Ehrenfeucht–Fraïssé games in finite set theory

Information Processing Letters 108 (2008) 3–9 www.elsevier.com/locate/ipl

Ehrenfeucht–Fraïssé games in finite set theory ✩ Xiang Zhou a,b,∗ a Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing 100190, China b Graduate School of the Chinese Academy of Sciences, 19 Yuquan Street, Beijing 100049, China

Available online 20 March 2008 Communicated by J. Chomicki

Abstract In this article, with Ehrenfeucht–Fraïssé games we prove that Δ1 = Δ0 on BF R, which implies Δ = Δ0 on BF R, and thus solve an open problem raised by Albert Atserias in his dissertation (Δ0 , Δ1 , Δ are fragments of first order logic and BF R is a class of finite sets which in essence is equivalent to a class of finite pure arithmetic structures with built-in BIT predicate). © 2008 Elsevier B.V. All rights reserved. Keywords: Specification languages; Arithmetic predicates; Finite set theory; Fragments of first order logic; Inexpressibility; Ehrenfeucht–Fraïssé games

1. Introduction The expressive power of logics on finite structures with arithmetic predicates is a subject extensively studied in the literature. Many results have been obtained (for a survey see [13]). Particularly interesting are inexpressibility results for logics on finite structures with arithmetic predicates. In general, to prove inexpressibility results for logics on finite structures, Ehrenfeucht– Fraïssé games are main tools. There have been a lot of successful applications of games (see [6,10] for introductions and related references). But on finite structures with arithmetic predicates, using games to prove ✩

Supported by the National Natural Science Foundation of China under Grant No. 60573012 and 60721061, and the National Grand Fundamental Research 973 Program of China under Grant No. 2002cb312200. * Address for correspondence: Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing 100190, China. E-mail address: [email protected]. 0020-0190/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2008.03.011

inexpressibility results becomes difficult. In the cases of powerful predicates, there are few successful examples. Ruhl’s game for first order logic with counting on finite structures with + predicate [12] may be the most remarkable one. In this article, we add another example: using games for some fragments of first order logic on finite structures with BIT predicate, where BIT = {(i, m) | i, m ∈ N and m/2i  ≡ 1 (mod 2)}. BIT is a very powerful predicate. [8] (Theorem 1.17) and [7] show that on finite structures with BIT, we can define predicates + and × with first order logic. It is more interesting that although the definition of BIT is highly arithmetical, there is a surprising reinterpretation of its meaning in terms of hereditarily finite sets. Hereditarily finite sets Vω are defined as follows: V0 = ∅, Vk+1  = P(Vk ) (here P(X) is the power set of X), Vω = k∈ω Vk . There is a bijection e : N → Vω defined by the recursion: e(0) = ∅, e(m) = {e(i) | BIT(i, m)}. The bijection actually is an isomorphism between (N, BIT) and (Vω , ∈), well known as Ackerman isomorphism [4]. Let BIT = {(N , BIT N ) | N  0},

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where N = {0, . . . , N} and BIT N = BIT ∩ N 2 . BIT is one class of finite pure arithmetic structures with BIT. Under Ackerman bijection, BIT can be reinterpreted in finite set theory by BFR = {(PN , ∈) | N  0}, where PN = {e(0), e(1), . . . , e(N )}. The set theoretical framework suggests new problems to consider. In set theory, the basic language is first order logic over {∈}, FO{∈}. Several fragments of this language are important for their fundamental roles: Δ0 , Σ1 , Π1 , Δ1 , Σ, Π, Δ, where Δ0 , Σ1 , Π1 , Δ1 are in the low levels of famous Lévy’s hierarchy [9] of FO{∈} and Σ, Π, Δ are tightly related to Δ0 . It is natural for us to consider problems related to these fragments on BFR. Albert Atserias raised two problems in his dissertation [3]. One is “Prove that Δ = Δ0 on BFR using Ehrenfeucht–Fraïssé games”. For this problem he commented “The solution may give some insight on the following important question: how do we design winning strategies for the Duplicator in Ehrenfeucht–Fraïssé games when both arithmetic built-in predicates + and × are available?”. In this article, we prove that Δ1 = Δ0 on BFR, which implies Δ = Δ0 on BFR, with Ehrenfeucht–Fraïssé games and thus solve the problem. The remainder of the article is structured as follows: in Section 2, some notations are introduced; and in Section 3, Ehrenfeucht–Fraïssé games for Δ0 are defined and the characterization theorem is established; in Section 4, one query on BFR for separation is shown with Ehrenfeucht–Fraïssé games to be inexpressible in Δ0 ; finally in Section 5, some conclusions are given. 2. Preliminaries 2.1. Basic notations We use N = {0, 1, 2, . . .} for the set of natural numbers, and N>0 for the set of positive natural numbers. For N ∈ N we write N to denote the initial segment {0, . . . , N} of N. 2.2. Signature, structure, isomorphism, and query A signature is a set τ containing finitely many relation symbols. Each relation symbol R ∈ τ has a fixed arity ar(R) ∈ N>0 . A τ -structure A = (A, τ A ) consists of a set A which is called the universe of A and a set τ A that contains an interpretation R A ⊆ Aar(R) for each R ∈ τ . A τ -structure is finite if its universe is finite. An isomorphism π between two τ -structures A = (A, τ A ) and B = (B, τ B ) is a bijection π : A → B ¯ iff R B (π(a)) ¯ for each R ∈ π and all such that R A (a)

