3 Expressive Power of First-Order Logic

(1)

Algorithmic Model Theory SS 2016

Prof. Dr. Erich Grädel and Dr. Wied Pakusa

(2)

c b n d

This work is licensed under:

http://creativecommons.org/licenses/by-nc-nd/3.0/de/

Dieses Werk ist lizenziert unter:

http://creativecommons.org/licenses/by-nc-nd/3.0/de/

http://www.logic.rwth-aachen.de

3 Expressive Power of First-Order Logic

In the whole chapter we restrict ourselves tofiniteandrelationalvocabu- lariesτ.

3.1 Ehrenfeucht-Fraïssé Theorem

LetAandBbe τ-structures witha∈ A^k andb ∈ B^k for somek≥ 0. Recall that we writeA,a≡B,bif no FO-formula can distinguish between(A,a)and(B,b), that is if for allφ(x)∈FO(τ)we have

A|=φ(a)⇔B|=φ(b).

For m ≥ 0 we write A,a ≡m B,b if the same holds for all FO(τ)- formulas of quantifier rank at mostm. We aim to develop an algebraic characterisation of≡m viaback-and-forth systemsand a game-theoretic characterisation viaEhrenfeucht-Fraïssé games.

Back-and-forth systems. Apartial isomorphismbetweenτ-structuresAand Bis a bijective function pwithfinitedomain dom(p)⊆Aand range rg(p)⊆Bsuch thatpis an isomorphism between the substructures of AandBinduced on dom(p)and rg(p), respectively, that is

p:A↾dom(p)∼=B↾rg(p).

Part(A,B)denotes the set of partial isomorphism betweenAandB.

For allA and B we have∅ ∈ Part(A,B). For p ∈ Part(A,B) we writep =a→bfor a∈ A^kandb∈B^kif dom(p) = {a1, . . . ,ak}and rg(p) ={b1, . . . ,b_k}and ifp(a_i) =b_ifor 1≤i≤k.

(4)

3 Expressive Power of First-Order Logic

Definition 3.1. Let I ⊆ Part(A,B) and p ∈ Part(A,B). Then p has back-and-forth extensionsinIif

∀a∈A∃b∈B:p∪ {(a,b)} ∈I (forth)

∀b∈B∃a∈A:p∪ {(a,b)} ∈I (back)

Accordingly, forI,J⊆Part(A,B)we say thatIhasback-and-forth exten- sionsinJ, if everyp∈Ihas back-and-forth extensions inJ.

Definition 3.2.Letm≥0. Aback-and-forth systemform-equivalence of (A,a)and(B,b)is a sequence(Ii)_i≤m of sets of partial isomorphisms I_i⊆Part(A,B)such that

•a→b∈Im, and

• for all 0<i≤m,Iihas back-and-forth extensions inIi−1. If such a system(Ii)_i_≤_mfor(A,a)and(B,b)exists, then we write

(Ii)_i≤m:(A,a)≃m(B,b),

and we say that(A,a)and(B,b)arem-isomorphic.

Lemma 3.3. For everym≥ 0, everyτ-structureAand everya∈ A^k, there exists an FO(τ)-formulaχ_A,a^m (x1, . . . ,xk)of quantifier rankmsuch that for allBandb∈B^kwe have

B|=χ^m_A,a(b)⇔A,a≃mB,b.

Moreover the number of different formulasχ^m_A,aonly depends onm,τ, andk, and not onAora(up to logical equivalence).

Proof. The construction is by induction onm≥0 (for allk≥0,A, and a∈A^kat the same time).

χ⁰_A,a(x₁, . . . ,x_k) =^{^}{φ(x₁, . . . ,x_k):φis an atomic or negated atomic FO(τ)-formula withA|=φ(x₁, . . . ,x_k)} We have thatA,a≃0 B,bif, and only if, a→ b∈Part(A,B)which means that(A,a)and(B,b)satisfy the same atomic formulas. Note that

44

3.1 Ehrenfeucht-Fraïssé Theorem the number of different atomic formulas inkvariables only depends on the vocabularyτand onk≥0.

Now letm>0. Then we setχ_A,a^m (x1, . . . ,xk) =

^ a^′∈A

∃xχ^m_A,a,a⁻¹′(x1, . . . ,xk,x)∧ ∀x ^_

a^′∈A

χ_A,a,a^m⁻¹′(x1, . . . ,xk,x).

