Hanf normal form for first-order logic with unary counting quantifiers

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1 Hanf normal form for first-order logic with unary counting quantifiers Lucas Heimberg Humboldt-Universität zu Berlin Dietrich Kuske Technische Universität Ilmenau Nicole Schweikardt Humboldt-Universität zu Berlin Abstract We study the existence of Hanf normal forms for extensions FO(Q of first-order logic by sets Q P(N of unary counting quantifiers. A formula is in Hanf normal form if it is a Boolean combination of formulas ζ(x describing the isomorphism type of a local neighbourhood around its free variables x and statements of the form the number of witnesses y of ψ(y belongs to (Q+k where Q Q, k N, and ψ describes the isomorphism type of a local neighbourhood around its unique free variable y. We show that a formula from FO(Q can be transformed into a formula in Hanf normal form that is equivalent on all structures of degree d if, and only if, all counting quantifiers occurring in the formula are ultimately periodic. This transformation can be carried out in worst-case optimal 3-fold exponential time. In particular, this yields an algorithmic version of Nurmonen s extension of Hanf s theorem for first-order logic with modulocounting quantifiers. As an immediate consequence, we obtain that on finite structures of degree d, model checking of first-order logic with modulo-counting quantifiers is fixed-parameter tractable. Categories and Subject Descriptors F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic Computational Logic, Model Theory General Terms Theory, Algorithms Keywords Extensions of first-order logic, Hanf locality, modulocounting quantifiers, unary counting quantifiers, ultimately periodic sets, structures of bounded degree, normal forms, model checking, elementary algorithms 1. Introduction Two elements of a given graph are indistinguishable by first-order formulas whenever some automorphism maps the first to the second. More generally, if the two elements have isomorphic neighbourhoods of radius 4 q, then they cannot be distinguished by first-order formulas of quantifier depth q. This and similar phenomena are summarized under the slogan first-order logic can only express local properties and formalized by the theorems by Hanf, by Gaifman, and by Schwentick and Barthelmann [5, 8, 9, 19]. All these results give rise to normal forms for first-order formulas. Hanf s and Gaifman s theorem have found various applications in algorithms and Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, contact the Owner/Author(s. Request permissions from permissions@acm.org or Publications Dept., ACM, Inc., fax +1 ( LICS 16, July 05-08, 2016, New York, NY, USA Copyright c 2016 held by owner/author(s. Publication rights licensed to ACM. ACM /16/07... $15.00 DOI: complexity (cf., e.g., [12, 13]. In particular, there are very general algorithmic meta-theorems stating that first-order model checking is fixed-parameter tractable for various classes of structures [7, 20], and that the results of first-order queries against various classes of databases can be enumerated with constant delay after a linear-time preprocessing phase [3, 11, 21]. In the context of such algorithms, questions about the efficiency of the normal forms have recently attracted interest (cf. e.g., [1, 2, 14]. Notions of locality have also been developed for extensions of first-order logic, and they have found application in proving inexpressibility results for these logics (cf., e.g., [10, 13, 18]. When restricting attention to classes of finite structures of bounded degree, these locality notions also give rise to normal forms for the respective logics. Let us focus on the particular case of Hanflocality: Hanf s locality theorem for first-order logic implies that for every first-order sentence ϕ over a finite relational signature σ, and for every degree bound d N, there exists a first-order sentence ψ that is equivalent to ϕ on all finite σ-structures of degree d, such that ψ is a Boolean combination of statements of the form the number of elements x whose r-neighbourhood has isomorphism type τ is k. Such a formula ψ is said to be in Hanf normal form. A worst-case optimal algorithm for constructing ψ when given ϕ and d has been developed in [1]. In [18], Nurmonen extended Hanf s locality theorem to the extension of first-order logic by modulo-counting quantifiers D p (for positive integers p, where a formula of the form D p y ψ(x, y states that the number of witnesses y for ψ(x, y is divisible by p. As an easy consequence of Nurmonen s theorem, one obtains that for every sentence ϕ of first-order logic with modulo-counting quantifiers, and for every degree bound d N there exists a firstorder sentence with modulo-counting quantifiers ψ that is equivalent to ϕ on all finite structures of degree d, such that ψ is a Boolean combination of statements of the form the number of elements x whose r-neighbourhood has isomorphism type τ is congruent k modulo p and statements of the form the number of elements x whose r-neighbourhood has isomorphism type τ is k. Again, we say that ψ is in Hanf normal form. For algorithmic applications, an effective procedure for computing ψ when given ϕ and d, would be highly desirable (cf., e.g., the use of Nurmonen s theorem in the full version of [17]. The proof of [18], however, does not lead to such an effective procedure. The two main questions which started the research whose results are presented in this paper are (1 Is there an algorithmic version of Nurmonen s result? (2 For which classes of unary counting quantifiers does an analogue of Nurmonen s result hold? Answering question (2, our first main result provides a precise characterisation: A class Q of unary counting quantifiers permits