a¯ ∈ Aar(R) . Two structures A and B are isomorphic, denoted by A ∼ = B, if there is an isomorphism between them. Let p be a map with do(p) ⊆ A and rg(p) ⊆ B, where do(p) and rg(p) denote respectively the domain and the range of p. Then p is said to be a partial iso¯ morphism from A to B if p is injective and R A (a) ¯ for each R ∈ π and all a¯ ∈ (do(p))ar(R) . iff R B (π(a)) We write Part(A, B) for the set of partial isomorphisms from A to B. Let a¯ = a1 . . . as ∈ As and b¯ = b1 . . . bs ∈ B s , s  1, if the clause p(ai ) = bi for i = 1, . . . , s de¯ If this map fines a map, we denote this map as a¯ → b. is also a partial isomorphism from A to B, we write a¯ → b¯ ∈ Part(A, B). Let C be a class of τ -structures and r  1. An rary query on C is a mapping Q that associates each structure A ∈ C with a subset of Ar , such that Q is closed under isomorphism: if A ∼ = B via isomorphism π : A → B, then Q(B) = π(Q(A)). 2.3. Structures with arithmetic Arithmetic predicates are relations on N essentially, for example: – the ternary addition predicate +, consisting of all triples (x, y, z) such that x + y = z, – the binary BIT predicate, consisting of all tuples (x, y) such that the xth bit in the binary representation of x is 1, i.e., y/2x  is odd. When speaking of finite structures with arithmetic predicates we consider a set AP of arithmetic predicates. Furthermore, we consider arbitrary signature τ and all τ -structures whose universe is an initial segment of N. Given such a τ -structure A = (N , τ A ) we enrich A by the arithmetic predicates in AP. Thus we move to the (τ ∪ AP)-structures (N , τ A , AP N ), where AP N is the collection of the relation P N = P ∩ N ar(P ) , for all P ∈ AP. In addition, finite pure arithmetic structures mean that we restrict our attention to structures where the signature τ is empty. Using Ackerman isomorphism, every finite τ -structure with BIT predicate can be reinterpreted in finite set theory and thus can be regarded as a τ ∪ {∈}-structure. BIT is a class of finite pure arithmetic structures. It can be replaced by BFR and thus regarded as a class of finite {∈}-structures. 2.4. First order logic Let τ , σ be signatures and σ = τ ∪ {∈}. We assume a countably infinite set of variables, denoted by x, y, z, . . .