Since the number of different formulasχ^m_A,a,a⁻¹′ (up to equivalence) only depends onm−1 andk+1 (by the induction hypothesis), also the number of different formulasχ^m_A,aonly depends onmandk(up to equivalence) and not onAora. This is of particular importance if one of the structures is infinite, because it guarantees that the conjunction and the disjunction inχ^m_A,aare finite. It holds

(A,a)≃m(B,b)

⇐⇒





∀a^′∈A∃b^′∈B:(A,a,a^′)≃m−1(B,b,b^′)

∀b^′∈B∃a^′∈A :(A,a,a^′)≃m−1(B,b,b^′)

⇐⇒^{(by (IH))}





∀a^′∈A∃b^′∈B:B|=χ^m_A,a,a⁻¹_′(b,b^′)

∀b^′∈B∃a^′∈A :B|=χ^m_A,a,a⁻¹_′(b,b^′)

⇐⇒ B|=χ^m_A,a(b). q.e.d.

Ehrenfeucht-Fraïssé games. The Ehrenfeucht-Fraïssé gameGm(A,a,B,b) is played by two players according to the following rules.

Thearenaconsists of the structures A andB. We assume that A∩B=∅. The players are calledSpoilerandDuplicator, and a play of Gm(A,a,B,b)consists ofmmoves.

The initial position isGm(A,a,B,b). In thei-th move, 1≤i≤m, the play proceeds from the position

Gm−i+1(A,a,c1, . . . ,ci−1,B,b,d1, . . . ,di−1).

Spoiler either chooses an elementc_i∈Aor an elementd_i∈B. Duplicator answers by choosing an elementci∈Aordi∈Bin the other structure.

The new position isGm−i(A,a,c1, . . . ,ci,B,b,d1, . . . ,di). Aftermmoves, elementsc1, . . . ,cmfromAandd1, . . . ,dmfromBare chosen. Duplicator

45

(5)

wins at a final positionG0(A,a,c1, . . . ,cm,B,b,d1, . . . ,dm)ifA,a,c≡0

B,b,d. Otherwise Spoiler wins.

Awinning strategyof Spoiler is a function which determines, for every reachable position, a move such that Spoiler wins each play which is consistent with this strategy, no matter how Duplicator plays. Winning strategies for Duplicator are defined analogously. We say thatSpoiler (respectively, Duplicator) wins the game Gm(A,a,B,b)if this player has a winning strategy forGm(A,a,B,b). By induction on the number of moves it is easy to show that for every (sub)game exactly one of the two players has a winning strategy.

Theorem 3.4(Ehrenfeucht, Fraïssé). LetA,Bbeτ-structures (recall,τ is finite and relational), leta∈A^kandb∈B^kand letm≥0. Then the following statements are equivalent:

(i)A,a≡mB,b.

(ii)A,a≃mB,b.

(iii)B|=χ^m_A,a(b).

(iv) Duplicator winsGm(A,a,B,b).

Proof. SinceA|=χ^m_A,a(a)and since qr(χ^m_A,a) ≤m, we have that(i) ⇒ (iii). By Lemma 3.3,(ii)⇔(iii). Recall from the introductory course that(iv)⇒(ii). The proof strategy was to show, by induction on the quantifier rankm≥0, that if a formulaφ(x)of quantifier rankmcan distinguish betweenA,aandB,b, then we can extract a winning strategy for Spoiler from this formula for the gameGm(A,a,B,b).

Hence, it suffices to prove(ii)⇒(iv). Let(I_i)_i≤m:(A,a)≃m(B,b). For m = 0 the claim holds, since a → b ∈ Im ⊆ Part(A,B). For m>0 assume that the Spoiler at positionGm(A,a,B,b)picks an element a^′ ∈ A. By the forth property Duplicator can pickb^′ ∈ Bsuch that (a,a^′) → (b,b^′) ∈ Im−1. Hence, (Ii)_i≤m−1 : (A,a,a^′) ≃m−1 (B,b,b^′). By the induction hypothesis, Duplicator winsGm−1(A,a,a^′,B,b,b^′). If Spoiler picks an elementb^′∈Bthe reasoning is analogous using the

back property. q.e.d.