2 Hanf normal forms (analogous to the ones obtained from Nurmonen s result if, and only if, all counting quantifiers in Q are ultimately periodic. Answering question (1, our second main result provides an algorithm which, when given a degree bound d and a formula ϕ of the extension of first-order logic with ultimately periodic unary counting quantifiers, transforms ϕ into a corresponding Hanf normal form ψ which is equivalent to ϕ on all structures of degree d. This algorithm uses 3-fold exponential time and is worst-case optimal. As an easy application of our algorithm, we obtain that Seese s and Frick/Grohe s [7, 20] fixed-parameter tractability result for first-order model checking on structures of degree d can be generalised to first-order logic with ultimately periodic unary counting quantifiers. The rest of the paper is structured as follows. Section 2 fixes basic notations used throughout the paper. Section 3 gives precise statements of our two main results. Sections 4 and 5 are devoted to the proof of the only if -direction and the if -direction, respectively, of our characterisation of the sets of unary counting quantifiers that permit Hanf normal forms. Section 6 contains the runtime analysis of our algorithm for transforming a given formula into Hanf normal form. Section 7 shows how to use our algorithm to achieve fixed-parameter tractability of the model checking problem. Section 8 concludes the paper and points out directions for future work. 2. Preliminaries We write P(S to denote the power set of a set S. We write N for the set of non-negative integers, and we let N 1 := N \ {0}. For all m, n N with m n, we write [m, n] for the set {i N : m i n}, and we let [m, n := [m, n] \ {n}. For a real number r > 0, we write log(r to denote the logarithm of r with respect to base 2. We say that a function f from N to the set R 0 of non-negative reals is at most k-fold exponential, for some k N, if there exists a number c > 0 such that for all sufficiently large n N we have f(n T (k, n c, where T (0, m = m and T (k+1, m = 2 T (k,m for all k, m 0 (i.e., T (k, m is a tower of 2s of height k with an m on top. For a finite word w {0, 1}, we write w to denote the length of w. For an ω-word w = w 0w 1w 2 {0, 1} ω and a number n N, we write w[n] to denote the letter w n in w at position n. For numbers i, j N with i j, we write w[i, j] for the (finite word w iw i+1 w j. Similarly, we write w(i, j] for the (finite word w i+1 w j. In particular, w(i, i] is the empty word ɛ, and w(j 1, j] = w[j]. Structures and formulas. A signature σ is a finite set of relation symbols and constant symbols. Associated with every relation symbol R is a positive integer ar(r called the arity of R. The size σ of a signature σ is the number of its constant symbols plus the sum of the arities of its relation symbols. We call a signature relational if it only contains relation symbols. A σ-structure A consists of a finite non-empty set A called the universe of A, a relation R A A ar(r for each relation symbol R σ, and an element c A A for each constant symbol c σ. Note that according to this definition, all signatures and all structures considered in this paper are finite. To indicate that two σ-structures A and B are isomorphic, we write A = B. We use the standard notation concerning first-order logic and extensions thereof, cf. [4, 13]. By FO[σ] we denote the class of all first-order formulas of signature σ, and by FO we denote the union of all FO[σ] for arbitrary signatures σ. By free(ϕ we denote the set of all free variables of ϕ. A sentence is a formula ϕ with free(ϕ =. We write ϕ(x, for x = (x 1,..., x n with n 0, to indicate that free(ϕ {x 1,..., x n}. If A is a σ-structure and a = (a 1,..., a n A n, we write A = ϕ[a] to indicate that the formula ϕ(x is satisfied in A when interpreting the free occurrences of the variables x 1,..., x n with the elements a 1,..., a n. For a class C of σ-structures, two formulas ϕ(x and ψ(x of signature σ are called equivalent on C (for short: C-equivalent if for all σ-structures A C and for all a A n we have A = ϕ[a] A = ψ[a]. Unary counting quantifiers. All quantifiers considered in this article are unary counting quantifiers (for short: quantifiers, i.e., subsets of N. We will use the terms set (of natural numbers and quantifier interchangeably. For a quantifier Q N and a formula ϕ(x, y over a signature σ, the formula Qy ϕ(x, y is satisfied by a σ-structure A and an interpretation a of the variables x if {b A : A = ϕ[a, b]} Q. For a set Q P(N of quantifiers and a relational signature σ, we write FO(Q[σ] to denote the extension of FO[σ] with the quantifiers from Q. That is, FO(Q[σ] is built from atomic formulas of the form x 1=x 2 and R(x 1,..., x ar(r, for R σ and variables x 1, x 2,..., x ar(r, and closed under Boolean connectives,, existential first-order quantifiers x, and unary counting quantifiers Q x, for any Q Q and any variable x. 1 By FO(Q we denote the union of all FO(Q[σ] for arbitrary relational signatures σ. The quantifier rank qr(ϕ of an FO(Q-formula ϕ is defined as the maximal nesting depth of all quantifiers. Since we only consider finite structures, the classical quantifier and the unary counting quantifier N 1 are equivalent. Thus, to avoid unnecessary special cases in proofs, we will often tacitly assume that the existential quantifier = N 1 is contained in the sets Q considered. For a number k 0 we write (Q+ky ϕ(x, y as a short hand for a formula expressing in a σ-structure A and for an interpretation a of the variables x that the number of elements b A such that A = ϕ[a, b] belongs to the set (Q+k := {n+k : n Q}. Clearly, (Q+ky ϕ(x, y can be expressed by the formula ( y 1 y k y i=y j 1 i<j k y ( y=y i ϕ(x, y 1 i k Qy ( ϕ(x, y y=y i. 1 i k For every k 1, we will write k y ϕ and =k y ϕ for the formulas ( +(k 1y ϕ and k y ϕ k+1 y ϕ, respectively. The displacement of a formula ψ is the smallest K 0 such that for every subformula of ψ of shape (Q+ky ϕ with Q N and k 0 it holds that k K. It is the aim of this paper to study the locality of the logics FO(Q in the sense of Hanf s theorem [4, 5, 9]. To define the according locality notion for this logic, we need the concepts introduced in the remainder of this section. Gaifman graph. Let A be a σ-structure. Its Gaifman graph G A is the undirected graph with vertex set A and an edge between two distinct vertices a, b A iff there exists R σ and a tuple (a 1,..., a ar(r R A such that a, b {a 1,..., a ar(r }. 1 As usual, x,,, will be used as abbreviations when constructing formulas. (1

