We study the complexity of reasoning tasks for logics in team semantics. Our main focus is on the data complexity of model checking but we also derive new results for logically defined counting and enumeration problems. Our approach is based on modular reductions of these problems into the corresponding problems of various classes of Boolean formulas. We illustrate our approach via several new tractability/intractability results.

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.

We study the data complexity of model checking for logics with team semantics. We focus on dependence, inclusion, and independence logic formulas under both strict and lax team semantics. Our results delineate a clear tractability/intractability frontiers in data complexity of both quantifier-free and quantified formulas for each of the logics. For inclusion logic under the lax semantics, we reduce the model-checking problem to the satisfiability problem of so-called dual-Horn Boolean formulas. Via this reduction, we give an alternative proof for the known result that the data complexity of inclusion logic is in PTIME.

Abstract We study some large cardinals in terms of reflection, establishing new connections between the model-theoretic and the set-theoretic approaches.

Abstract We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

Abstract We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality $\aleph _2 $ such that the second player has a winning strategy in the Ehrenfeucht–Fraïssé-game of length ω1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

Independence-Friendly Logic: A Game-Theoretic Approach, Allen L. Mann , Gabriel Sandu and Merlijn Sevenster , Cambridge University Press, 2011. Paperback, ISBN 9780521149341, 216 pp. - Volume 14 Issue 1

Abstract We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ/D is not λ++-universal and elementarily equivalent models M and N of size λ for which Mλ/D and Nλ/D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1].

Lindstrom theorems characterize logics in terms of model-theoretic conditions such as Compactness and the Lowenheim-Skolem property. Most existing Lindstrom theorems concern extensions of first-order logic. On the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindstrom theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindstrom characterizations for a number of fragments of first-order logic. These include the k-variable fragments for k > 2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindstrom proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems. Characterizing the 2-variable fragment or the full guarded fragment remain open problems.

Abstract Let be the class of structures 〈λ, <, A〉, where A ⊆ λ is disjoint from a club, and let be the class of structures 〈λ, <, A), where A ⊆ λ contains a club. We prove that if λ = λ<κ is regular, then no sentence of Lλ + κ separates and On the other hand, we prove that if λ = μ+ , μ = μ<μ, and a forcing axiom holds (and if μ = ℵ0), then there is a sentence of Lλλ which separates and .

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.

Abstract We study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of . We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space . We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2). Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.

Abstract Let σ( ) denote the number of automorphisms of a model of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an with . We study the sufficiency of some conditions for . These conditions are analogous to conditions studied by D. Kueker in connection with countable models.

Generalized quantifiers in the realm of finite structures are studied and combined with an infinitary logic L/sub infinity omega //sup omega / to obtain new logics that can be used to express polynomial-time properties that are not definable in the original logic. It is shown that equivalence of finite structures relative to the new logics can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht-Fraisse games. This time-theoretic characterization is combined with sophisticated combinatorial tools in order to investigate the scopes and limits of generalized quantifiers in finite model theory.>

Abstract A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Härtig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted byLI, is in some sense very natural and has in consequence special interest. Properties ofLIare studied in many papers. In [BF, Chapter VI] there is a short survey of some known results aboutLI. We feel that a more extensive exposition of these results is needed. The aim of this paper is to give an overview of the present knowledge about the languageLIand list a selection of open problems concerning it. After the Introduction (§1), in §§2 and 3 we give the fundamental results aboutLI. In §4 the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories inLI. In §6 the spectra of sentences ofLIare discussed, and §7 is devoted to properties ofLIwhich depend on set-theoretic assumptions. The paper finishes with a list of open problem and an extensive bibliography. The bibliography contains not only papers we refer to but also all papers known to us containing results about the language with Härtig quantifier. Contents. §1. Introduction. §2. Preliminaries. §3. Basic results. §4. Model-theoretic properties ofLI. §5. Decidability of theories withI. §6. Spectra ofLI-sentences. §7. Independence results. §8. What is not yet known aboutLI. Bibliography.

Abstract Let and be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal α with the property that i.e. and are L∞ω-equivalent up to quantifier-rank α but not up to α + 1. In this paper we consider models and of cardinality ω1 and construct trees which have a similar relation to and as a above. For this purpose we introduce a new ordering T ≪ T′ of trees, which may have some independent interest of its own. It turns out that the above ordinal α has two qualities which coincide in countable models but will differ in uncountable models. Respectively, two kinds of trees emerge from α. We call them Scott trees and Karp trees, respectively. The definition and existence of these trees is based on an examination of the Ehrenfeucht game of length ω1 between and . We construct two models of power ω1 with mutually noncomparable Scott trees.

Abstract The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.