Corollary 3.5. For allk ≥0, the relation≡minduces an equivalence relation on pairs(A,a)ofτ-structuresAanda∈A^kof finite index.

3.2 Hanf’s technique Corollary 3.6.A classKofτ-structures is FO-definable if, and only if, there existsm≥0 such that for allτ-structuresAandBwithA≡mB it holds thatA∈ K ⇔B∈ K.

3.2 Hanf’s technique

Describing winning strategies in Ehrenfeucht-Fraïssé games can be diffi- cult. In this section we want to establish sufficient criteria for structures AandBwhich guarantee that Duplicator has a winning strategy in the gameGm(A,B). The following approach goes back to Hanf who gave a similar criterion to characterise≡(equivalance in full first-order logic).

However, since we are mainly interested in properties offinitestructures,

≡is far too powerful (two finite structuresA,Bare isomorphic if, and only if,A≡B).

Gaifman graph. Let A be a τ-structure. The Gaifman-graph G(A) = (V^G(A),E^G(A))ofAis defined as the undirected graph over the vertex setV^G^(A)=Awith the edge relation

E^G^(A)={(a,b):a̸=band the elementsa,boccur together in some tuplec∈R^Afor a relationR∈τ}. The Gaifman graph allows us to define a notion of distance between the elements of the structureA: we defined^A:A²→^N∪ {∞}as the usual distance function in the Gaifman graphG(A)ofA.

Letr ≥ 0. The r-neighbourhoodof an element a ∈ Ais the set N_A^r(a) = N^r(a) = {b∈A :d^A(a,b)≤r}. In particular,N⁰(a) = {a}. For a tuplea= (a1, . . . ,ak)∈A^kwe set

N^r(a) = ^[

1≤i≤k

N^r(a_i).

Ther-isomorphism typeof an elementa∈Ais the isomorphism type ιof the structure(A↾N^r(a),a)(that is of the substructure ofAinduced on ther-neighbourhood of aextended by a new constant symbol to distinguish the elementa). This means that forτ-structuresA,B, two

(6)

elementsa∈Aandb∈Bhave the samer-isomorphism typeif there is an isomorphismπ:A↾^N^r(a)→B↾^N^r(b)withπ(a) =b.

Definition 3.7. Letr ≥ 0 and t ≥ 0. Two τ-structuresAandBare (r,t)-Hanf equivalentif for all isomorphism typesιof structures(C,c) (whereCis aτ-structure andc ∈C is a distinguished constant) the number ofa∈Awithr-isomorphism typeιis the same as the number ofb∈Bwithr-isomorphism typeιor both numbers exceed thethreshold t.

Remark3.8. IfAandBare(r,t)-Hanf equivalent, then they also are (r^′,t)-Hanf equivalent for allr^′≤r.

Theorem 3.9(Hanf’s Theorem). Letm ≥ 0 and letAandBbe two τ-structures such that all 3^m-neighbourhoods inAandBhave at most e≥0 many elements.

IfAandBare(3^m,m·e)-Hanf equivalent, thenA≡mB.

Proof. Fori≥0 we obtain a back-and-forth system form-equivalence of AandBby setting

I_m−i={a→b∈Part(A,B):|a|=|b|=i,

A↾^N³^m−i(a),a∼=B↾^N³^m−i(b),b}. We haveIm={∅}, so leti≥1. Without loss of generality, it suffices to show thatIm−ihas forth-extensions inIm−i−1. Leta= (a1, . . . ,ai)and b= (b₁, . . . ,b_i)andρbe such thatρ:A↾^N³^m−i(a),a∼=B↾^N³^m−i(b),b.

Leta∈A. We have to findb∈Bsuch thatA↾N³^m−i−1(a,a),a,a∼=B↾ N³^m−i−1(b,b),b,b.

Case 1 (close to a). Ifa∈ N^2·3^m−i−1(a), then we chooseb= ρ(a)∈ N^2·3^m−i−1(b). This is a valid choice since we haveρ:A↾N³^m−i(a),a,a∼= B↾^N³^m−i(b),b,b.