3 The distance dist A (a, b between two elements a, b A is the minimal length (i.e., the number of edges of a path from a to b in G A (if no such path exists, we set dist A (a, b =. For r 0 and a A, the r-neighbourhood of a in A is the set Nr A (a := {b A : dist A (a, b r}. For a tuple a = (a 1,..., a n A n, we write Nr A (a for the union of the sets Nr A (a i for all i [1, n]. Types and spheres. Let σ be a relational signature and let c 1, c 2,... be a sequence of pairwise distinct constant symbols. For every n 1 we write σ n for the signature σ {c 1,..., c n}. For every r 0 and n 1, a type with n centres and radius r (for short: r-type with n centres is a σ n-structure (A, a 1,..., a n, where A is a σ-structure, the constant symbols c 1,..., c n are interpreted by the elements a 1,..., a n A, and A = Nr A (a 1,..., a n. The elements a 1,..., a n are called the centres of the r-type. For a σ-structure A and a non-empty set B A, we write A[B] to denote the restriction of the structure A to the universe B A. For a tuple a = (a 1,..., a n A n, the r-sphere of a in A is defined as the r-type Nr A (a := (A[Nr A (a], a. Bounded structures. The degree of a σ-structure A is the degree of its Gaifman graph G A. If this degree is d, then we call A d-bounded. Two formulas ϕ(x and ψ(x of signature σ are called d-equivalent if they are C d -equivalent, where C d is the class of all d-bounded σ-structures. Note that for all d 2, r 0, and n 1, every d-bounded r-type τ with n centres contains at most n d r+1 elements. Thus, given τ, one can construct an FO[σ]-formula sph τ (x such that for every σ-structure A and every tuple a A n we have A = sph τ [a] N A r (a = τ. The formula sph τ (x has size (n d r+1 O( σ ; and sph τ (x is called a sphere-formula (over σ and x. Hanf normal form for first-order logic. An FO-sentence over a relational signature σ is said to be in Hanf normal form (cf. [1] if it is a Boolean combination 2 of so-called Hanf-sentences. A Hanfsentence is a sentence of the form k y sph τ (y, expressing that there exist at least k elements whose r-sphere is isomorphic to τ, for a given r-type τ. It is known that every first-order sentence is d-equivalent to a sentence in Hanf normal form [4] and that such a sentence in Hanf normal form can be computed in 3-fold exponential time [1]. The main result of the present paper is that both these results generalize to formulas from FO(Q if, and only if, all quantifiers in Q are ultimately periodic which motivates the last definitions of this section. Ultimately periodic sets. Let Q N be a unary counting quantifier. It is ultimately periodic (cf., e.g., [16] if there exist numbers p, n 0 N with p 1, such that for all n n 0 we have n Q n+p Q. (2 The minimal number p 1 for which there exists an n 0 such that statement (2 is true is called the period of Q, and n 0 is called an offset of Q. Examples of ultimately periodic sets are the existential quantifier := N 1 (with period 1 and offset 1 and the divisibility quantifier D p := { p m : m N } for every p N with p 2 (with period p and offset 0. The characteristic sequence χ Q of Q N is the ω-word w = w 0w 1 {0, 1} ω with Q = {n N : w n = 1}. 2 Throughout this paper, whenever we speak of Boolean combinations, we mean finite Boolean combinations. Fact 2.1. Let Q N. If Q is ultimately periodic with period p and offset n 0, then there exist finite words α {0, 1} and π {0, 1} + such that χ Q = α π ω, π = p, and α = n 0. If, conversely, χ Q = α π ω, then Q is ultimately periodic, its period divides π, and α is an offset. We can thus represent an ultimately periodic set Q by the finite word rep(q := α#π, where χ Q = α π ω. To make this definition unambiguous, we demand that p := π is the period of Q, and n 0 := α is the smallest offset of Q. The size Q of Q is defined as the length of rep(q. Let Q P(N be a set of ultimately periodic sets. The size ϕ of an FO(Q-formula ϕ of signature σ is its length when viewed as a word over the alphabet σ Var {, } {=,,,, (,, 0, 1, #}, where Var is a countable set of variable symbols, and where each quantifier Q Q is represented by the word rep(q. 3. Main results In the following, we denote by σ a relational signature. We generalise the notion of Hanf normal form to extensions of first-order logic by unary counting quantifiers in the following way. A Hanf-sentence (over σ is a sentence of the form (Q+ky sph τ (y, where Q N, k N, and τ is an r-type with 1 centre (of signature σ; its locality radius is r. The sentence expresses that the number of interpretations for y such that the r-sphere of y is isomorphic to τ, belongs to the set (Q+k. I.e., for every σ- structure A, we have A = (Q+ky sph τ (y if, and only if, {b A : N A r (b = τ} (Q+k. A sphere-formula (over σ and x is a formula of the form sph τ (x, where x = (x 1,..., x n is a non-empty tuple of pairwise distinct variables, and τ is an r-type with n centres (of signature σ, for some r N. The number r is called the locality radius of the sphere-formula. An FO(Q[σ]-formula ψ(x in Hanf normal form (HNF, for short is a Boolean combination of Hanf-sentences and sphereformulas over σ. Accordingly, a sentence in HNF is a Boolean combination of Hanf-sentences. The locality radius of a formula in HNF is the maximum of the locality radii of its Hanf-formulas and its sphere-formulas. Definition 3.1. A set Q P(N of unary counting quantifiers permits Hanf normal forms if for every relational signature σ and every degree bound d 0, every FO(Q[σ]-formula is d-equivalent to an FO(Q[σ]-formula in HNF. Our first main result characterises the sets Q that permit Hanf normal forms in terms of ultimately periodic sets: Theorem 3.2. A set Q P(N of unary counting quantifiers permits Hanf normal forms if, and only if, every quantifier Q Q is ultimately periodic. For the only if direction of Theorem 3.2, we consider an arbitrary unary counting quantifier S Q that is not ultimately periodic, and we show that already for the signature σ P := {P } with P unary, no sentence in HNF can express A S. I.e., we show that the formula Sy y=y is not equivalent to any FO(Q[σ P ]- sentence in HNF. The proof s key point is that no finite factor of the characteristic sequence χ S of S determines the remainder of χ S, see Section 4. For the if direction, consider a set Q P(N of ultimately periodic sets. Let D P(N be the set of all divisibility quantifiers D p for which there exists Q Q with period p. The proof then proceeds by showing the following:

