Miika Hannula

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.

Complexity of Propositional Independence and Inclusion Logic

We classify the computational complexity of the satisfiability, validity and model-checking problems for propositional independence and inclusion logic and their extensions by the classical negation.

A Finite Axiomatization of Conditional Independence and Inclusion Dependencies

We present a complete finite axiomatization of the unrestricted implication problem for inclusion and conditional independence atoms in the context of dependence logic. For databases, our result implies a finite axiomatization of the unrestricted implication problem for inclusion, functional, and embedded multivalued dependencies in the unirelational case.

On the finite and general implication problems of independence atoms and keys

We investigate implication problems for keys and independence atoms in relational databases. For keys and unary independence atoms we show that finite implication is not finitely axiomatizable, and establish a finite axiomatization for general implication. The same axiomatization is also sound and complete for finite and general implication of unary keys and independence atoms, which coincide. We show that the general implication of keys and unary independence atoms and of unary keys and general independence atoms is decidable in polynomial time. For these two classes we also show how to construct Armstrong relations. Finally, we establish tractable conditions that are sufficient for certain classes of keys and independence atoms not to interact. For unary independence atoms and keys, general implication is finitely axiomatizable but finite implication is not.Finite and general implication of unary keys and independence atoms coincide and enjoy a finite axiomatization.General implication of unary independence atoms and keys is decidable in polynomial time.General implication of unary keys and independence atoms is decidable in polynomial time.There are tractable conditions that are sufficient for some classes of keys and independence atoms not to interact.

Axiomatizing first-order consequences in independence logic

Independence logic cannot be effectively axiomatized. However, first-order consequences of independence logic sentences can be axiomatized. In this article we give an explicit axiomatization and prove that it is complete in this sense. The proof is a generalization of the similar result for dependence logic.

Hierarchies in independence and inclusion logic with strict semantics

We study the expressive power of fragments of inclusion and independence logic defined by restricting the number k of universal quantifiers in formulas. Assuming the so-called strict semantics for these logics, we relate these fragments of inclusion and independence logic to sublogics ESO_f(k\forall) of existential second-order logic, which in turn are known to capture the complexity classes NTIME_{RAM}(n^k).

On Independence Atoms and Keys

Uniqueness and independence are two fundamental properties of data. Their enforcement in knowledge systems can lead to higher quality data, faster data service response time, better data-driven decision making and knowledge discovery from data. The applications can be effectively unlocked by providing efficient solutions to the underlying implication problems of keys and independence atoms. Indeed, for the sole class of keys and the sole class of independence atoms the associated finite and general implication problems coincide and enjoy simple axiomatizations. However, the situation changes drastically when keys and independence atoms are combined. We show that the finite and the general implication problems are already different for keys and unary independence atoms. Furthermore, we establish a finite axiomatization for the general implication problem, and show that the finite implication problem does not enjoy a k-ary axiomatization for any k.

A finite axiomatization of conditional independence and inclusion dependencies

We present a complete finite axiomatization of the unrestricted implication problem for inclusion and conditional independence atoms in the context of dependence logic. For databases, this result implies a finite axiomatization of the unrestricted implication problem for inclusion, functional, and embedded multivalued dependencies in the unirelational case. We also indicate the generality of our approach by showing the analogous result for inclusion and embedded join dependencies.

Reasoning About Embedded Dependencies Using Inclusion Dependencies

The implication problem for the class of embedded dependencies is undecidable. However, this does not imply lackness of a proof procedure as exemplified by the chase algorithm. In this paper we present a complete axiomatization of embedded dependencies that is based on the chase and uses inclusion dependencies and implicit existential quantification in the intermediate steps of deductions.

Approximation and Dependence via Multiteam Semantics

We define a variant of team semantics called multiteam semantics based on multisets and study the properties of various logics in this framework. In particular, we define natural probabilistic versions of inclusion and independence atoms and certain approximation operators motivated by approximate dependence atoms of Vaananen.