An optimal construction of Hanf sentences

arXiv:1105.5487v2 [cs.LO] 6 Jun 2011 An optimal construction of Hanf sentences Benedikt Bollig1 and Dietrich Kuske2 1 Laboratoire Sp´ecification et V´erification, ´Ecole Normale Sup´erieure de Cachan & Centre National de la Recherche Scientifique 2 Institut f¨ur Theoretische Informatik, TU Ilmenau Abstract. We give a new construction of formulas in Hanf normal form that are equivalent to first- order formulas over structures of bounded degree. This is the first algorithm whose running time is shown to be elementary. The triply exponential upper bound is complemented by a matching lower bound. 1 Introduction Various syntactical normal forms for semantical properties of structures are known. For ex- ample, every first-order definable property that is preserved under extensions of structures is definable by an existential first-order sentence ( Lo´s-Tarski [23, 20]). Gaifman’s normal form is another example that formalizes the observation that first-order logic can only ex- press local properties [10]. A third example in this line is Hanf’s theorem, giving another formalization of locality of first-order logic (at least for structures of bounded degree) [12, 6]. Gaifman’s and Hanf’s theorems have found applications in finite model theory and in particular in parametrized complexity. Namely, they lead to efficient parametrized algo- rithms deciding whether a formula holds in a (finite) structure [22, 17, 8, 9, 14, 3, 19, 15, 16] and even for more general algorithms that list all the satisfying assignments [5, 13]. Hanf’s theorem was also used in the transformation of logical formulas into different automata models [24, 11, 2, 1]. In [4], it was shown that passing from arbitrary formulas to those in Lo´s-Tarski or Gaifman normal form leads to a non-elementary blowup. The same paper also proves that for structures of bounded degree, the blowup for Gaifman’s normal form is between 2- and 4-fold exponential, and that for Lo´s-Tarski normal forms (for a restricted class of structures) is between 2- and 5-fold exponential. This paper shows that Hanf’s normal form can be computed in three-fold exponential time and that this is optimal since there is a necessary blowup of three exponentials when passing from general first-order formulas to their Hanf normal form. We remark (as already observed by Seese [22]) that the first construction of Hanf normal forms [6] is not effective since satisfiability of first-order formulas in graphs of bounded degree is undecidable, also when we restrict to finite structures [25]. Only Seese [22] gave a small additional argu- ment showing that Hanf normal forms can indeed be computed. But his algorithm is not primitive recursive. This was improved later to a primitive-recursive algorithm by Durand and Grandjean [5] and (independently) by Lindell [19]. Their papers do not give an upper bound for the construction of Hanf normal forms, but on the face of it, the algorithm seems not to be elementary.3 Their algorithm is a quantifier-alternation procedure that only works if the signature consists of finitely many injective functions (following Seese, one can bi-interprete every structure of bounded degree in such a structure, so this is no real restriction of the algorithm). Differently, our algorithm follows the original proof of Hanf’s theorem very closely by examining spheres of bounded diameter, but avoiding the detour via Ehrenfeucht-Fra¨ıss´e-games. Acknowledgement We would like to thank Luc Segoufin and Arnaud Durand for comments on an earlier version and for hints to the literature that improved this paper. 2 Definitions and background Throughout this paper, let L be a finite relational signature and let Lm denote the extension of L by the constants c1, c2, . . . , cm. Let A be an Lm-structure. We write a ∈A when we mean that a is an element of the universe of A. Furthermore, ¯a denotes a tuple (a1, . . . , an) of length n of elements of some structure A and ¯x is the list of variables (x1, . . . , xn). In both cases, n will be determined by the context. Finally, we define a distance (from N ∪{∞}) on the universe of A setting distA(a, b) = 0 iffa = b and distA(a, c) = d + 1 if there exists b ∈A with distA(a, b) ≤d, there is some tuple in some of the relations of A that contains both, b and c, and there is no such b ∈A with distA(a, b) < d, and distA(a, b) = ∞if distA(a, b) ̸= d for all d ∈N. Next, the degree of a ∈A is the number of elements b ∈A with distA(a, b) = 1, the degree of A is the supremum of the degrees of a ∈A. Let A be an L-structure, ¯a = (a1, . . . , an) ∈A, and d > 0. Then BA d (¯a) is the set of elements b ∈A with distA(ai, b) < d for some 1 ≤i ≤n. The d-sphere around ¯a is the Ln-structure SA d (¯a) = (A ↾BA d (¯a), ¯a) . A d-sphere (with n centers) is an Ln-structure (A, ¯a) with BA d (¯a) = A. The Ln-structure (A, ¯a) is a sphere if there exists d > 0 such that (A, ¯a) is a d-sphere; the least such d is denoted d(τ) and is the radius of (A, ¯a). The d-sphere τ is realised by ¯a in A if τ ∼=

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