Case 2 (far from a). If a ̸∈ N^2·3^m−i−1(a), then N³^m−i−1(a)∩

N³^m−i−1(aj) = ∅ for all 1 ≤ j ≤ i. Hence, it suffices to findb ∈ B

with the same 3^m−i−1-isomorphism type asa(call thisι) and the property thatN³^m−i−1(b)∩N³^m−i−1(bj) =∅for all 1≤j≤i.

We know that inAandBthere are the same numbers of realisations of ι or more thanm·emany. By our assumption, we know that in

48

3.3 Gaifman’s Theorem

N^2·3^m−i−1(a)there are at mostm·erealisations, and the same number of

realisations can be found inN²^·³^m−i−1(b)(because ofρ). Hence, we can

find ab∈Bas claimed. q.e.d.

Corollary 3.10.Letm≥0 and letAandBbeτ-structures such that the maximal degree in the Gaifman graphsG(A)andG(B)isd≥0. IfA andBare(3^m,m·d³^m)equivalent, thenA≡mB.

Corollary 3.11. Connectivity of finite graphs is not definable in first- order logic.

Proof. LetAnbe a cycle of length 2nand letBnbe the disjoint union of two cycles of lengthn. Formwe can setn=3^m+1. ThenA_nandB_nare (3^m,∞)-Hanf equivalent butAnis connected whileBnis not.

q.e.d.

3.3 Gaifman’s Theorem

Hanf’s technique shows that first-order logic can essentially express local properties only: if two structures realise the same number of f(m)-neighbourhood types, then no first-order sentence with quantifier rank≤mcan distinguish between both structures. Gaifman’s Theorem makes this observation more precise by showing that every FO-sentence is equivalent to an FO-sentence which only speaks about neighbourhoods of elements of a bounded radius (and this semantic property is guaranteed by the syntactic structure of the sentence). To formally introduce thisGaifman normal formfor first-order logic we first have to introduce the notions oflocal formulasandlocal sentences.

First of all, for everyr≥ 0 we can find an FO-formulaϑ≤r(x,y) which defines in each structureAthe pairs of elements(a,b)∈A²whose distance in the Gaifman graphG(A)ofAis at mostr, that is

ϑ^A_≤_r={(a,b):d^A(a,b)≤r}.

In formulas we will usually write d(x,y) ≤ r as a shorthand for ϑ≤r(x,y). Also we write d(x,y) ≤ r for a tuple of variables

49

(7)

x= (x1, . . . ,x_k)to abbreviate the formula d(x,y)≤r= ^_

1≤i≤k

d(xi,y)≤r.

Local formulas. A formulaφ(x)isr-local if its evaluation in a structure Awith respect to a tuplea∈A^konly depends on ther-neighbourhood of a. To capture this formally, we inductively define therelativisation φ^N^r^(x)(x,y)of a formulaφ(x,y)to ther-neighbourhoodN^r(x)ofx(for the construction we assume that no variable inxis bound inφ):

φ^N^r^(x)=φ for atomic formulasφ

φ^N^r^(x)=ψ^N^r^(x)◦ϑ^N^r^(x) forφ=ψ◦ϑ,◦ ∈ {∧,∨}

φ^N^r^(x)=¬ψ^N^r^(x) forφ=¬ψ

φ^N^r^(x)=∃z(d(x,z)≤r∧ψ^N^r^(x)) forφ=∃zψ φ^N^r^(x)=∀z(d(x,z)≤r→ψ^N^r^(x)) forφ=∀zψ

Lemma 3.12.For allr≥0,A,a∈A^kandb∈(N^r(a))^ℓwe have A↾^N^r(a)|=φ(a,b) ⇔ A|=φ^N^r^(x)(a,b).

Definition 3.13.A formulaφ(x)is calledr-localifφ(x)≡φ^N^r^(x)(x), that is if for allAanda∈A^kwe have

A|=φ(a) ⇔ A|=φ^N^r^(x)(a) ⇔ A↾^N^r(a)|=φ(a).

Note thatr-locality is a semantic property of formulas. However, it is easy to see that all formulasφ^N^r^(x)(x)arer-local (in other words, the syntatic transformations guarantee that we obtain a local formula, but of course there are local formulas which do not have this syntactic form). Moreover, it is not hard to verify that every formulaφ(x)which is r-local is alsor^′-local for allr^′ ≥ r. For a formula φ(x) we write φ^r(x) =φ^N^r^(x)(x)to denote ther-local version of the formulaφ(x).