4 (1 Any FO(Q-formula ϕ can be translated into an FO(D- formula that is equivalent to ϕ on all finite structures (irrespective of their degree. (2 For t N 1 let σ be the signature consisting of the unary relation symbols P 1,..., P t and restrict attention to σ -structures A whose universe is a disjoint union of the relations P1 A,..., Pt A. The properties A ( +k and A (D p+k can be expressed by Boolean combinations of formulas of the form ( +ly P s(y and (D p+ly P s(y (with s [1, t] and all l N. (3 The set D permits Hanf normal forms effectively. (4 If Q N is ultimately periodic with period p 2 and ψ = (D p+ky ϕ, then there is a Boolean combination of formulas (Q+ly ϕ and ( +ly ϕ for suitable numbers l N that is equivalent to ψ on all finite structures (irrespective of their degree. Step (1 is straightforward. Steps (2 and (3 are the crucial steps (and (2 is used for proving (3. Step (4 is obtained by an application of a basic result on word combinatorics. Note that the if direction of Theorem 3.2 is an immediate consequence of steps (1, (3, and (4. Proof details for the steps (1 (4 are given in Section 5. Our second main result provides a worst-case optimal algorithm for transforming formulas into Hanf normal form: Theorem 3.3. There is an algorithm which receives as input a degree bound d N and a formula ϕ FO(Q, where Q P(N is a set of ultimately periodic sets, and constructs a d-equivalent HNF ψ FO(Q of the same signature and with the same free variables as ϕ. For any d 2, the formula ψ has locality radius 4 qr(ϕ and displacement in d 2O( ϕ, and the algorithm s runtime is in 2 d2o( ϕ. Our proof of the if direction of Theorem 3.2 allows it to be read as the algorithm of Theorem 3.3. An upper bound on the algorithm s runtime is obtained by a careful analysis of the time required for performing each of the steps of that proof; see Section 6 for details. For obtaining the 3-fold exponential upper bound, it is crucial that the main construction within our proof of Step (2 is done via a divide and conquer approach (a more straightforward brute-force approach only yields a 4-fold exponential upper bound. A matching 3-fold exponential lower bound (for d = 3 was shown in [1] already for plain first-oder logic, i.e., for the special case where Q = { }. As an easy application of Theorem 3.3, we obtain that Seese s and Frick/Grohe s [7, 20] fixed-parameter tractability result for firstorder model checking on structures of degree d can be generalised to first-order logic with ultimately periodic unary counting quantifiers. Precisely, we obtain the following, where A denotes the size of a reasonable encoding of a σ-structure A (as defined, e.g., in [6]. Theorem 3.4. There is an algorithm which receives as input a formula ϕ(x FO(Q[σ], where Q P(N is the set of all ultimately periodic sets, and where σ consists of precisely the relation symbols that occur in ϕ, and a finite σ-structure A and a tuple a A x, and decides whether A = ϕ[a] in time 2 d2o( ϕ A, where d 2 is an upper bound on the degree of A. 4. Proof of the only if direction of Theorem 3.2 In this section we show that, whenever a set Q of quantifiers contains some quantifier that is not ultimately periodic, then Q does not permit Hanf normal forms. For this, it suffices to consider a signature consisting of a single unary relation symbol. Lemma 4.1. Let σ P := {P } be the signature with P unary. Let Q P(N be a set of unary counting quantifiers which contains a quantifier S N that is not ultimately periodic. There is no FO(Q[σ P ]-sentence δ in Hanf normal form, such that for all σ P - structures A we have A = δ A S. Proof. For contradiction, assume that δ is an FO(Q[σ P ]-sentence in HNF expressing A S. Since P is unary, any r-neighbourhood of an element a in a σ P - structure A consists of its centre a, only. Consequently, P (y and P (y are the only formulas sph τ (y, where τ is a type with one centre. Hence, there are a finite set Q Q and a natural number k 1 such that δ is a Boolean combination of sentences of the form (Q+ly P (y or (Q+ly P (y, where Q Q and l [0, k 1]. Let Q 1,..., Q j be a list of all Q Q. For each a N with a k consider the word w a of length k j defined as the concatenation of the bitstrings χ Qi (a k, a] for i = 1,..., j. Clearly, there exist natural numbers b > a > k with w a = w b, i.e., χ Q (a k, a] = χ Q (b k, b] for all Q Q. If, for all c 0, we have a + c S b + c S, then S is ultimately periodic (with period dividing b a and offset a. Since this is not the case, there is a c 0 with a + c S b + c / S. Now consider σ P -structures A and B with A = a+c, B = b+c, and P A = P B = c. By choice of a, b, c, we have A S B / S, and thus, A = δ B = δ. Nevertheless, A and B cannot be distinguished by any of the Hanf-sentences that occur in δ: To this end, let Q Q and l [0, k 1]. Then A = (Q+ly P (y iff P A (Q+l. This is equivalent to P B (Q+l since P A = P B, and therefore to B = (Q+ly P (y. On the other hand, A = (Q+ly P (y iff a l Q, since A \ P A = a. This is equivalent to b l Q, since χ Q (a k, a] = χ Q (b k, b]. Finally, b l Q is equivalent to B = (Q+ly P (y, since B \ P B = b. In summary, the structures A and B satisfy the same Hanf-sentences that occur in δ. As δ is a Boolean combination of these Hanf-sentences, we obtain A = δ B = δ. This is a contradiction, completing the proof of Lemma 4.1. The only if direction of Theorem 3.2 is an immediate consequence: For S Q not ultimately periodic, the FO(Q[σ P ]- sentence Sy y=y expresses A S which cannot be expressed by any FO(Q[σ P ]-sentence in HNF. 5. Proof of the if direction of Theorem 3.2 For the proof of the if direction of Theorem 3.2, we have to show that for every set Q P(N of ultimately periodic sets, for every relational signature σ, for every degree bound d 0, and for every FO(Q[σ]-formula ϕ, there is a d-equivalent FO(Q[σ]-formula ψ in Hanf normal form. For this, we introduce the notion of the generalised quantifier rank gqr(ϕ of a formula ϕ, which is defined in the same way as the quantifier rank, with the only exception that a formula ψ of the shape (Q+ky ϕ has generalised quantifier rank gqr(ψ = gqr(ϕ + 1 (in contrast, ψ has quantifier rank qr(ψ = qr(ϕ + k + 1. For the remainder of this section we denote by Q P(N a set of ultimately periodic quantifiers. Furthermore, we let D P(N