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with subscripts. σ -terms are variables. Atomic σ -formulas are formulas of the form R(t1 , . . . , tm ), where R ∈ σ is of arity m and t1 , . . . , tm are σ -terms. First order σ -formulas, FO(σ )-formulas, are built up as usual from atomic σ -formulas with the logical connectives ∨, ∧, ¬ and the existential quantification ∃x and the universal quantification ∀x. As usual, we use the standard shorthand ϕ → ψ for ¬ϕ ∨ ψ and ϕ ↔ ψ for (ϕ → ψ) ∧ (ψ → ϕ). The collection of Δ0 -formulas of σ is the smallest class of formulas that contains all atomic σ -formulas and is closed under ∨, ∧, ¬ and bounded quantification: If ϕ is a Δ0 -formula, then (∃x ∈ y)ϕ and (∀x ∈ y)ϕ are also Δ0 -formulas, in which, (∃x ∈ y)ϕ stands for ∃x(x ∈ y ∧ ϕ), and (∀x ∈ y)ϕ for ∀x(x ∈ y → ϕ). Σ1 is the collection of formulas of the form ∃x1 · · · ∃xm ψ, ψ ∈ Δ0 . Π1 is the collection of formulas of the form ∀y1 · · · ∀yn θ, θ ∈ Δ0 . Σ is the smallest collection of formulas, containing the Δ0 -formulas and closed under ∨, ∧, bounded quantification (∃x ∈ y), (∀x ∈ y) and existential quantification ∃x. Π is defined dually with Σ by allowing closure under universal quantification ∀x. With free(ϕ) we denote the set of all variables that occur free in ϕ. In bounded quantification ∃x ∈ y or ∀x ∈ y, only x is bounded. We write ϕ(x1 , . . . , xm ) to indicate that free(ϕ) ⊆ {x1 , . . . , xm }. Let C be a class of σ -structures, let Q be an r-ary query on C, and L(σ ) be a class of formulas of σ , where L ∈ {Δ0 , Σ1 , Π1 , Σ, Π, FO}. We say that Q is L-definable on C if there is L(σ )-formula ϕ(x1 , . . . , xr ) such that Q(A) = {(a1 , . . . , ar ) ∈ Ar |A |= ϕ(a1 , . . . , ar )} for every A ∈ C. A query Q is Δ1 (Δ)-definable on C if it is Σ1 (Σ)-definable and Π1 (Π)-definable on C. Let L1 , L2 ∈ {Δ0 , Σ1 , Π1 , Δ1 , Σ, Π, Δ, FO} and C be a class of σ -structures. L1 ⊆ L2 on C if every L1 definable query on C is also L2 -definable. L1 = L2 on C if L1 ⊆ L2 and L2 ⊆ L1 on C. L1 = L2 on C if there is a query on C which is L1 -definable but not L2 definable or conversely. 3. Ehrenfeucht–Fraïssé games for Δ0 In general, to prove inexpressibility results for logics on finite structures, various Ehrenfeucht–Fraïssé games are the main tools. Especially for fragments of FO, variants of games have been well-studied in the community of modal logic, although there they usually appear in the forms of various bisimulations (see [5] for an introduction). Δ0 is a fragment of FO, and in fact it is in the third form of bounded fragments of FO, introduced by Andréka et al. in [1]. Meanwhile Δ0 has its own partic-

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ularity for its form of bounded quantifications: ∃x ∈ y and ∀x ∈ y. Thus we should adapt games in [1] for Δ0 . We start with a description of Ehrenfeucht–Fraïssé games for Δ0 . Assume σ be a signature containing ∈, let A = (A, σ A ) and B = (B, σ B ) be σ -structures, a¯ ∈ As , Δ ¯ is ¯ B, b) b¯ ∈ B s , s  1 and m ∈ N. The game Gm0 (A, a, played by two players called the spoiler and the duplicator: • For m = 0, if a¯ → b¯ ∈ Part(A, B), then the duplicator wins the game, otherwise the spoiler wins. • For m > 0, whether a¯ → b¯ is a partial isomorphism is an initial condition. ◦ a¯ → b¯ ∈ / Part(A, B): The spoiler wins the game. ◦ a¯ → b¯ ∈ Part(A, B): The game has many possible plays. Every play has at most m rounds. Assume a play has finished ith round, i < m, and yet no player wins the play (when i = 0, we regard no one wins the play), thus an (s + i)-tuple v¯ in A is the distinguished output: vj = aj , for 1  j  s, and vj = ek for j = s + k, 1  k  i, where ek is selected by one of the players in kth round; similarly a corresponding tuple w¯ exists in B. Then the (i + 1)th round of the play consists of the following steps: (i) The spoiler tries to select a structure (e.g., A), and one element in this structure which is a ∈-predecessor of some element in the distinguished tuple of this structure (e.g., an element ei+1 in A which is a ∈-predecessor of the qth element in v: ¯ ei+1 ∈A vq , 1  q  s + i). If the spoiler fails to make such selection, then the play ends and the duplicator wins the play, otherwise the round of this play continues into step (ii). (ii) The duplicator tries to respond in the other structure (e.g., B). He has to select one element in that structure which is a ∈predecessor of the element on the same position of the corresponding tuple (e.g., an element fi+1 in B such that fi+1 ∈B wq ). And what is more, the two new tuples must still form a partial isomorphism (e.g., ve ¯ i+1 → wf ¯ i+1 ∈ Part(A, B)). If the duplicator cannot respond as required above, then the play ends and the spoiler wins the play. Otherwise, if i + 1 = m, the play ends and the duplicator wins the play, or if i + 1 < m, the play moves into next round with the two new tuples.