3.3 Gaifman’s Theorem Local sentences. Anℓ-tuple of elements a = (a1, . . . ,aℓ) ∈ A^ℓ in a structureAis calledr-scatteredifd(a_i,a_j)>2rfor alla_ianda_j,i̸=j, that is if ther-neighbourhoodsN^r(ai), 1≤i≤ℓ, are pairwise disjoint. A basic local sentenceofGaifman rank(r,m,ℓ)is a sentence of the form

∃x1· · · ∃xℓ



^{^}

i̸=j

d(xi,xj)>2r∧^{^}

i

ψ^r(xi)



,

where qr(ψ) =m, which expresses the existence of anr-scattered tuple of lengthℓsuch that every point in this tuple satisfies anr-local property which is specified by a formulaψof quantifier-rankm. Alocal sentence is Boolean combination of basic local sentences.

Theorem 3.14(Gaifman). Every first-order sentence is equivalent to a local sentence.

To prove Gaifman’s Theorem it suffices to show the following lemma.

Lemma 3.15. IfAandBsatisfy the same basic local sentences, then A≡B.

Proof (of Gaifman’s Theorem using the preceeding lemma). LetΦdenote the set of all basic local sentences. Letφbe an FO-sentence and letK= Mod(φ)be the class of models ofφ. ForA∈ Kwe define

Φ(A) ={φ:φ∈Φ,A|=φ} ∪ {¬φ:φ∈Φ,A|=¬φ}

Then for allA∈ Kwe haveΦ(A)|=φ, because ifB|=Φ(A), then AandBagree on all sentences fromΦand thus, by the preceeding lemma, we have thatA≡B. By the compactness theorem, we can find finite setsΦ0(A)⊆Φ(A)such thatΦ0(A)|=φfor allA∈ K.

We claim that for a finite subclass K0 ⊆ K, the sentence φ is equivalent to^WA∈K0

VΦ0(A) (which is a local sentence). We know that^WA∈K0

VΦ0(A) |= φ, so assume that for every finite subclass of structuresK0⊆ Kthe set{φ} ∪ {¬^VΦ₀(A):A∈ K0}would be satisfiable. Then, by compactness, also{φ} ∪ {¬^VΦ0(A):A∈ K}would be satisfiable which is impossible sinceA|=^VΦ0(A)for allA∈ K. q.e.d.

(8)

Proof (of Lemma 3.15). For all m ≥ 0, we prove by induction on j = m, . . . , 0 that one can find valuesg(0),g(1), . . . ,g(m)such that

Ij={a→b:|a|=|b|=m−j,(A↾^N⁷^j(a),a)≡g(j)(B↾^N⁷^j(b),b)} defines a back-and-forth system form-equivalence ofAandB. Sufficient criteria for the valuesg(0), . . . ,g(m)are collected in the course of the proof (and it will be obvious that we can find values which satisfy all contraints). Note thatIm={∅}.

Let 0≤j<mand leta→b∈Ij+1. Then we know that (A↾^N⁷^j+1(a),a)≡g(j+1)(B↾^N⁷^j+1(b),b).

By symmetry, it suffices to show thata→bhas a forth-extension inIj. Leta∈A. We have to findb∈Bsuch that

(A↾N⁷^j(aa),aa)≡_g(j)(B↾N⁷^j(bb),bb).

To this end we consider theg(j)-types of the 7^j-neighbourhoods of tuples inAandB. Recall from Lemma 3.3 that we can describe these types by a first-order formula. More precisely, for a structureDand a tupledinDwe set

ψ_d^j(x) =

χ^g(j)

(D↾N⁷^j(d),d)(x) N⁷^j(x)

.

Then ψ^j_d(x) is a 7^j-local formula such that C |= ψ^j_d(c) if the 7^j- neighbourhood ofcinC(with distinguished tuplec) isg(j)-equivalent to the 7^j-neighbourhood ofdinD(with distinguished tupled). To find an appropriateb∈Bwe distinguish between the following cases.

Case 1 (a is close to a).Assume thata∈N^2·7^j(a). Then (A↾^N⁷^j+1(a),a)|=∃z(d(a,z)≤2·7^j∧ψ^j_aa(a,z)).