5 be the set which contains for each Q Q with period p 2 the divisibility quantifier D p. The proof follows the four steps outlined in Section Step (1 Step (1 is established by the following lemma. Lemma 5.1. Let Q N be ultimately periodic with period p and offset n 0. Every formula of the shape Qy ϕ is equivalent to a Boolean combination of formulas of the form (D p+ly ϕ and ( +ly ϕ, for l < n 0+p. This Boolean combination has size O((n 0+p 3 ϕ, and it has the same generalised quantifier rank as Qy ϕ. Proof. Let n 1 N be the (unique number in [n 0, n 0+p that is divisible by p. Clearly, Q is also ultimately periodic with period p and offset n 1. Let Q 1 := Q [0, n 1 and R := {r [0, p : n 1+r Q}. It is straightforward to verify that Qy ϕ is equivalent to the formula ( ( =l y ϕ n 1 y ϕ (D p+ry ϕ l Q 1 r R. (3 Clearly, this formula has the same generalised quantifier rank as Qy ϕ, and it is of size O((n 0+p 3 ϕ. 5.2 Step (2 For every t N 1 let σ be the signature consisting of t unary relation symbols P 1,..., P t. Step (2 consists of proving the following lemma. Lemma 5.2. There is an algorithm which receives as input numbers i, j, t, k N with 1 i j t, and a quantifier Q { } D with period p 1, and constructs a Boolean combination δ (Q+k of sentences of the form (R+ly P s(y with R {, Q}, l max{k, p}, and s [i, j], such that for every σ -structure B where the relations P1 B,..., Pt B are mutually disjoint, B = δ (Q+k The displacement of δ (Q+k j P B s is max{k, p}. (Q+k. To ensure that our algorithm provided by Theorem 3.3 is worstcase optimal, we need the formula δ (Q+k to be sufficiently small. Therefore, our proof of Lemma 5.2 proceeds in a divide-and-conquer manner, such that its runtime meets the bound stated in Corollary 6.2 below. Proof of Lemma 5.2. Let C denote the class of all σ -structures B whose relations Ps B, for s [1, t], are mutually disjoint. Observe that for every B C, we have j Ps B j (Q+k B = (Q+ky P s(y. Thus, if i = j, we are done. If i < j, the algorithm proceeds by a recursive subdivision of the interval [i, j]. For this, let h := i+j 2. Case 1: Q =. The formula ( +ky j Ps(y, that is, the formula k+1 y j Ps(y, is C-equivalent (but not equivalent on all structures to the formula h k+1 y P s(y k+1 y k l=1 ( l y j P s(y s=h+1 h P s(y k l+1 y j s=h+1 P s(y. The algorithm proceeds recursively by decomposing the quantified subformulas in the same manner, and arrives at a Boolean combination δ ( +k of formulas of the shape ( +ly P s(y with 0 l k. Case 2: Q = D p with p 2 and k < p. The formula (D j p+ky Ps(y is C-equivalent (but not equivalent on all structures to ( h j (D p+k 1y P s(y (D p+k 2y P s(y. k 1,k 2 [0,p, k 1 +k 2 k mod p s=h+1 In the same way as in Case 1, the algorithm proceeds recursively and arrives at a Boolean combination δ (Dp+k of formulas of the shape (D p+ly P s(y for l < p. Case 3: Q = D p with p 2 and k p. Let k [0, p with k k mod p. For every B C, we have j Ps B j (Q+k P B s ( +(k 1 (Q+k Therefore, the algorithm can output the sentence where δ ( +(k 1 δ (Dp+k := δ ( +(k 1 δ (Dp+k, and δ (Dp+k are sentences constructed according to Case 1 and Case 2, respectively. Note that in this case δ (Dp+k has displacement max{k, p}. 5.3 Step (3 In the following, we let σ be a relational signature. For every d N, we write C d for the class of all d-bounded σ-structures. For each n 1 and each r 0 we write T d r (n for a set of all (up to isomorphism d-bounded r-types with n centres (of signature σ. I.e., for every d-bounded r-type τ with n centres (of signature σ, there is precisely one τ T d r (n with τ = τ. We write for a fixed tautological FO[σ]-sentence in HNF; e.g., we can choose := y sph τ (y y sph τ (y, where τ is an arbitrary, fixed type of radius 0 with one centre. We let := be the corresponding unsatisfiable sentence in HNF. This subsection s aim is to show that any set D of divisibility quantifiers permits Hanf normal forms. As a warm-up, let us first show how to transform a formula of the form (R+ly sph τ (x, y into a d-equivalent formula in HNF. The crucial point is the separation of the variable y from the variables x. Lemma 5.3. Let R N, let l, d, r, n N with r 1, let τ T d r (n+1, and let α(x := (R+ly sph τ (x, y. There is an FO({R}[σ]-formula α (x in HNF that is d-equivalent to α(x. The formula α (x has locality radius 4r and displacement l + n d (2r+1+1. Furthermore, for ultimately periodic R there is an algorithm which constructs α (x when given α(x and d. Proof. If n = 0 then α := α is in HNF and we are done. For n 1, let x = (x 1,..., x n be the free variables of α(x, and let τ = (T, c, c, for the n+1 centres c, c = (c 1,..., c n, c.

6 For each ρ T d 4r(n, we will construct an FO({R}[σ]-sentence α ρ in HNF which, for all d-bounded σ-structures A and all a A n with N A 4r(a = ρ, satisfies Then, obviously, A = α ρ A = α[a]. (4 α (x := ρ T d 4r (n ( sphρ (x α ρ is in HNF and is d-equivalent to α(x. Let ρ = (R, e T d 4r(n be an arbitrary d-bounded 4r-type with n centres e = (e 1,..., e n. The construction of α ρ proceeds by the following case distinction: Case 1: c N2r+1(c. τ a A n we have Then, for every σ-structure A and all {b A : N A r (a, b = τ} = {b N A 2r+1(a : N A r (a, b = τ}. Furthermore, N τ r (c N τ 3r+1(c. Since 2r+1+r = 3r+1 4r, this implies that for every σ-structure A and every tuple a A n with N A 4r(a = ρ, {b N A 2r+1(a : N A r (a, b = τ} (5 = {f N2r+1(e ρ : Nr ρ (e, f = τ}. (6 }{{} =: k ρ Hence, putting (5 and (6 together, we let α ρ := if k ρ (R+l, and α ρ := if k ρ (R+l. Clearly, α ρ satisfies (4. Case 2: c N2r+1(c. τ Then, the sets Nr τ (c and Nr τ (c are disjoint and there are no edges in the Gaifman graph of τ between the nodes from Nr τ (c and the nodes from Nr τ (c. Case 2.1: N τ r (c and N ρ r (e are not isomorphic. Then, for every σ-structure A and for each a A n with N A 4r(a = ρ, we have {b A : N A r (a, b = τ} = 0. Hence, if 0 (R+l we let α ρ :=, and if 0 (R+l we let α ρ :=. Clearly, α ρ satisfies (4. Case 2.2: N τ r (c and N ρ r (e are isomorphic. Then, for every σ- structure A and every a A n with N A 4r(a = ρ, we have {b A : N A r (a, b = τ} = {b A \ N A 2r+1(a : N A r (b = N τ r (c }. Since 2r+1+r = 3r+1 4r, we have {b N A 2r+1(a : N A r (b = N τ r (c } (7 = {f N2r+1(e ρ : Nr ρ (f = Nr τ (c }. (8 }{{} =: l ρ Hence, putting (7 and (8 together, we know that {b A : N A r (a, b = τ} = {b A : N A r (b = N τ r (c } l ρ. It follows that the Hanf-sentence α ρ := (R+(l+l ρy sph N τ r (c (y satisfies (4. Observe that l ρ n d (2r+1+1 and hence, α ρ has displacement l + n d (2r+1+1. Observe that for each ρ T d 4r(n the HNF-sentence α ρ has locality radius r and displacement l + n d (2r+1+1. Consequently, the HNF-formula α (x has locality radius 4r and displacement l + n d (2r+1+1. Furthermore, for ultimately periodic R, it can easily be decided for a given number m if m (R+l. In Case 1 and Case 2.1, this is used for m = k ρ and m = 0, respectively, and leads to an algorithm which constructs the HNF-formula α (x. The following Lemma 5.4 states this subsection s main result; in particular, it implies that