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In the case of the game that there are many possible plays, we say that a player, the duplicator or the spoiler, Δ ¯ if it is possible for him wins the game Gm0 (A, a, ¯ B, b), to win each play whatever choices are made by his opponent. Now we have finished the description of games for Δ0 . It is time to understand why these games can be used to prove inexpressibility results of Δ0 . We need a definition of quantifier rank firstly. Definition 1. Let σ be a signature containing ∈. The quantifier rank of a Δ0 (σ )-formula, qr(ϕ), is its depth of bounded quantifier nesting. That is: • • • •

qr(ϕ) = 0, if ϕ is atomic, qr(ϕ1 ∨ ϕ2 ) = qr(ϕ1 ∧ ϕ2 ) = max{qr(ϕ1 ), qr(ϕ2 )}, qr(¬ϕ) = qr(ϕ), qr((∃x ∈ y)ϕ) = qr((∀x ∈ y)ϕ) = qr(ϕ) + 1.

The following lemma is the base case of our characterization theorem for Δ0 . Lemma 2. Let σ be a signature containing ∈. Given two σ -structures A and B, a¯ ∈ As and b¯ ∈ B s , for s  1, the following are equivalent: ¯ (i) The duplicator wins G0 0 (A, a, ¯ B, b); ¯ (ii) a¯ → b ∈ Part(A, B); (iii) For every quantifier free formula ϕ(x1 , . . . , xs ): ¯ A |= ϕ[a] ¯ iff B |= ϕ[b]; (iv) For every atomic formula ϕ(x1 , . . . , xs ): A |= ϕ[a] ¯ ¯ iff B |= ϕ[b]. Δ

Proof. (i) ⇔ (ii) according to the description of the game; (ii) ⇔ (iv) according to the definition of partial isomorphism; (iii) ⇔ (iv) since every quantifier free formula is a boolean combination of atomic formulas. 2 Here is our characterization theorem for Δ0 . Theorem 3. Let σ be a signature containing ∈. Given m  0 and two σ -structures A and B, a¯ ∈ As and b¯ ∈ B s for s  1, the following are equivalent: Δ ¯ ¯ B, b). (i) The duplicator wins Gm0 (A, a, ¯ (ii) a¯ and b satisfy the same Δ0 -formulas of quantifier rank  m, that is, if ϕ(x1 , . . . , xs ) is a Δ0 -formula of quantifier rank  m, then

A |= ϕ[a] ¯

¯ iff B |= ϕ[b].

(∗)