We assume that the quantifier rank of this formula, which only depends onjandg(j), is at mostg(j+1)(this gives a first condition ong(j+1)).

52

3.3 Gaifman’s Theorem

But then, by our precondition, we can findb∈N^2·7^j(b)such that (B↾^N⁷^j(b))|=ψ_aa^j (b,b),

which implies thataa→bb∈Ij.

Case 2 (a is far from a). Assume that a ̸∈ N²^·⁷^j(a). Then the 7^j- neighbourhoods ofaandaare disjoint, i.e.N⁷^j(a)∩N⁷^j(a) =∅. Hence it suffices to find ab∈Bwhose 7^j-neighbourhood is disjoint with the 7^j-neighbourhood ofband such that the 7^j-neighbourhood ofainAand ofbinBhave the sameg(j)-type. Formally the requirements forb∈B are:

N⁷^j(b)∩N⁷^j(b) =∅ B↾^N⁷^j(b)|=ψ^ja(b).

Fors≥1 we define a formulaδs(x1, . . . ,xs)which expresses the existence of a 2·7^j-scattered tuple of elements whose 7^j-neighbourhood has the sameg(j)-type as the 7^j-neighbourhood ofainA:

δs(x₁, . . . ,xs) = ^{^}

ℓ̸=k

d(xℓ,x_k)>4·7^j∧^{^}

k

ψ^j_a(x_k).

We now determine the maximal lenght e of such tuples which are realised inAand the maximal lenghtiof such tuples which are realised inA↾N^2·7^j(a), that isiandeare determined such that

(A↾^N⁷^j+1^,^a)|=∃x1· · · ∃x_i(^{^}

k

d(a,x_k)≤2·7^j∧δ_i) (3.1) (A↾N⁷^j+1,a)̸|=∃x1· · · ∃xi+1(^{^}

k

d(a,xk)≤2·7^j∧δ_i+1) (3.2) A|=∃x1· · · ∃xeδe (3.3) A̸|=∃x1· · · ∃xe+1δ_e+1. (3.4) Of course,i≤e. Moreover,i≤m−j= |a|=|b|. We claim that the corresponding values determined inBare the same. For 3.1 and 3.2 we guarantee this by choosingg(j+1)large enough. Note that the quantifier rank of the formulas in 3.1 and 3.2 only depends onm(becauseiis

53

(9)

bounded bym),jandg(j)(we obtain a second condition ong(j+1)).

For 3.3 and 3.4 this follows since these are basic local sentences andA andBsatisfy the same basic local sentences by our assumption.

Case 2.1 (i=e). Then we claim thatall c∈Awhose 7^j-neighbourhood has the sameg(j)-type asaare contained inN⁶^·⁷^j(a). Indeed, we could extend each 2·7^j-scattered tuple of such elements inN^2·7^j(a)by each such elementc∈Awithd(a,c)>6·7^j. Sincea̸∈N^2·7^j(a)we have

(A↾^N⁷^j+1(a),a)|=∃z(2·7^j<d(a,z)≤6·7^j∧ψ^j_a(z)∧ψ_a^j(a)). We assume thatg(j+1)is larger than the quantifier rank of this formula (this gives a third condition ong(j+1)). Then by our assumption we have that

(B↾^N⁷^j+1(b),b)|=∃z(2·7^j<d(b,z)≤6·7^j∧ψ^ja(z)∧ψ_a^j(b)). This in turn shows that we can find an appropriateb∈B.

Case 2.2 (i< e). In this case we know thatB|=∃x1· · · ∃xi+1δ_i+1 which implies that we can findb∈Bsuch thatN⁷^j(b)∩N⁷^j(b) =∅and

such thatB|=ψ_a^j(b). q.e.d.

3.4 Lower bound for the size of local sentences

Gaifman’s Theorem states that for every FO-sentence there is an equivalent local one. In the following we show that the local sentence can be much longer than the original one, as captured by

Theorem 3.16.For everyh≥1 there is an FO(E)-sentenceφ_h∈ O(h⁴) such that every FO(E)-sentence in Gaifman normal form, i.e. every local sentence, that is equivalent toφ_hhas size at leastTower(h).

Here,Tower:N→^Nis the function defined byTower(0):=1 and Tower(n):=2^Tower(n−1)forn>0. In order to prove this theorem we first introduce and analyse an encoding of natural numbers by trees.

