7,824 research outputs found
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
The Expressive Power of k-ary Exclusion Logic
In this paper we study the expressive power of k-ary exclusion logic, EXC[k],
that is obtained by extending first order logic with k-ary exclusion atoms. It
is known that without arity bounds exclusion logic is equivalent with
dependence logic. By observing the translations, we see that the expressive
power of EXC[k] lies in between k-ary and (k+1)-ary dependence logics. We will
show that, at least in the case of k=1, the both of these inclusions are
proper.
In a recent work by the author it was shown that k-ary inclusion-exclusion
logic is equivalent with k-ary existential second order logic, ESO[k]. We will
show that, on the level of sentences, it is possible to simulate inclusion
atoms with exclusion atoms, and this way express ESO[k]-sentences by using only
k-ary exclusion atoms. For this translation we also need to introduce a novel
method for "unifying" the values of certain variables in a team. As a
consequence, EXC[k] captures ESO[k] on the level of sentences, and we get a
strict arity hierarchy for exclusion logic. It also follows that k-ary
inclusion logic is strictly weaker than EXC[k].
Finally we will use similar techniques to formulate a translation from ESO[k]
to k-ary inclusion logic with strict semantics. Consequently, for any arity
fragment of inclusion logic, strict semantics is more expressive than lax
semantics.Comment: Preprint of a paper in the special issue of WoLLIC2016 in Annals of
Pure and Applied Logic, 170(9):1070-1099, 201
Capturing k-ary Existential Second Order Logic with k-ary Inclusion-Exclusion Logic
In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which is
obtained by extending first order logic with k-ary inclusion and exclusion
atoms. We show that every formula of INEX[k] can be expressed with a formula of
k-ary existential second order logic, ESO[k]. Conversely, every formula of
ESO[k] with at most k-ary free relation variables can be expressed with a
formula of INEX[k]. From this it follows that, on the level of sentences,
INEX[k] captures the expressive power of ESO[k].
We also introduce several useful operators that can be expressed in INEX[k].
In particular, we define inclusion and exclusion quantifiers and so-called term
value preserving disjunction which is essential for the proofs of the main
results in this paper. Furthermore, we present a novel method of relativization
for team semantics and analyze the duality of inclusion and exclusion atoms.Comment: Extended version of a paper published in Annals of Pure and Applied
Logic 169 (3), 177-21
Characterizing downwards closed, strongly first order, relativizable dependencies
In Team Semantics, a dependency notion is strongly first order if every
sentence of the logic obtained by adding the corresponding atoms to First Order
Logic is equivalent to some first order sentence. In this work it is shown that
all nontrivial dependency atoms that are strongly first order, downwards
closed, and relativizable (in the sense that the relativizations of the
corresponding atoms with respect to some unary predicate are expressible in
terms of them) are definable in terms of constancy atoms.
Additionally, it is shown that any strongly first order dependency is safe
for any family of downwards closed dependencies, in the sense that every
sentence of the logic obtained by adding to First Order Logic both the strongly
first order dependency and the downwards closed dependencies is equivalent to
some sentence of the logic obtained by adding only the downwards closed
dependencies
Inclusion and Exclusion Dependencies in Team Semantics: On Some Logics of Imperfect Information
We introduce some new logics of imperfect information by adding atomic
formulas corresponding to inclusion and exclusion dependencies to the language
of first order logic. The properties of these logics and their relationships
with other logics of imperfect information are then studied. Furthermore, a
game theoretic semantics for these logics is developed. As a corollary of these
results, we characterize the expressive power of independence logic, thus
answering an open problem posed in (Gr\"adel and V\"a\"an\"anen, 2010)
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