12 research outputs found
Decidability of predicate logics with team semantics
We study the complexity of predicate logics based on team semantics. We show
that the satisfiability problems of two-variable independence logic and
inclusion logic are both NEXPTIME-complete. Furthermore, we show that the
validity problem of two-variable dependence logic is undecidable, thereby
solving an open problem from the team semantics literature. We also briefly
analyse the complexity of the Bernays-Sch\"onfinkel-Ramsey prefix classes of
dependence logic.Comment: Extended version of a MFCS 2016 article. Changes on the earlier arXiv
version: title changed, added the result on validity of two-variable
dependence logic, restructurin
Safe Dependency Atoms and Possibility Operators in Team Semantics
I consider the question of which dependencies are safe for a Team
Semantics-based logic FO(D), in the sense that they do not increase its
expressive power over sentences when added to it. I show that some
dependencies, like totality, non-constancy and non-emptiness, are safe for all
logics FO(D), and that other dependencies, like constancy, are not safe for
FO(D) for some choices of D despite being strongly first order. I furthermore
show that the possibility operator, which holds in a team if and only if its
argument holds in some nonempty subteam, can be added to any logic FO(D)
without increasing its expressive power over sentences.Comment: In Proceedings GandALF 2018, arXiv:1809.0241
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
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
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
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
Propositional union closed team logics
In this paper, we study several propositional team logics that are closed under unions, including propositional inclusion logic. We show that all these logics are expressively complete, and we introduce sound and complete systems of natural deduction for these logics. We also discuss the locality property and its connection with interpolation in these logics. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Peer reviewe
On definability of team relations with k-invariant atoms
We study the expressive power of logics whose truth is defined over sets of assignments, called teams, instead of single assignments. Given a team X, any k-tuple of variables in the domain of X defines a corresponding k-ary team relation. Thus the expressive power of a logic L with team semantics amounts to the set of properties of team relations which L-formulas can define. We introduce a concept of k-invariance which is a natural semantic restriction on any atomic formulae with team semantics. Then we develop a novel proof method to show that, if L is an extension of FO with any k-invariant atoms, then there are such properties of (k+1)-ary team relations which cannot be defined in L. This method can be applied e.g. for arity fragments of various logics with team semantics to prove undefinability results. In particular, we make some interesting observations on the definability of binary team relations with unary inclusion-exclusion logic.publishedVersionPeer reviewe