Definition 3.17.For natural numbersi,nwe writebit(i,n)to denote the i-th bit in the binary representation ofn, i.e., bit(i,n) =0 if⌊₂ⁿi⌋is even,

3.4 Lower bound for the size of local sentences and bit(i,n) = 1 if⌊₂ⁿi⌋is odd. Inductively we define a directed and rooted treeT(n)for each natural numbernas follows:

•T(0)is the one-node tree.

• Forn> 0 the treeT(n) is obtained by creating a new root and attaching to it all treesT(i)for allisuch that bit(i,n) =1.

The following figure illustrates these trees.

T(0) T(1) T(2) T(3) T(10) T(2²¹⁰)

It is straightforward to see that

for allh,n≥0, height(T(n))≤h ⇐⇒ n<Tower(h). Recall that the height of a tree is the length of its longest path.

For a graph G = (V,E) and some node v ∈ V, let Gv be the subgraph induced on the set of nodes reachable fromv. Now, we show that important properties of these tree encodings of natural numbers can be expressed by small FO(E)-formulas in the sense of the following three Lemmata.

Lemma 3.18. For eachh ≥ 0 there is a formulaeqh(x,y)∈FO(E)of lengthO(h)such that for all graphsG= (V,E)we have that: if there areu,v∈Vandm,n<Tower(h)withGu∼=T(n)andGv∼=T(m), then G|=eq_h(u,v)⇔n=m.

Proof. • Ifh=0, seteq_h(x,y):=true.

(10)

• Ifh>0,eq_h(x,y)has to be equivalent to

∀z(Exz→ ∃w(Eyw∧eqh−1(z,w)))∧

∀w(Eyw→ ∃z(Exz∧eq_h−1(z,w))).

The length of the formula we get by this recursive definition would be exponential inh. However, we can rewrite it as follows:

eqh(x,y):=(∃zExz↔ ∃wEyw)∧

∀z(Exz→ ∃w(Eyw∧ ∀w^′(Eyw^′→ ∃z^′(Exz^′∧

∀u∀v((u=z∧v=w)∨(u=z^′∧v=w^′)→ eq_h−1(u,v)))))).

q.e.d.

Lemma 3.19. Forh≥0 there is a formulacodeh(x)∈FO(E)of length O(h²)such that for all graphsG= (V,E)andv∈V:

G|=codeh(v) ⇐⇒ Gv∼=T(i)for somei<Tower(h). Proof. • Ifh=0, setcode_h(x):=¬∃yExy.

• Ifh>0, set

code_h(x):=∀y(Exy→code_h−1(y))∧

∀y1∀y2(Exy1∧Exy2∧eqh−1(y1,y2)→y1=y2). Observe that

∥codeh(x)∥=∥codeh−1(x)∥+∥eqh−1(x,y)∥+O(1)

≤c·(1+2+· · ·+h)for somec≥1, implying that∥codeh(x)∥ ∈ O(h²).

q.e.d.

Lemma 3.20.Forh≥0 there are formulas (1)bith(x,y)of lengthO(h),

(2)less_h(x,y)of lengthO(h²),

56

3.4 Lower bound for the size of local sentences (3)min(x)of lengthO(1),

(4)succ_h(x,y)of lengthO(h³), (5)maxh(x)of lengthO(h⁴),

such that for allG= (V,E)and nodesu,v∈VwithGu∼=T(m)and Gv∼=T(n), wherem,n<Tower(h):

Proof. (1)bith(x,y):=∃z(Eyz∧eqh(x,z)), (2) • Ifh=0, setless_h(x,y):=f alse.

• Ifh>0, set

lessh(x,y):=∃y^′(Eyy^′∧ ∀x^′(Exx^′→ ¬eqh−1(x^′,y^′))∧

∀x^′′(Exx^′′∧less_h−1(y^′,x^′′)→

∃y^′′(Eyy^′′∧eq_h−1(y^′′,x^′′))) (3)min(x):=¬∃yExy.

(4) • Ifh=0, setsucc_h(x,y):= f alse.

