1,333 research outputs found
On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics
We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella
Guarded Teams: The Horizontally Guarded Case
Team semantics admits reasoning about large sets of data, modelled by sets of assignments (called teams), with first-order syntax. This leads to high expressive power and complexity, particularly in the presence of atomic dependency properties for such data sets. It is therefore interesting to explore fragments and variants of logic with team semantics that permit model-theoretic tools and algorithmic methods to control this explosion in expressive power and complexity.
We combine here the study of team semantics with the notion of guarded logics, which are well-understood in the case of classical Tarski semantics, and known to strike a good balance between expressive power and algorithmic manageability. In fact there are two strains of guardedness for teams. Horizontal guardedness requires the individual assignments of the team to be guarded in the usual sense of guarded logics. Vertical guardedness, on the other hand, posits an additional (or definable) hypergraph structure on relational structures in order to interpret a constraint on the component-wise variability of assignments within teams.
In this paper we investigate the horizontally guarded case. We study horizontally guarded logics for teams and appropriate notions of guarded team bisimulation. In particular, we establish characterisation theorems that relate invariance under guarded team bisimulation with guarded team logics, but also with logics under classical Tarski semantics
Upwards Closed Dependencies in Team Semantics
We prove that adding upwards closed first-order dependency atoms to
first-order logic with team semantics does not increase its expressive power
(with respect to sentences), and that the same remains true if we also add
constancy atoms. As a consequence, the negations of functional dependence,
conditional independence, inclusion and exclusion atoms can all be added to
first-order logic without increasing its expressive power.
Furthermore, we define a class of bounded upwards closed dependencies and we
prove that unbounded dependencies cannot be defined in terms of bounded ones.Comment: In Proceedings GandALF 2013, arXiv:1307.416
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
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
Counting in Team Semantics
We explore several counting constructs for logics with team semantics. Counting is an important task in numerous applications, but with a somewhat delicate relationship to logic. Team semantics on the other side is the mathematical basis of modern logics of dependence and independence, in which formulae are evaluated not for a single assignment of values to variables, but for a set of such assignments. It is therefore interesting to ask what kind of counting constructs are adequate in this context, and how such constructs influence the expressive power, and the model-theoretic and algorithmic properties of logics with team semantics. Due to the second-order features of team semantics there is a rich variety of potential counting constructs. Here we study variations of two main ideas: forking atoms and counting quantifiers.
Forking counts how many different values for a tuple w occur in assignments with coinciding values for v. We call this the forking degree of bar v with respect to bar w. Forking is powerful enough to capture many of the previously studied atomic dependency properties. In particular we exhibit logics with forking atoms that have, respectively, precisely the power of dependence logic and independence logic.
Our second approach uses counting quantifiers E^{geq mu} of a similar kind as used in logics with Tarski semantics. The difference is that these quantifiers are now applied to teams of assignments that may give different values to mu. We show that, on finite structures, there is an intimate connection between inclusion logic with counting quantifiers and FPC, fixed-point logic with counting, which is a logic of fundamental importance for descriptive complexity theory. For sentences, the two logics have the same expressive power. Our analysis is based on a new variant of model-checking games, called threshold safety games, on a trap condition for such games, and on game interpretations
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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