On the Complexity of Team Logic and Its Two-Variable Fragment

We study the logic FO(~), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown to be axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO^2(~) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(~) with respect to the number of variables and the quantifier rank, and prove a dichotomy between PSPACE- and ATIME-ALT(exp, poly)-complete fragments. For the lower bounds, we propose a translation from modal team logic MTL to FO^2(~) that extends the well-known standard translation from modal logic ML to FO^2. For the upper bounds, we translate FO(~) to fragments of second-order logic with PSPACE-complete and ATIME-ALT(exp, poly)-complete model checking, respectively

On the Complexity of Team Logic and its Two-Variable Fragment

We study the logic FO(~), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO2(~) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(~) with respect to the number of variables and the quantifier rank, and prove a dichotomy between PSPACE- and ATIME-ALT(exp, poly)-completeness. To achieve the lower bounds, we propose a translation from modal team logic MTL to FO2(~) that extends the well-known standard translation from modal logic ML to FO2. For the upper bounds, we translate to a fragment of second-order logic

A Fragment of Dependence Logic Capturing Polynomial Time

In this paper we study the expressive power of Horn-formulae in dependence logic and show that they can express NP-complete problems. Therefore we define an even smaller fragment D-Horn* and show that over finite successor structures it captures the complexity class P of all sets decidable in polynomial time. Furthermore we study the question which of our results can ge generalized to the case of open formulae of D-Horn* and so-called downwards monotone polynomial time properties of teams

Decidability of predicate logics with team semantics

We study the complexity of predicate logics based on team semantics. We show that the satisfiability problems of two-variable independence logic and inclusion logic are both NEXPTIME-complete. Furthermore, we show that the validity problem of two-variable dependence logic is undecidable, thereby solving an open problem from the team semantics literature. We also briefly analyse the complexity of the Bernays-Sch\"onfinkel-Ramsey prefix classes of dependence logic.Comment: Extended version of a MFCS 2016 article. Changes on the earlier arXiv version: title changed, added the result on validity of two-variable dependence logic, restructurin

On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella

Guarded Teams: The Horizontally Guarded Case

Team semantics admits reasoning about large sets of data, modelled by sets of assignments (called teams), with first-order syntax. This leads to high expressive power and complexity, particularly in the presence of atomic dependency properties for such data sets. It is therefore interesting to explore fragments and variants of logic with team semantics that permit model-theoretic tools and algorithmic methods to control this explosion in expressive power and complexity. We combine here the study of team semantics with the notion of guarded logics, which are well-understood in the case of classical Tarski semantics, and known to strike a good balance between expressive power and algorithmic manageability. In fact there are two strains of guardedness for teams. Horizontal guardedness requires the individual assignments of the team to be guarded in the usual sense of guarded logics. Vertical guardedness, on the other hand, posits an additional (or definable) hypergraph structure on relational structures in order to interpret a constraint on the component-wise variability of assignments within teams. In this paper we investigate the horizontally guarded case. We study horizontally guarded logics for teams and appropriate notions of guarded team bisimulation. In particular, we establish characterisation theorems that relate invariance under guarded team bisimulation with guarded team logics, but also with logics under classical Tarski semantics

Boolean Dependence Logic and Partially-Ordered Connectives

We introduce a new variant of dependence logic called Boolean dependence logic. In Boolean dependence logic dependence atoms are of the type =(x_1,...,x_n,\alpha), where \alpha is a Boolean variable. Intuitively, with Boolean dependence atoms one can express quantification of relations, while standard dependence atoms express quantification over functions. We compare the expressive power of Boolean dependence logic to dependence logic and first-order logic enriched by partially-ordered connectives. We show that the expressive power of Boolean dependence logic and dependence logic coincide. We define natural syntactic fragments of Boolean dependence logic and show that they coincide with the corresponding fragments of first-order logic enriched by partially-ordered connectives with respect to expressive power. We then show that the fragments form a strict hierarchy.Comment: 41 page