In this article, we consider the interplay of generalized quantifiers and built-in relations over finite structures, in particular, in the range of logics capturing the circuit complexity classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) . It is well known that for capturing \(\mathrm{AC^{0}}\) first-order logic has to be equipped with order and, e.g., predicates for addition and multiplication, whereas for \(\mathrm{TC^{0}}\) generalized quantifiers such as majority quantifiers are necessary. The sharp division between the classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) can be explained by the fact that \(\mathrm{AC^{0}}\) is not closed under restricting \(\mathrm{AC^{0}}\) -computable queries into simple subsequences of the input, whereas \(\mathrm{TC^{0}}\) is closed under such relativization as its queries can be expressed in terms of first-order formulas using universe-independent generalized quantifiers and order as the only built-in relation. In the terminology of abstract logics, the above means that logics capturing \(\mathrm{AC^{0}}\) do not have the relativization property, and hence, they are not regular logics unlike the logics capturing \(\mathrm{TC^{0}}\) . This weakness of \(\mathrm{AC^{0}}\) has been also elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington et al.) was that if a language \( L \) has a neutral letter, then \( L \) can be defined in \(\operatorname{FO}_{\mathcal{A}}\) , first-order logic with the collection of all numerical built-in relations \(\mathcal{A}\) , if and only if \( L \) can be already defined in \(\operatorname{FO}_{\leq}\) . Our approach is two-fold. First, we study universe-independent cardinality quantifiers \(\operatorname{\mathsf{Q}}\) defined by a parameter set \(S\subseteq\mathbb{N}\) and formulate a combinatorial criterion for \( S \) implying that all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) can be defined in \(\operatorname{FO}_{\leq}(\operatorname{\mathsf{Q}})\) . For instance, this criterion is satisfied if \( S \) is the range of some polynomial with positive integer coefficients of degree at least two. Second, by adapting the key properties of abstract logics to accommodate built-in relations, we define the regular interior \(\operatorname{\mathcal{R}-int}(\mathcal{L})\) (the largest regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}^{*}\subseteq\mathcal{L}\) ) and regular closure \(\operatorname{\mathcal{R}-cl}(\mathcal{L})\) (the least regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}\subseteq\mathcal{L}^{*}\) ), of a logic \(\mathcal{L}\) with built-in relations, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of \(\mathcal{L}\) . By extending the results of Barrington et al., we further show that if \(\mathcal{B}=\{+\}\) , or \(\mathcal{B}\) contains only unary relations besides \(\leq\) , then \(\operatorname{\mathcal{R}-int}(\operatorname{FO}_{\mathcal{B}})\equiv \operatorname{FO}_{\leq}\)

Abstract A game that characterizes equivalence of structures with respect to all first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. We define three other games and prove that they are all equivalent to the Immerman game, and hence also give a characterization for the number of quantifiers needed for separating structures. In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the last two of our games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but two of our games are genuine two-player games.

Abstract We introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets, we study the behavior of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. To this end we associate each atom with a natural notion or arity. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of $k+1$ -ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any k-ary atoms, whence the $k+1$ -ary atoms are not definable in terms of any atoms of smaller arity.

This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree d receives messages through d input ports and it sends messages through d output ports, both numbered with 1,2,...,d. In this work, VVc is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of VVc:

Linear orders are of inherent interest infinite model theory, especially in descriptive complexity theory. Here, the class of ordered structures is approached from a novel point of view, using generalized quantifiers as a means of analysis. The main technical result is a characterization of the cardinality quantifiers which can express equicardinality on ordered structures. This result can be viewed as a dichotomy: the cardinality quantifier either shows a lot of periodicity, or is quite non-periodic, the equicardinality quantifier being definable only in the latter case. The main result shows, once more, that there is a drastic difference between definability among ordered structures and definability on unordered structures. Connections of the result to the descriptive complexity of low-level complexity classes are discussed.

Abstract A combinatorial criterium is given when a monadic quantifier is expressible by means of universe-independent monadic quantifiers of width n. It is proved that the corresponding hierarchy does not collapse. As an application, it is shown that the second resumption (or vectorization) of the Härtig quantifier is not definable by monadic quantifiers. The techniques rely on Ramsey theory.

Abstract The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend his method to arbitrary similarity types.

We describe on a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory such as fixpoint logic (FP), partial fixpoint logic (PFP), infinitary logic /spl Lscrsub /spl infin/wsup w/ with a finite number of variables, as well as the closures of these logics under implicit definitions. Moreover, we show that the upper and lower bounds established here can not be improved dramatically, unless outstanding conjectures in complexity theory are resolved at the same time.>

Abstract We prove that if ℵα is uncountable and regular, then the Beth-closure of ℒωω(Qα) is not a sublogic of ℒαω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(ℒωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindström quantifiers such that B(ℒωω(Qα)) ≤ ℒωω(Q).