Juha Kontinen

On Definability in Dependence Logic

We study the expressive power of open formulas of dependence logic introduced in Väänänen [Dependence logic (Vol. 70 of London Mathematical Society Student Texts), 2007]. In particular, we answer a question raised by Wilfrid Hodges: how to characterize the sets of teams definable by means of identity only in dependence logic, or equivalently in independence friendly logic.

Hierarchies in Dependence Logic

We study fragments <i>D</i>(<i>k</i>∀) and <i>D</i>(<i>k</i>-dep) of dependence logic defined either by restricting the number <i>k</i> of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic. We also show that, for any fixed signature, the fragments <i>D</i>(<i>k</i>∀) give rise to an infinite hierarchy with respect to expressive power. On the other hand, for the fragments <i>D</i>(<i>k</i>-dep), a hierarchy theorem is otained only in the case the signature is also allowed to vary. For any fixed signature, this question is open and is related to the so-called Spectrum Arity Hierarchy Conjecture.

Team Logic and Second-Order Logic

Team logic is a new logic, introduced by Vaananen [1], extending dependence logic by classical negation. Dependence logic adds to first-order logic atomic formulas expressing functional dependence of variables on each other. It is known that on the level of sentences dependence logic and team logic are equivalent with existential second-order logic and full second-order logic, respectively. In this article we show that, in a sense that we make explicit, team logic and second-order logic are also equivalent with respect to open formulas. A similar earlier result relating open formulas of dependence logic to the negative fragment of existential second-order logic was proved in [2].

Axiomatizing first-order consequences in dependence logic

Dependence logic, introduced in [8], cannot be axiomatized. However, first-order consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem.

Modal Independence Logic

This paper introduces modal independence logic MIL, a modal logic that can explicitly talk about independence among propositional variables. Formulas of MIL are not evaluated in worlds but in sets of worlds, so called teams. In this vein, MIL can be seen as a variant of Vaananen’s modal dependence logic MDL. We show that MIL embeds MDL and is strictly more expressive. However, on singleton teams, MIL is shown to be not more expressive than usual modal logic, but MIL is exponentially more succinct. Making use of a new form of bisimulation, we extend these expressivity results to modal logics extended by various generalized dependence atoms. We demonstrate the expressive power of MIL by giving a specification of the anonymity requirement of the dining cryptographers protocol in MIL. We also study complexity issues of MIL and show that, though it is more expressive, its satisfiability and model checking problem have the same complexity as for MDL.

Hierarchies in independence logic

We study the expressive power of fragments of inclusion and independence logic defined either by restricting the number of universal quantifiers or the arity of inclusion and independence atoms in formulas. Assuming the so-called lax semantics for these logics, we relate these fragments of inclusion and independence logic to familiar sublogics of existential second-order logic. We also show that, with respect to the stronger strict semantics, inclusion logic is equivalent to existential second-order logic.

Independence in Database Relations

We investigate the implication problem for independence atoms

A Van Benthem Theorem for Modal Team Semantics

X \bot Y

Complexity of Propositional Independence and Inclusion Logic

of disjoint attribute sets X and Y on database schemata. A relation satisfies

A Finite Axiomatization of Conditional Independence and Inclusion Dependencies

X \bot Y