the set D permits Hanf normal forms. For proving Lemma 5.4, we will use the Lemmas 5.2 and 5.3. Lemma 5.4. There is an algorithm which receives as input a degree bound d N and a formula ϕ FO(D[σ] (where σ is a relational signature, and constructs a HNF ψ FO(D[σ] that is d-equivalent to ϕ. Furthermore, free(ψ = free(ϕ, and ψ has locality radius 4 q and displacement max{k, P } + ϕ d 4q, where q 0 is the generalised quantifier rank of ϕ, K 0 is the displacement of ϕ, and P 1 is an upper bound on the periods of the quantifiers occurring in ϕ. The below proof of Lemma 5.4 proceeds by induction on the shape of FO(D[σ]-formulas. While the case of quantifier-free formulas turns out to be straightforward, much more work is needed to transform a formula ϕ(x of the shape (Q+ky ψ (x, y into a d-equivalent HNF. Suppose that ψ (x, y is already in HNF. Our construction is carried out along the following steps: For each structure A C d and each tuple a A n, we let B A,a := { b A : A = ψ [a, b] }. Clearly, A = ϕ[a] B A,a (Q+k. Suppose that ϕ(x has n 0 free variables and that the HNF ψ (x, y has locality radius r 0. Observe that for every b B A,a, there is exactly one τ Tr d (n+1 such that Nr A (a, b = τ. Let τ 1,..., τ t, for t := Tr d (n+1, be an enumeration of Tr d (n+1. Let B be the σ -structure (B, P1 B,..., Pt B with universe B := B A,a and Ps B := {b B : Nr A (a, b = τ s}, for each s [1, t]. Clearly, B is the disjoint union of the sets P1 B,..., Pt B. Hence, for the sentence δ (Q+k FO(D[σ ], obtained from Lemma 5.2 in Step (2, we have that B (Q+k B = δ (Q+k. Recall from Lemma 5.2 that δ (Q+k is a Boolean combination of formulas of the form (R+ly P s(y, with R {, Q}, s [1, t], and l max{k, P }. For every such formula (R+ly P s(y, we construct a HNF ψ s (R+l (x FO(D[σ], such that for every A C d, for every tuple a A n, and for the σ -structure B := B A,a, we have that A = ψ s (R+l [a] Ps B (R+l. Thus, we interpret the σ -structure B A,a in (A, a. In the sentence δ (Q+k, we replace every formula of the shape (R+ly P s(y by the HNF ψ s (R+l (x. Clearly, the resulting FO(D[σ]-formula ψ(x is in HNF and, furthermore, d-equivalent to ϕ(x. The details of the above construction are carried out as follows. Proof of Lemma 5.4. We describe the algorithm on input of a degree bound d 2 and an FO(D[σ]-formula ϕ(x over a finite relational signature σ. Let n = x 0 be the number of free variables of ϕ. Let q 0 be the generalised quantifier rank of ϕ. Let K 0 be the

7 displacement of ϕ, and let P 1 be an upper bound on the periods of the quantifiers occurring in ϕ. The algorithm proceeds by induction on the shape of ϕ(x. We will show that for the HNF ψ(x FO(D[σ], which the algorithm constructs, the following claim holds: Claim 5.5. (a ψ(x has locality radius 4 q. (b ψ(x is d-equivalent to ϕ(x. (c ψ(x has displacement max{k, P } + ϕ d 4q. Suppose that ϕ is quantifier-free, i.e., q = 0. In this case, n 1, and ϕ is d-equivalent to the disjunction over all types in T d 0 (n that satisfy ϕ(x. The algorithm proceeds as follows: (1 Compute the set T d 0 (n. (2 Let c = (c 1,..., c n and compute the set T T d 0 (n that contains precisely those types τ = (T, c from T d 0 (n with T = ϕ[c]. (3 If T is the empty set, then ϕ is unsatisfiable and, hence, equivalent to the HNF ψ(x := sph τ (x sph τ (x, where τ is an arbitrary element in T0 d (n. Otherwise, let ψ(x := sph τ (x. τ T It is clear that in both cases, ψ(x is a HNF which satisfies Claim 5.5 (a (c. Suppose that ϕ is a Boolean combination with generalised quantifier rank q 1. If ϕ = ϕ then let ψ be an FO(D[σ]-formula in HNF that is d-equivalent to ϕ and let ψ := ψ. If ϕ = (ϕ 1 ϕ 2, then let ψ 1 and ψ 2 be FO(D[σ]-formulas in HNF that are d- equivalent to ϕ 1 and ϕ 2, respectively, and let ψ := (ψ 1 ψ 2. In both cases, Claim 5.5 (a (c is obviously satisfied. Suppose that ϕ(x = (Q+ky ϕ (x, y. Let p [1, P ] be the period of Q. Recall that k K. The algorithm proceeds as follows: (4 By Claim 5.5 (a (c, there is a HNF ψ (x, y FO(D[σ] that is d-equivalent to ϕ (x, y and which has locality radius r := 4 q 1 and displacement at most k := max{k, P } + ϕ d 4q 1. (5 Let t := T d r (n+1, and let τ 1,..., τ t be an enumeration of the set T d r (n+1. Recall that for every A C d and every tuple (a, b A n+1, there is exactly one s [1, t] such that N A r (a, b = τ s. (6 Since ψ (x, y is in HNF, it is a Boolean combination of sphereformulas over x, y and of Hanf-sentences. For every s [1, t] consider the type τ s and let ψ s be the HNF-sentence obtained from ψ as follows: Letting (T, c := τ s, check for every sphere-formula sph ρ (x, y in ψ, whether T = sph ρ [c]. If so, replace every occurrence of sph ρ (x, y in ψ with the tautological HNF-sentence ; otherwise replace it with the unsatisfiable HNF-sentence. Since ψ (x, y has locality radius r, clearly ψ (x, y is d- equivalent to t ( sphτs (x, y ψ s. s=1 For any fixed A C d and a A n, consider the σ -structure B := B A,a of universe t s=1 P B s where P B s := { b A : (A, a, b = sph τs (x, y ψ s }, for any s [1, t]. (7 By Lemma 5.2, there is an FO(D[σ ]-sentence δ (Q+k such that B = δ (Q+k B (Q+k. Furthermore, δ (Q+k is a Boolean combination of sentences of the shape (R+ly P s(y with R {, Q}, s [1, t], and l max{k, p} max{k, P }. (8 For every sentence of the shape (R+ly P s(y in δ (Q+k, let the formula ϕ (R+l s (x be defined as follows: If 0 (R+l, then let ϕ (R+l s (x := ψ s (R+ly sph τs (x, y ; otherwise let ϕ (R+l s (x := ψ s (R+ly sph τs (x, y. It is straightforward to verify that for every d-bounded σ- structure A and every a A n we have A = ϕ (R+l s [a] B A,a = (R+ly P s(y. (9 Let ψ s (R+l (x be the HNF-formula obtained from ϕ (R+l s (x by replacing (R+ly sph τs (x, y with the d-equivalent HNFformula obtained from Lemma 5.3. Since τ s has radius r = 4 q 1, and l max{k, P }, the formula ψ s (R+l (x has locality radius 4r = 4 q and displacement max{k, P } + n d 4q. (9 Let ψ(x be the formula obtained from δ (Q+k as follows: For every s [1, t], replace every occurrence of a formula of the shape (R+ly P s(y with the HNF-formula ψ s (R+l (x. Note that ψ(x is in HNF, has locality radius 4 q, and has displacement max{k, P } + ϕ d 4q. By construction, the following is true for every d-bounded σ-structure A, every tuple a A n, and the σ -structure B := B A,a: A = ϕ[a] (A, a = (Q+ky ψ (x, y B (Q+k B = δ (Q+k A = ψ[a]. The last equivalence follows from the construction of ψ(x and equivalence (9. In summary, we obtain that Claim 5.5 (a (c is satisfied. This completes the proof of Lemma Step (4 Step (4 is established by the following lemma. Lemma 5.6. Let Q N be ultimately periodic with period p 2 and offset n 0. Every formula of the shape (D p+ky ϕ, for a k 0, is equivalent to a Boolean combination of formulas of the shape (Q+ly ϕ and ( +ly ϕ with l < n 0+k+2p. Furthermore, such a Boolean combination can be computed in time O(( Q +k 3 ϕ. Proof. Let n 1 N 1 be the smallest number max{n 0, k} such that n 1 1 k mod p. Clearly, Q is ultimately periodic with period p also for the offset n 1. From Fact 2.1 we obtain bitstrings α and π of length α = n 1 and π = p such that χ Q = α π ω. I.e., χ Q [0, n 1 1] = α, and χ Q [n 1, n 1+p 1] = π = χ Q [n 1+(s 1p, n 1+sp 1], (10

