On Variants of Dependence Logic : Axiomatizability and Expressiveness

Dependence logic is a novel logical formalism that has connections to database theory, statistics, linguistics, social choice theory, and physics. Its aim is to provide a systematic and mathematically rigorous tool for studying notions of dependence and independence in different areas. Recently many variants of dependence logic have been studied in the contexts of first-order, modal, and propositional logic. In this thesis we examine independence and inclusion logic that are variants of dependence logic extending first-order logic with so-called independence or inclusion atoms, respectively. The work consists of two parts in which we study either axiomatizability or expressivity hierarchies regarding these logics. In the first part we examine whether there exist some natural parameters of independence and inclusion logic that give rise to infinite expressivity or complexity hierarchies. Two main parameters are considered. These are arity of a dependency atom and number of universal quantifiers. We show that for both logics, the notion of arity gives rise to strict expressivity hierarchies. With respect to number of universal quantifiers however, strictness or collapse of the corresponding hierarchies turns out to be relative to the choice of semantics. In the second part we turn attention to axiomatizations. Due to their complexity, dependence and independence logic cannot have a complete recursively enumerable axiomatization. Hence, restricting attention to partial solutions, we first axiomatize all first-order consequences of independence logic sentences, thus extending an analogous result for dependence logic. We also consider the class of independence and inclusion atoms, and show that it can be axiomatized using implicit existential quantification. For relational databases this implies a sound and complete axiomatization of embedded multivalued and inclusion dependencies taken together. Lastly, we consider keys together with so-called pure independence atoms and prove both positive and negative results regarding their finite axiomatizability.Riippuvuuslogiikka on formalismi, joka tutkii muodollisen logiikan viitekehyksessä riippuvuuden ja riippumattomuuden käsitteitä. Koska nämä käsitteet ilmenevät myös monilla muilla eri tieteenaloilla, riippuvuuslogiikan tutkimus kytkeytyy muun muassa tietokantateoriaan, tilastotieteeseen, kielitieteeseen, sosiaalisen valinnan teoriaan ja fysiikkaan. Ideana riippuvuuslogiikassa on laajentaa tunnettuja muodollisen logiikan kieliä erilaisilla riippuvuuden käsitteillä. Propositio-, modaali- ja predikaattilogiikoille voidaan kaikille määritellä laajennoksia, joissa riippuvuuksia ilmaistaan uusien atomikaavojen avulla. Tämä väitöskirja tarkastelee kahta tällaista ensimmäisen kertaluvun predikaattilogiikan laajennosta. Toisessa uudet atomikaavat kuvaavat riippumattomuuden, ja toisessa sisältyvyyden käsitteitä. Saatuja laajennoksia kutsutaan riippumattomuuslogiikaksi ja inkluusiologiikaksi. Tutkielma jakautuu kahteen osaan. Ensimmäisessä osassa tarkastellaan edellä mainittujen logiikoiden ilmaisuvoimaan ja laskennalliseen vaativuuteen liittyviä hierarkioita. Kyseiset hierarkiat saadaan rajoittamalla joko uusien atomikaavojen kokoa tai universaalikvanttorien lukumäärää. Toisessa osassa tutkitaan riippumattomuus- ja inkluusiologiikan muodollista päättelyä. Tarkastelun kohteena on muodollisen päättelyn kehittäminen riippumattomuuslogiikan ensimmäisen kertaluvun seurauksille sekä erilaisille kokoelmille uusia atomikaavoja. Jälkimmäiseen tapaukseen kehitetty muodollisen päättelyn teoria soveltuu erityisesti relationaalisten tietokantojen riippuvuuskäsitteiden implikaatio-ongelmiin

Asymptotic properties of keys and functional dependencies in random databases

AbstractPractical database applications give the impression that sets of constraints are rather small and that large sets are unusual and are caused by bad design decisions. Theoretical investigations, however, show that minimal constraint sets are potentially very large. Their size can be estimated to be exponential in terms of the number of attributes. The gap between observation in practice and theory results in the rejection of theoretical results. However, practice is related to average cases and is not related to worst cases.The theory used until now considered the worst-case complexity. This paper aims to develop a theory for the average-case complexity. Several probabilistic models and asymptotics of corresponding probabilities are investigated for random databases formed by independent random tuples with a common discrete distribution. Poisson approximations are studied for the distributions of some characteristics for such databases where the number of tuples is sufficiently large. We intend to prove that the exponential complexity of key sets and sets of functional dependencies is rather unusual and almost all minimal keys in a relation have a length which depends mainly on the size of the relation

On the Discovery of Semantically Meaningful SQL Constraints from Armstrong Samples: Foundations, Implementation, and Evaluation

A database is said to be C-Armstrong for a finite set Σ of data dependencies in a class C if the database satisfies all data dependencies in Σ and violates all data dependencies in C that are not implied by Σ. Therefore, Armstrong databases are concise, user-friendly representations of abstract data dependencies that can be used to judge, justify, convey, and test the understanding of database design choices. Indeed, an Armstrong database satisfies exactly those data dependencies that are considered meaningful by the current design choice Σ. Structural and computational properties of Armstrong databases have been deeply investigated in Codd’s Turing Award winning relational model of data. Armstrong databases have been incorporated in approaches towards relational database design. They have also been found useful for the elicitation of requirements, the semantic sampling of existing databases, and the specification of schema mappings. This research establishes a toolbox of Armstrong databases for SQL data. This is challenging as SQL data can contain null marker occurrences in columns declared NULL, and may contain duplicate rows. Thus, the existing theory of Armstrong databases only applies to idealized instances of SQL data, that is, instances without null marker occurrences and without duplicate rows. For the thesis, two popular interpretations of null markers are considered: the no information interpretation used in SQL, and the exists but unknown interpretation by Codd. Furthermore, the study is limited to the popular class C of functional dependencies. However, the presence of duplicate rows means that the class of uniqueness constraints is no longer subsumed by the class of functional dependencies, in contrast to the relational model of data. As a first contribution a provably-correct algorithm is developed that computes Armstrong databases for an arbitrarily given finite set of uniqueness constraints and functional dependencies. This contribution is based on axiomatic, algorithmic and logical characterizations of the associated implication problem that are also established in this thesis. While the problem to decide whether a given database is Armstrong for a given set of such constraints is precisely exponential, our algorithm computes an Armstrong database with a number of rows that is at most quadratic in the number of rows of a minimum-sized Armstrong database. As a second contribution the algorithms are implemented in the form of a design tool. Users of the tool can therefore inspect Armstrong databases to analyze their current design choice Σ. Intuitively, Armstrong databases are useful for the acquisition of semantically meaningful constraints, if the users can recognize the actual meaningfulness of constraints that they incorrectly perceived as meaningless before the inspection of an Armstrong database. As a final contribution, measures are introduced that formalize the term “useful” and it is shown by some detailed experiments that Armstrong tables, as computed by the tool, are indeed useful. In summary, this research establishes a toolbox of Armstrong databases that can be applied by database designers to concisely visualize constraints on SQL data. Such support can lead to database designs that guarantee efficient data management in practice