in G is fixed
parameter tractable, then FPT=W[1].
6
Proof. An r-subdivision of a graph G is the graph obtained from G by subdi-
vidingeveryedgeexactlyr times. SinceG is notnowhere-dense,thereexistsan
integer r such that G contains anr-subdivision of every graph[24]. By hypoth-
esisthereexistsanFPTalgorithmAtodecidethe existenceofanr-subdivision
of the complete graph K in an input graph from G, where the latter problem
m
isparametrizedbym. Thisimpliesthattestingtheexistenceofacompletesub-
graphofordermisfixedparametertractableforgeneralgraphsG,becauseitis
equivalentto testing whether the r-subdivision ofG hasa subgraphisomorphic
to the r-subdivision of K , and the latter can be tested using the algorithm
m
A. But testing the existence of a K subgraph is a well-known W[1]-complete
m
problem [4], and hence FPT=W[1], as desired.
DawarandKreutzer[3]provedtherelatedresultthatifG failstobenowhere
denseinan“effective”wayanddecidingFOLpropertiesinG isfixedparameter
tractable, then FPT=AW[∗].
2 Classes of graphs with bounded expansion
In this section, we survey results on classes of graphs with bounded expansion
and classes of nowhere-dense graphs that we need in the paper. We say that
the expansion of a graph G is bounded by a function g : N → R+ if for every
integerr ≥0 everyr-shallowminorofG hasedge-densityatmostg(r). We say
that a class of graphs G has expansion bounded by g if every member of G has
expansion bounded by g. Thus G is a class of graphs of bounded expansion if
and only if its expansion is bounded by some function N→R+.
A rooted forest F is a directed graph such that every weak component is
an out-branching (a rooted tree with every edge directed away from the root).
The depth of F is the number of vertices of a longest directed path in F. The
closureofF is the undirectedgraphwithvertex-setV(F)andedge-setallpairs
of vertices joined by a directed path in F. The tree-depth of an (undirected)
graph G is the smallest integer s such that G is a subgraph of the closure of
a rooted forest of depth s. For an integer d ≥ 1 a vertex coloring of a graph
G is a low tree-depth coloring of order d if for every s = 1,2,...,d the union
of any s color classes induces a subgraph of G of tree-depth at most s. Thus,
in particular, every low tree-depth coloring of order d of G is a proper coloring
of G.
The existence ofa lowtree-depthcoloringwithbounded number ofcolorsis
one of the major structural results for graphs with bounded expansion. Before
we state this result formally, we have to introduce more definitions.
Let D be a directed graph, and and let G′ be the graph obtained from the
underlying undirected graph of D by adding all edges xy such that:
• there exists a vertex z such that D contains an edge oriented from x to z
and an edge oriented from z to y (transitivity), or
• there exists a vertex z such that D contains an edge oriented from x to z
and an edge oriented from y to z (fraternality).
7
We call G′ the augmentation of D. The following is a mild strengthening of a
result of Neˇsetˇriland Ossona de Mendez [21], but the proof is adequate.
Theorem6. Thereexistfunctionsf1 andf2 withthefollowing property. LetD
be a directed graph with maximum in-degree at most ∆ such that its underlying
undirected graph has expansion bounded by g, and let G′ be the augmentation of
D. Then the expansion of G′ is bounded by thefunction g′(r)=f1(g(f2(r)),∆).
Let G be a graph from a class G with bounded expansion. We denote by
∇0(G) the maximum edge density among the subgraphs of G. Thus every sub-
graphofG hasa vertexofdegreeatmost2∇0(G), andhence G hasanorienta-
tionwithmaximumin-degreeatmost2∇0(G). Considerthefollowingsequence
ofgraphs: LetG0 :=GandassumethatwehaveconstructedG0,G1,...,Gk−1.
Let D be a directed graph of maximum in-degree at most 2∇0(Gk−1)+2 such
that Gk−1 is its underlying undirected graph, and let Gk be the augmentation
ofD. We say that G is a k-th augmentation of G. By Theorem6, the class G
k k
consisting of the graphs G obtained as described above starting from graphs
k
in G has bounded expansion. It follows that 2∇0(Gk)≤ d for some constant d
dependingonlyontheclassG andk,andthusG isd-degenerate. Inparticular,
k
the chromatic number of a k-th augmentation of a graph G∈G is bounded by
a constant that depends on G only.
IfG isanorientationofagraph,thenanypropercoloringofits (3d2+1)-th
augmentation G′ of G is a low tree-depth coloring of order d [21], also see [7]
for further details. Moreover, the subgraph H of G′ induced by the vertices of
any s ≤ d color classes contains the underlying undirected graph of a rooted
forest F of depth at most s such that H is a subgraph of the closure of F.
We refer to this property of the augmentation as depth-certifying and call F a
depth-certifying forest. The next theorem from [21] follows.
Theorem 7. Let G be a class of graphs with bounded expansion and d ≥ 1
an integer. There exist integers k and K such that any proper coloring of a
k-th augmentation of a graph G∈G is a low tree-depth coloring of order d and
the k-th augmentation is depth-certifying. In particular, G has a low tree-depth
coloring of order d with at most K colors. Moreover, such a coloring of G∈G
and the corresponding depth-certifying forests can be found in linear time.
Similarly, for classes of nowhere dense graphs, we obtain [24]:
Theorem 8. Let G be a class of nowhere-dense graphs and d ≥ 1 an integer.
There exists k such that any proper coloring of a k-th augmentation of a graph
G ∈ G is a low tree-depth coloring of order d and the k-th augmentation is
depth-certifying. In particular, for every ǫ>0 the graph G has a low tree-depth
coloring of order d with at most O(nε) colors. Moreover, such a coloring of
G ∈ G and the corresponding depth-certifying forests can be found in almost
linear time.