8 for every s 1. Claim 5.7. For all n N with n n 1+p 1 we have n (D p+k χ Q (n p, n] = π. Before presenting the proof of the claim, let us first show how the claim can be used to prove Lemma 5.6. Letting π 0, π 1,..., π p 1 {0, 1} be such that π = π p 1 π 1 π 0, it is straightforward to see that for all n N with n n 1+p 1 we have χ Q (n p, n] = π n (Q + i n (Q + j. i [0,p : π i =1 j [0,p : π j =0 Thus, the formula (D p+ky ϕ is equivalent to the formula ( =l y ϕ n1+p 1 y ϕ l S (Q+iy ϕ (Q+jy ϕ, i [0,p : π i =1 j [0,p : π j =0 (11 where S is the set of all n (D p+k with n < n 1+p 1. Since n 1 max{n 0, k}+p, each of the quantifiers that explicitly occur in the formula (11 has displacement < n 0+k+2p 2 Q + k. Using this, it is straightforward to see that the formula (11 has size O(( Q +k 3 ϕ and can easily be computed within the same time bound. To complete the proof of Lemma 5.6, it only remains to prove Claim 5.7. Proof of Claim 5.7. Fix an arbitrary n N with n n 1+p 1. The direction = is an immediate consequence of equation (10. Note that for proving the direction =, it suffices to prove the following: If χ Q = β π ω for some word β {0, 1} with α β, then α β mod p (in our case, β = χ Q [0, n 1]. Since α is not longer than β, there exist i N, a word u {0, 1} with u < π, and a word v {0, 1} + with απ i u = βπ and βπv = απ i+1. Hence απ i uv = απ i+1, and thus π = uv. Consequently, βπvu = απ i+1 u is a prefix of απ ω = βπ ω. This implies that vu is a prefix of π ω of length vu = uv = π, i.e., vu = π. Since u and v commute (i.e., uv = vu, a basic result in word combinatorics (see [15, Proposition 1.3.2] implies the existence of some word w {0, 1} + such that u, v w and therefore π = uv w. But then, χ Q = απ ω = αw ω implies π = w, since p = π was the period of χ Q, and hence minimal. Hence, w = π = uv. Consequently, u = ε and therefore β = απ i. This ensures α β mod p. This completes the proof of Claim 5.7 and therefore the proof of Lemma 5.6. Note that the if direction of Theorem 3.2 is an immediate consequence of the Lemmas 5.1, 5.4, and Complexity This section is devoted to the proof of Theorem 3.3. We proceed along the steps outlined in Section 3 and provide a runtime analysis of the algorithm obtained from the proof presented in Section 5. Due to space restrictions, some proofs had to be deferred to the paper s full version. The following Corollary 6.1 shows how to use Lemma 5.1 to compute, for a given FO(Q-formula, an equivalent FO(D- formula. Here, we let w(ϕ := 1 if ϕ is quantifier-free; otherwise, w(ϕ is the largest number Q for all quantifiers Q occurring in ϕ. Corollary 6.1. There is an algorithm which receives as input a ϕ FO(Q of quantifier rank q, and constructs in time ϕ w(ϕ O(q an equivalent ψ FO(D of the same signature and with the same free variables as ϕ. The formula ψ has generalised quantifier rank q and displacement w(ϕ. The proof, as well as the proof of the following result, is straightforward. Corollary 6.2. The algorithm from Lemma 5.2 runs in time max{2k+2, 2p} O(log(j i+1. We proceed with a runtime analysis of the algorithms provided by the Lemmas 5.3 and 5.4. Two basic tasks that are repeatedly used within both algorithms are (a to test two d-bounded r-types for isomorphism and (b to compute a set Tr d (n of representatives of the isomorphism classes of d-bounded r-types with n centres. Both can be accomplished by brute-force algorithms: Recall that for d 2, the universe of every d-bounded r-type with n centres has size Ñ := n dr+1. For every bijection between the elements of two such r-types, it can be checked in time O(n+Ñ σ Ñ O( σ whether it is indeed an isomorphism. Since there are at most Ñ Ñ bijections, the test whether the two r- types are isomorphic can be performed in time a0 2Ñ, where a 0 1 is a suitable number of size O( σ. For computing the set Tr d (n, we can assume that the universe of each element in Tr d (n is a subset of {1,..., Ñ}. Hence, the relations of an arbitrary r-type in Tr d (n can be represented by a bitstring of length at most Ñ σ. Furthermore, each of the r-type s n centres is interpreted by an element from {1,..., Ñ}. Therefore, there are at most Ñ n σ 2Ñ σ +2 2Ñ candidates for elements in Tr d (n. For any two such candidates, we use the above described algorithm for testing whether they are isomorpic. In summary, Tr d (n can be constructed in time ( 2Ñ σ Ñ a 0 where a 1 > a 0 is a suitable number of size O( σ. Using this, we obtain the following. 2Ñ a 1, (12 Corollary 6.3. For d 2, the algorithm from Lemma 5.3 runs in time O(l 2 2 (n d4r+1 O( σ. Proof. We use the same notation as in the proof of Lemma 5.3, and we let N := n d 4r+1. According to equation (12 the set T d 4r(n can be constructed in time 2 N a1, for a suitable number a 1 of size O( σ. For each ρ T d 4r(n, it takes time N O( σ to construct the formula sph ρ (x. For the construction of α ρ, we have to make a case distinction depending on whether c N τ 2r+1(c. To determine which of the two cases actually applies for the given τ, recall that the universe of τ has size at most (n+1 d r+1. Thus, the time needed do decide whether c N τ 2r+1(c is in N O( σ. Case 1: If c N τ 2r+1(c, we compute the number k ρ defined in equation (6. This requires us to check for at most n d (2r+1+1 N d-bounded r-types with (n+1 centres, each with a universe of size (n+1 d r+1 N, whether they are isomorphic to τ. By using the brute-force isomorphism test described at the beginning of Section 6, the number k ρ can be computed in time 2 N O( σ. And testing whether k ρ (R+l is done in time O(k ρ, for ultimately periodic R. Hence, in Case 1, the algorithm uses time 2 N O( σ to construct the formula α ρ.