Proof. (i) ⇒ (ii) The proof proceeds by induction on m. The case m = 0 holds according to Lemma 2. Let m > 0 Δ ¯ ¯ B, b). and suppose that the duplicator wins Gm0 (A, a, Clearly, the set of formulas ϕ(x1 , . . . , xs ) satisfying (∗) is closed under ¬, ∨ and ∧. Thus it suffices to show that a¯ and b¯ satisfy the same Δ0 formulas of the form ¯ y), 1  i  s, and qr(ϕ)  m. ϕ(x) ¯ = (∃y ∈ xi )ψ(x, Assume, for instance, A |= ϕ[a]. ¯ Then there is a in A ¯ a]. As, by (i), the dusuch that a ∈A ai and A |= ψ[a, Δ ¯ there is b in B such that plicator wins Gm0 (A, a, ¯ B, b), Δ0 ¯ b ∈B bi and the duplicator wins Gm−1 (A, aa, ¯ B, bb). Since qr(ψ)  m − 1, the induction hypothesis yields ¯ b], hence B |= ϕ[b]. ¯ The proof of the conB |= ψ[b, ¯ verse, that B |= ϕ[b] implies A |= ϕ[a], ¯ is similar. (ii) ⇒ (i) The proof proceeds by induction on m too. The case m = 0 holds according to Lemma 2. Let m > 0 and suppose that a¯ and b¯ satisfy the same Δ0 formulas of quantifier rank  m. We explain how the duplicaΔ ¯ Obviously the tor can win the game Gm0 (A, a, ¯ B, b). initial condition a¯ → b¯ ∈ Part(A, B) is satisfied. If the spoiler cannot make any selection, then the duplicator wins the game. Otherwise, assume the spoiler selects an element a in A such that a ∈A ai for some i, 1  i  s. ¯ a] and qr(θ )  Set Ψ = {θ (x1 , . . . , xs , y) | A |=θ [a, ¯ It is easy to m − 1}. Clearly A |= (∃y ∈ xi )( Ψ )[a]. verify that for any r  1 and n  0 there are only finitely many non-equivalent Δ0formulas ψ(x1 , . . . , xr ) of quantifier rank  n. Thus Ψ is actually equivalent to a finite conjunction which has quantifier rank   ¯ also m − 1. Then, by (ii), B |= (∃y ∈ xi )( Ψ )[b] holds. This implies there is b in B  such that b ∈B bi ¯ b]. Since B |= ( Ψ )[b, ¯ b] means and B |= ( Ψ )[b, ¯ that aa ¯ and bb satisfy the same Δ0 formulas of quantifier rank  m − 1, the induction hypothesis yields the Δ0 ¯ duplicator wins Gm−1 (A, aa, ¯ B, bb). So if the duplicator select b and then follow his winning strategy in Δ0 ¯ he is guaranteed to win. Similarly Gm−1 (A, aa, ¯ B, bb), in the case that the spoiler makes his selection in B. 2 The characterization theorem offers methods for us to prove inexpressibility results of Δ0 . Corollary 4. Let σ be a signature containing ∈, C be a class of σ -structures and Q be r-ary query on C, r  1. Then Q is not Δ0 -definable on C if for every m ∈ N, there exist two structures in C, Am and Bm , and two r-tuples a¯ and b¯ in them such that: ¯ • the duplicator wins Gm0 (Am , a, ¯ Bm , b), / Q(Bm ). • a¯ ∈ Q(Am ) and b¯ ∈ Δ

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Proof. By contradiction. Suppose Q is Δ0 -definable, that is to say, there is a Δ0 -formula ϕ(x1 , . . . , xr ) such ¯ for every M that Q(M) = {(e1 , . . . , er ) | M |= ϕ[e]} in C. Set m = qr(ϕ). According to the condition, there are two structures in C, Am and Bm and two tuples a¯ and b¯ in them such that a¯ ∈ Q(A) and b¯ ∈ / Q(B) and the Δ ¯ b¯ ∈ ¯ Bm , b). / Q(Bm ) imduplicator wins Gm0 (Am , a, Δ ¯ plies Bm |= ϕ[b]. Since the duplicator wins Gm0 (Am , ¯ by Theorem 3, a¯ and b¯ satisfy the same a, ¯ Bm , b), Δ0 formulas of quantifier rank  m, specially, Am |= ¯ a¯ ∈ Q(Bm ) implies Am |= ϕ[a], ¯ ϕ[a] ¯ iff Bm |= ϕ[b]. ¯ Now we have a contradiction. 2 and thus Bm |= ϕ[b].

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nite ordinals and α + β = γ }, for every M ∈ BFR. Qadd is Δ1 -definable, and its Δ0 -inexpressibility is implied by the Δ0 -inexpressibility of another query Qord even , where Qord (M) = {e | e ∈ M and e is a finite ordinal even and the cardinality of e is even} for every M ∈ BFR. The following proposition gives proofs for these two facts. Proposition 6. (i) Qadd is Δ1 -definable on BFR; (ii) if Qord even is not Δ0 -definable on BFR, then Qadd is also not Δ0 -definable on BFR.

4. Main results It is obvious that Δ0 ⊆ Δ1 ⊆ Δ on BFR, according to their definitions. We are interested in whether these inclusion relations are proper. Atserias has established that Δ0 ⊂ Δ on BFR in his dissertation [3]. There he constructed one Δ-definable query Qeven on BFR and proved that Qeven is not Δ0 -definable using the wellknown result of Furst, Saxe and Sipser [11] that AC0 circuits cannot compute the parity of its input. Obviously the use of the circuit lower bound to prove that Δ = Δ0 is an overkill, so Atserias raised the problem that “Prove that Δ = Δ0 on BFR using Ehrenfeucht– Fraïssé games”. Here using the games we have developed for Δ0 , we prove that Δ1 = Δ0 on BFR, which implies Δ = Δ0 on BFR. Our proofs use some special sets in Vω : finite ordinals, which are regarded as the standard encoding of natural numbers in set theory. There is a map h : N → Vω defined by the recursion: h(0) = ∅, h(n + 1) = h(n) ∪ {h(n)}. Then the image of h is the set of all finite ordinals. The ordinal addition is defined as: α + β = γ , if there is n1 , n2 , n3 ∈ N such that n1 + n2 = n3 and h(n1 ) = α, h(n2 ) = β, h(n3 ) = γ . Some properties of finite ordinals are: Proposition 5. (i) Every element of a finite ordinal is also a finite ordinal; (ii) For any finite ordinals α and β, if α ∈ β, then α ⊂ β; (iii) For any finite ordinals β and γ , one and only one of β ∈ γ , γ ∈ β and β = γ holds. Proof. According to the definition of h.

