Functions as types or the "Hoare logic" of functional dependencies

Inspired by the trend on unifying theories of programming, this paper shows how the algebraic treatment of standard data dependency theory equips relational data with functional types and an associated type system which is useful for type checking database operations and for query optimization. Such a typed approach to database programming is then shown to be of the same family as other programming logics such as eg. Hoare logic or that of strongest invariant functions which has been used in the analysis of while statements. The prospect of using automated deduction systems such as Prover9 for type-checking and query optimization on top of such an algebraic approach is considered.Fundação para a Ciência e a Tecnologia (FCT

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

ON CONSISTENT EXTENSIONS TO THE RELATIONAL DATABASE MODEL

Information Systems Working Papers Serie

Genuine Process Logic

The Genuine Process Logic described here (abbreviation: GPL) places the object-bound process itself at the center of formalism. It should be suitable for everyday use, i.e. it is not primarily intended for the formalization of computer programs, but instead, as a counter-conception to the classical state logics. The new and central operator of the GPL is an action symbol replacing the classical state symbols, e.g. of equivalence or identity. The complete renunciation of object-language state expressions also results in a completely new metalinguistic framework, both regarding the axioms and the expressive possibilities of this system. A mixture with state logical terms is readily possible

Talking about Forests: an Example of Sharing Information Expressed with Vague Terms

Most natural language terms do not have precise universally agreed definitions that fix their meanings. Even when conversation participants share the same vocabulary and agree on taxonomic relationships (such as subsumption and mutual exclusivity, which might be encoded in an ontology), they may differ greatly in the specific semantics they give to the terms. We illustrate this with the example of `forest', for which the problematic arising of the assignation of different meanings is repeatedly reported in the literature. This is especially the case in the context of an unprecedented scale of publicly available geographic data, where information and databases, even when tagged to ontologies, may present a substantial semantic variation, which challenges interoperability and knowledge exchange. Our research addresses the issue of conceptual vagueness in ontology by providing a framework based on supervaluation semantics that explicitly represents the semantic variability of a concept as a set of admissible precise interpretations. Moreover, we describe the tools that support the conceptual negotiation between an agent and the system, and the specification and reasoning within standpoints

ON CONSISTENT EXTENSIONS TO THE RELATIONAL DATABASE MODEL

Information Systems Working Papers Serie

Pointfree foundations for (generic) lossless decomposition

This report presents a typed, “pointfree” generalization of relational data depen- dency theory expressed not in the standard set-theoretic way, “a` la Codd”, but in the calculus of binary relations which, initiated by De Morgan in the 1860s, is the core of modern algebra of programming. Contrary to the intuition that a binary relation is just a particular case of an n- ary relation, this report shows the effectiveness of the former in “explaining” and reasoning about the latter. Data dependency theory, which is central to relational database design, is addressed in pointfree calculational style instead of reasoning about (sets of) tuples in conventional “implication-first” logic style. It turns out that the theory becomes more general, more structured and sim- pler. Elegant expressions replace lengthy formulæ and easy-to-follow calculations replace pointwise proofs with lots of “· · ·” notation, case analysis and natural lan- guage explanations for “obvious” steps. In particular, attributes are generalized to arbitrary (observation) functions and the principle of lossless decomposition is established for arbitrary such functions. The report concludes by showing how the proposed generalization of data dependency theory paves the way to interesting synergies with other branches of computer science, namely formal modeling and transition systems theory. A number of open topics for research in the field are proposed as future work.Fundação para a Ciência e a Tecnologia (FCT

FORMAL SEMANTICS FOR TIME IN DATABASES

The concept of an historical database is introduced as a tool for modelling the dynamic nature of some part of the real world. Just as first-order logic has been shown to be a useful formalism for expressing and understanding the underlying semantics of the relational database model, intensional logic is presented as an analogous formalism for expressing and understanding the temporal semantics involved in an historical database. The various components of the relational model, as extended to include historical relations, are discussed in terms of the model theory for the logic ILs, a variation of the logic IL formulated by Richard Montague. The modal concepts of intensional and extensional data constraints and queries are introduced and contrasted. Finally, the potential application of these ideas to the problem of Natural Language Database Querying is discussed.Information Systems Working Papers Serie