9 Case 2: If c N τ 2r+1(c, we have to check whether the two r- types N τ r (c and N ρ r (e (each with N elements are isomorphic. If they are not, we use time O(l to check whether 0 (R+l and construct α ρ accordingly. Otherwise, i.e., if N τ r (c and N ρ r (e are isomorphic, we have to compute the number l ρ defined in equation (8. This requires us to check for at most n d (2r+1+1 N d-bounded r-types with one centre, each with a universe of at most d r+1 N elements, whether they are isomorphic to N τ r (c. By using the brute-force isomorphism test described at the beginning of Section 6, the number l ρ can be computed in time 2 N O( σ. Afterwards, we construct the formula α ρ := (R+(l+l ρy sph N τ r (c (y. For this, we use time N O( σ to construct the formula sph N τ r (c (y; and resolving the quantification (R+(l+l ρ via the quantifiers R and involves an extra factor of (l+l ρ 2 (l+n 2. Altogether, we obtain that in Case 2 the algorithm uses time at most O(l 2 2 N O( σ to construct the formula α ρ. Finally, the algorithm outputs the formula α (x, which is the disjunction of the formulas ( sph ρ (x α ρ, for all ρ T d 4r (n. The overall time used for constructing this formula is 2 N a 1 (N O( σ + O(l 2 2 N O( σ O(l 2 2 N O( σ. By a similar reasoning, a runtime analysis of the algorithm provided by Lemma 5.4 leads to the following result. Corollary 6.4. For d 2, the algorithm from Lemma 5.4 runs in time max{2k+2, 2P } ( ϕ d4q +1 O( σ. The proof of Theorem 3.3 now follows by combining Corollary 6.1, Lemma 5.4, Corollary 6.4, and Lemma 5.6: Proof of Theorem 3.3. We analyse the algorithm s runtime when receiving as input a degree bound d 2 and a formula ϕ FO(Q. Let σ denote the relational signature consisting of all relation symbols that occur in ϕ, and let q 0 be the quantifier rank of ϕ. W.l.o.g., we can assume that Q consists of exactly those unary counting quantifiers that occur in the formula ϕ. In Step (1, the algorithm of Corollary 6.1 computes an FO(D[σ]-formula ϕ that is equivalent to ϕ and that has displacement w(ϕ and generalised quantifier rank q. This takes time ϕ w(ϕ O(q 2 ϕ O(1. For the latter inequality, recall that q, w(ϕ, σ ϕ. In Step (3, the algorithm of Lemma 5.4 computes a HNF ψ FO(D[σ] that is d-equivalent to ϕ. According to Corollary 6.4, for K, P ϕ this takes time max{2k+2, 2P } ( ϕ d 4q +1 O( σ 2 d2o( ϕ. Note that ψ has locality radius 4 q and displacement max{k, P } + ϕ d 4q d 2O( ϕ. In Step (4, the algorithm of Lemma 5.6 computes for each Hanfsentence in ψ an equivalent FO(Q[σ]-sentence in HNF. For each Hanf-sentence in ψ of the shape (D p+ky γ, this takes time O(( Q +k 3 γ, where Q Q is ultimately periodic with period p. Since Q ϕ and k d 2O( ϕ and γ ψ, the time for computing ψ from ψ is at most ψ ( ( ϕ + d 2 O( ϕ 3 ψ 2 d2o( ϕ. ψ has locality radius 4 q and displacement in d 2O( ϕ. Altogether, ψ is an FO(Q[σ]-formula in HNF, ψ is d-equivalent to ϕ, and ψ be computed in time 2 d2o( ϕ. This completes the proof of Theorem Model-checking Proof of Theorem 3.4. Let ϕ(x, A, and a be the algorithm s input, where σ is the relational signature that consists of precisely the relation symbols occurring in ϕ, A is a σ-structure, and Q is the set of ultimately periodic unary counting quantifiers that occur in ϕ. For checking whether A = ϕ[a], the algorithm proceeds as follows: (1 Compute an upper bound d 2 on the degree of A. (2 Use the algorithm from Theorem 3.3 to transform ϕ(x into a d-equivalent FO(Q[σ]-formula ψ(x in HNF. (3 For each sphere-formula α that occurs in ψ, check if A = α[a], and replace each occurrence of α in ψ with the Boolean constant 1 if A = α[a], and with the Boolean constant 0 otherwise. (4 For each Hanf-sentence χ that occurs in ψ, check if A = χ, and replace each occurrence of χ in ψ with the Boolean constant 1 if A = χ, and with the Boolean constant 0 otherwise. (5 After having performed the steps (1 (4, ψ is a Boolean combination of the Boolean constants 0 and 1. Evaluate this Boolean combination and output yes if the result is 1, and output no if the result is 0. Obviously, the algorithm s output is yes if, and only if, A = ϕ[a]. For analysing the algorithm s runtime, we use the same conventions as in [6]. I.e., we use random-access machines with a uniform cost measure, and the input structure A is given by an adjacency list representation of size linear in A := σ + A + R σ R A ar(r. We let n := x and let x = (x 1,..., x n and a = (a 1,..., a n. For the following runtime analysis of each of the steps (1 (5, note that n, σ, and Q are smaller than ϕ, for each Q Q. (1 To compute d, compute an adjacency list representation of A s Gaifman-graph G A: For each R σ and each occurrence of an element of A in a tuple of R A, we have to add less than ar(r σ < ϕ edges to G A. For each edge, this takes time O(d, since A is d-bounded. In summary, computing G A and d takes time at most O( A ϕ d. (2 According to Theorem 3.3, this takes time 2 d2o( ϕ. Recall that ψ(x is a Boolean combination of sphere-formulas of the form sph ρ (x and Hanf-sentences (Q+ky sph τ (y, where x is a tuple of free variables from x, ρ is a d-bounded type with radius 4 qr(ϕ and x centres, Q Q, k N, and τ is a d-bounded type with radius 4 qr(ϕ and one centre. (3 Consider a sphere-formula α(x := sph ρ (x, where x = (x i1,..., x im for 1 m n and i 1,..., i m [1, n], and where ρ is a d-bounded type with radius r 4 qr(ϕ and m centres.