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We define a query Qadd for separating Δ1 from Δ0 : Qadd (M) = {(α, β, γ ) | α, β, γ ∈ M and α, β, γ are fi-

Proof. For (i), we refer to Example 1 and Theorem 2 in [2]; For (ii), we can equivalently show if Qadd is Δ0 definable on BFR then Qord even is also Δ0 -definable on BFR. Suppose ϕ is a Δ0 -definition of Qadd , then Qord even can be defined by a Δ0 -formula: ∀z ∈ x(z = z) ∨ ∃z ∈ xϕ(z, z, x). 2 The next proposition discusses some conditions for the duplicator to win games for Δ0 . In essence, these conditions come from the games for first order logic on linear orders (Theorem 3.6 in [10] or Example 2.3.6 in [6]). Proposition 7. Let m > 0 and a = h(n1 ), b = h(n2 ) such that n1 , n2  2m and A, B be structures in BFR satisfying that a is in A and b is in B. Then the dupliΔ cator wins the game Gm0 (A, a, B, b). Δ

Proof. For convenience we consider the game Gm0 (A, ∅a, B, ∅b). This does not matter since if the duplicaΔ tor can win Gm0 (A, ∅a, B, ∅b), then he also can win Δ0 Gm (A, a, B, b). We define a distance function d between any two finite ordinals by d(α, β) := |l − n|, where α = h(l) and β = h(n). And, for j  0, we introduce the “truncated” j -distance function dj on finite ordinals by  j dj (α, β) := d(α, β) if d(α, β) < 2 , ∞ else. Now, for j  m, set Ij := {p ∈ Part(A, B)| do(p), rg(p) contain only finite ordinals and p(∅) = ∅, p(a) = b and ∀η ∈ do(p)(η = a ∨ η ∈ a) and ∀ζ ∈ rg(p)(ζ = b ∨ ζ ∈ b) and dj (α, β) = dj (p(α), p(β)) for α, β ∈ do(p)}. According to Proposition 5, we know that Ij is well defined. Trivially ∅a → ∅b ∈ Im . Thus if (Ij )j m satisfies the following properties:

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(1) (Forth property) For every j < m, p ∈ Ij +1 , and α  ∈ do(p), α in A such that α ∈ α  , there is q ∈ Ij such that q ⊇ p and α ∈ do(q), (2) (Back property) For every j < m, p ∈ Ij +1 , and β  ∈ rg(p), β in B such that β ∈ β  , there is q ∈ Ij such that q ⊇ p and β ∈ rg(q) Δ