• Ifh>0, set

succ_h(x,y) =∃y^′(Eyy^′∧

∀y^′′(Eyy^′′∧y^′̸=y^′′→lessh−1(y^′,y^′′)∧

∀x^′(Exx^′→ ¬eq_h−1(x^′,y^′)∧

∀y^′′(Eyy^′′∧lessh−1(y^′,y^′′)→

∃x^′′(Exx^′′∧eq_h−1(y^′′,x^′′)))∧

∀x^′′(Exx^′′∧lessh−1(y^′,x^′′)→

∃y^′′(Eyy^′′∧eq_h−1(y^′′,x^′′)))∧

¬min(y^′)→(∃x^′(Exx^′∧min(x^′))∧

∀x^′(Exx^′∧less_h−1(x^′,y^′)→

∃z(succh−1(x^′,z)∧(z=y^′∨Exz)))).

57

(11)

3 Expressive Power of First-Order Logic (5) • Ifh=0, setmax_h(x):=¬∃yExy.

• Ifh>0, set

max_h(x):=∃y(Exy∧min(y))∧ ∀y(Exy→ (maxh−1(y)∨ ∃z(Exz∧succh−1(y,z))). This formula is correct sincex=Tower(h)−1=2^Tower(h−1)−1 implies thatT(Tower(h)−1)has a subtreeT(i)for anyi≤ Tower(h−1)−1.

q.e.d.

Finally, we use these three lemmata to prove a last lemma of which Theorem 3.16 is a corollary.

Lemma 3.21. For allh≥1 there is a formulaφ_h∈FO(E)with∥φ_h∥ ∈ O(h⁴)such that every local sentenceψwhich is equivalent toφ_hon the class of forests of height less or equal tohhas size∥ψ∥ ≥Tower(h). Proof. Let Fh be the forest consisting of all treesT(i) with 0 ≤ i <

Tower(h)and letF_h⁻ⁱbe the forest F_hwithout the tree T(i)for some 0≤i<Tower(h). Furthermore,root(x):=¬∃yEyx. Now, define

φ_h:=∃x(root(x)∧min(x))∧

∀x(root(x)∧ ¬maxh(x)→ ∃y(root(y)∧succh(x,y))). Observe that∥φ_h∥ ∈ O(h⁴)andF_h|=φ_has well asF_h⁻ⁱ̸|= φ_hfor each 0≤i<Tower(h).

Letψbe a local sentence which is equivalent toφhon the class of all forests of height less or equal toh. We want to show that∥ψ∥ ≥ Tower(h).

ψis a Boolean combination of basic local sentencesχ1, . . . ,χLwith χ_ℓ=∃x1. . .∃xkℓ(^{^}

i̸=j

d(xi,xj)>2·rℓ∧^{^}

i

ψ_ℓ^r^ℓ(xi)).

W.l.o.g. there is somem≤Lsuch thatFh|=χ_ℓfor allℓ≤mandFh̸|=χ_ℓ for allm< ℓ≤L. Hence we can find for allℓ≤mnodesuℓ,1, . . . ,uℓ,k_ℓ

inF_hsuch thatF_h|=d(u_ℓ,i,u_ℓ,j)>2·rℓ∧ψ^r_ℓ^ℓ(u_ℓ,i)for alli̸=j. The setU

3.4 Lower bound for the size of local sentences consisting of all these nodes contains at mostk1+· · ·+km≤ ∥ψ∥many nodes.

Towards a contradiction assume that∥ψ∥ < Tower(h). Since Fh

containsTower(h)many disjoint trees, there is at least onej<Tower(h) such thatT(j)inFhcontains noU-node. We claim thatF_h^−j|=ψ(which would yield the desired contradiction).

•F_h⁻^j |= χ_ℓ wherel ≤ m: the local properties around the nodes uℓ,1, . . . ,uℓ,k_ℓ also hold in F_h^−j since the neighbourhoods are not changed by removing the treeT(j).

•F_h^−j|=χ_ℓwherem< ℓ≤L: clear, sinceF_h^−jis a substructure ofFh. q.e.d.

3 Expressive Power of First-Order Logic

Algorithmic Model Theory SS 2016

Prof. Dr. Erich Grädel and Dr. Wied Pakusa

c b n d

Contents

3 Expressive Power of First-Order Logic

3.1 Ehrenfeucht-Fraïssé Theorem

3.2 Hanf’s technique

3.3 Gaifman’s Theorem

3.4 Lower bound for the size of local sentences