7,524 research outputs found
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
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
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
Dynamic Complexity of Parity Exists Queries
Given a graph whose nodes may be coloured red, the parity of the number of red nodes can easily be maintained with first-order update rules in the dynamic complexity framework DynFO of Patnaik and Immerman. Can this be generalised to other or even all queries that are definable in first-order logic extended by parity quantifiers? We consider the query that asks whether the number of nodes that have an edge to a red node is odd. Already this simple query of quantifier structure parity-exists is a major roadblock for dynamically capturing extensions of first-order logic.
We show that this query cannot be maintained with quantifier-free first-order update rules, and that variants induce a hierarchy for such update rules with respect to the arity of the maintained auxiliary relations. Towards maintaining the query with full first-order update rules, it is shown that degree-restricted variants can be maintained
The expressive power of modal logic with inclusion atoms
Modal inclusion logic is the extension of basic modal logic with inclusion
atoms, and its semantics is defined on Kripke models with teams. A team of a
Kripke model is just a subset of its domain. In this paper we give a complete
characterisation for the expressive power of modal inclusion logic: a class of
Kripke models with teams is definable in modal inclusion logic if and only if
it is closed under k-bisimulation for some integer k, it is closed under
unions, and it has the empty team property. We also prove that the same
expressive power can be obtained by adding a single unary nonemptiness operator
to modal logic. Furthermore, we establish an exponential lower bound for the
size of the translation from modal inclusion logic to modal logic with the
nonemptiness operator.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
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
Characterizing Quantifier Extensions of Dependence Logic
We characterize the expressive power of extensions of Dependence Logic and
Independence Logic by monotone generalized quantifiers in terms of quantifier
extensions of existential second-order logic.Comment: 9 page
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