8
3 Deciding FOL properties in linear time
Inthis section,we proveTheorem3. We startwith a lemma whichallowsus to
remove quantifiers from an FOL formula (Lemma 13). However, we need more
definitions. LetLbealanguageandletX beasetofL-terms. AnX-templateT
isarootedforestwithvertexsetV(T)equippedwithamappingα :X →V(T)
T
such that α−1(w) 6= ∅ for every vertex w of T with no descendants. If ϕ is a
T
quantifier-free L-formula, then a ϕ-template is an X-template where X is the
set of all terms appearing in ϕ. The depth of T is the maximum depth of a
tree of T. Two X-templates T and T′ are isomorphic if there exists a bijection
f :V(T)→V(T′) such that
• f is an isomorphism of T and T′ as rooted forests; in particular, w is a
root of T if and only if f(w) is a root of T′, and
• f(αT(t))=αT′(t) for every L-term t∈X.
The number of non-isomorphic X-templates of a given depth is finite, as
stated in the next proposition. The proof is straightforward and is left to the
reader.
Proposition 9. For every finite set of terms X and every integer d, there exist
only finitely many non-isomorphic X-templates of depth at most d.
LetLbealanguageandletX beasetofL-termswithvariables{x1,...,xn}.
AnembeddingofanX-templateT inarootedforestF isamappingν :V(T)→
V(F) such that ν(r) is a root of F for every root r of a tree of T and ν is an
isomorphism of T and the subforest of F induced by ν(V(T)). Let S be an
L-structure guarded by the closure of F, and v1,...,vn ∈ V(S). We say that
the embedding ν is (v1,...,vn)-admissible for S if for every term t ∈ X, we
have ν(αT(t)) = t(v1,...,vn), where t(v1,...,vn) denotes the element of V(S)
obtained by substituting v for x in the term t and evaluating the functions
i i
forming the term t (in particular,if x ∈X, then ν(x )=v ). We will need the
i i i
following lemma.
Lemma 10. Let d≥1be an integer, let F be arooted forest of depth at most d,
let L be a language, let ϕ(x1,...,xn) be an L-formula, let S be an L-structure
guarded by the closure of F, and let v1,...,vn ∈ V(S). Then there exists a
ϕ-template T and an embedding of T into F that is (v1,...,vn)-admissible for
S.
Proof. Let X be the set of all L-terms that appear in ϕ, and let Y be the set
of all evaluations t(v1,...,vn) of all terms t(x1,...,xn) from X. Let T be the
smallest subforest of F that includes all vertices from Y and the root of every
component of F that includes an element of Y. For a term t(x1,...,xn) in X
let αT(t):=t(v1,...,vn), and let ν be the identity mapping. Then T andν are
as desired.
9
If F is a rooted forest, then the function p : V(F) → V(F) is the F-
predecessor function if p(v) is the parent of v unless v is a root of F; if v is a
root of F, p(v) is set to be equal to v.
We now show that it can be tested by a quantifier-free formula whether an
embedding is admissible.
Lemma 11 (Testing admissibility). Let d ≥ 0 be an integer, L a language in-
cludingafunctionsymbolpandX afinitesetoftermswithvariables x1,...,xn.
If T is an X-template of depth at most d, then there exists a quantifier-free for-
mula ξT(x1,...,xn) such that for every rooted forest F and every L-structureS
guarded by the closure of F suchthat pS is the F-predecessor function in S, and
for every n-tuple v1,...,vn ∈V(S), the L-structure S satisfies ξT(v1,...,vn) if
and only if there exists an embedding of T in the forest F that is (v1,...,vn)-
admissible for S.
Proof. Let q :V(T)→V(T) be the T-predecessor function. Set ξT(x1,...,xn)
to be the conjunction of all formulas
• pk(t)=pk′(t′) if qk(α (t))=qk′(α (t′)), and
T T
• pk(t)6=pk′(t′) if qk(α (t))6=qk′(α (t′)),
T T
for all pairs of not necessarilydistinct terms t,t′ ∈X and allpairs of integers k
andk′,0≤k,k′ ≤d+1(note that including the formulaswitht=t′ allowsfor
testing the depth of t in F). Here pk denotes the function p iterated k times.
It is straightforward to show that for v1,...,vn ∈ V(S), an (v1,...,vn)-
admissibleembeddingforSofT inF existsifandonlyifS |=ξT(v1,...,vn).
The following lemma will be extremely useful in the proof of Lemma 13.
Recall that an L-term is simple if it is a variable or a function image of a
variable, and an L-formula is simple if all terms appearing in it are simple.
Lemma 12. Let d ≥ 0 be an integer, L a language, ϕ(x0,...,xn) a simple
quantifier-free L-formula that is a conjunction of atomic formulas and their
negations, and T a ϕ-template. There exist a language L that extends L and a
quantifier-free L-formula ϕT(x1,...,xn) such that the following holds:
• L is obtained from L by adding a function symbol p and finitely many
relation symbols U1,...,Uk of arity at most one,
• ϕT is quantifier-free and the variables x1,...,xn are the only variables
that appear in ϕ , but ϕ need not be simple, and
T T
• for every rooted forest F of depth at most d and every L-structure S
guarded by the closure of F, there exists an L-structure S such that S
is an expansion of S and for every v1,...,vn ∈V(S),
S |=ϕ(v0,v1,...,vn) for some v0 ∈V(S) such that there exists a
(v0,...,vn)-admissible embedding of T in F for S if and only if
S |=ϕT(v1,...,vn),
10