then the duplicator has a winning strategy for Gm0 (A, ∅a, B, ∅b) according to (Ij )j m . Now we try to verify that (Ij )j m satisfies the forth and back properties. We only give a proof of the forth property of (Ij )j m (the back property can be proven analogously). Suppose j < m, p ∈ Ij +1 , and α  ∈ do(p), α in B such that α ∈ α  . If α ∈ do(p), then p is just the required partial isomorphism since p is also in Ij . Otherwise, according to Proposition 5, we can list α  and its ∈-predecessors in do(p) as follows: ∅ = η1 ∈ η2 . . . ηr−1 ∈ ηr = α  , and then α must lie in one interval: ηi ∈ α ∈ ηi+1 for some i. Since p is a partial isomorphism, we also know that the interval between p(ηi ) and p(ηi+1 ) contains no elements of rg(p). To look for one suitable partial isomorphism q, there are two cases to consider: • d(ηi , ηi+1 ) < 2j +1 . Then d(ηi , ηi+1 ) = d(p(ηi ), p(ηi+1 )). Thus we simply find one finite ordinal β in B such that d(ηi , α) = d(p(ηi ), β) and d(α, ηi+1 ) = d(β, p(ηi+1 )). Clearly, q = p ∪ {(α, β)} is the required partial isomorphism. • d(ηi , ηi+1 )  2j +1 . In this case d(p(ηi ), p(ηi+1 ))  2j +1 . We have three possibilities: 1. d(ηi , α) < 2j . Then d(α, ηi+1 )  2j , and we can choose one finite ordinal β in B such that d(ηi , α) = d(p(ηi ), β) and d(β, p(ηi+1 ))  2j . Then q = p ∪ {(α, β)} is the required partial isomorphism. 2. d(α, ηi+1 ) < 2j . This case is similar to the previous one. 3. d(ηi , α)  2j , d(α, ηi+1 )  2j . Since d(p(ηi ), p(ηi+1 ))  2j +1 , by choosing the finite ordinal β in B to be in the middle of the interval between p(ηi ) and p(ηi+1 ), we ensure that d(p(ηi ), β)  2j and d(β, p(ηi+1 ))  2j . Thus q = p ∪ {(α, β)} is the required partial isomorphism. 2 The fact that Qord even is not Δ0 -definable on BFR is established in the following proposition. Proposition 8. Qord even is not Δ0 -definable on BFR.

Proof. For m = 0, let a = h(0), b = h(1); For m > 0, let a = h(2m ), b = h(2m + 1). For every m, let Am , Bm be any structures in BFR satisfying that a is in Am and b is in Bm . Obviously a ∈ Qord / even (Am ) and b ∈ Qord (B ) for every m. Moreover, the duplicator wins m even Δ the game Gm0 (Am , a, Bm , b) for every m according to the definition of games and Proposition 7. Thus from Corollary 4, we can conclude that Qord even is not Δ0 -definable on BFR. 2 Finally, the results we obtained in this article are Theorem 9. Δ0 ⊂ Δ1 ⊆ Δ on BFR. Proof. From Proposition 8 and Proposition 6(ii), we know that Qadd is not Δ0 -definable, whilst Proposition 6(i) tells us that Qadd is Δ1 -definable, thus Δ0 ⊂ Δ1 on BFR. Δ1 ⊆ Δ on BFR is trivial according to their definitions. 2 5. Conclusions With Ehrenfeucht–Fraïsse games for Δ0 and the query Qadd , we show that Δ1 = Δ0 on BFR. There are some questions remain. One is “Is Δ1 = Δ on BFR?”. There is another question in Atserias’s dissertation “Is Σ1 = Σ on BFR?”. Whether positive or negative answers to these two questions may have some important implications as remarked by Atserias in [3]. Thus to solve these questions, we need more insights and more powerful techniques. References [1] H. Andréka, J. van Benthem, I. Németi, Modal language and bounded fragments of predicate logic, Journal of Philosophical Logic 27 (1998) 217–274. [2] A. Atserias, Ph.G. Kolaitis, First-order logic vs. fixed-point logic in finite set theory, in: 14th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 275–284. [3] A. Atserias, Fixed-point logics, descriptive complexity, and random satisfiability, PhD thesis, University of California, Santa Cruz, Department of Computer Science, 2002. [4] J. Barwise, Admissible Sets and Structures, Springer-Verlag, 1975. [5] P. Blackburn, M. de Rijke, Y. Venema, Modal Logic, Cambridge University Press, 2001. [6] H.D. Ebbinghaus, J. Flum, Finite Model Theory, second ed., Springer-Verlag, 1999. [7] A. Dawar, K. Doets, S. Lindell, S. Weinstein, Elementary properties of finite ranks, Mathematical Logic Quarterly 44 (1998) 349–353. [8] N. Immerman, Descriptive Complexity, Springer-Verlag, 1999.

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[9] A. Lévy, A Hierarchy of Formulas in Set Theory, Memories Amer. Math. Soc., vol. 57, Amer. Math. Soc., 1965. [10] L. Libkin, Elements of Finite Model Theory, Springer-Verlag, 2004. [11] M. Furst, J. Saxe, M. Sipser, Parity, circuits and the polynomial time hierarchy, Mathematical Systems Theory 17 (1984) 13–27.

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[12] M. Ruhl, Counting and addition cannot express deterministic transitive closure, in: 14th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 326–334. [13] N. Schweikardt, Arithmetic, first-order logic, and counting quantifiers, ACM Transactions on Computational Logic 6 (3) (2005) 634–671.