904 research outputs found
Dependence Logic with Generalized Quantifiers: Axiomatizations
We prove two completeness results, one for the extension of dependence logic
by a monotone generalized quantifier Q with weak interpretation, weak in the
meaning that the interpretation of Q varies with the structures. The second
result considers the extension of dependence logic where Q is interpreted as
"there exists uncountable many." Both of the axiomatizations are shown to be
sound and complete for FO(Q) consequences.Comment: 17 page
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
Team Semantics and Recursive Enumerability
It is well known that dependence logic captures the complexity class NP, and
it has recently been shown that inclusion logic captures P on ordered models.
These results demonstrate that team semantics offers interesting new
possibilities for descriptive complexity theory. In order to properly
understand the connection between team semantics and descriptive complexity, we
introduce an extension D* of dependence logic that can define exactly all
recursively enumerable classes of finite models. Thus D* provides an approach
to computation alternative to Turing machines. The essential novel feature in
D* is an operator that can extend the domain of the considered model by a
finite number of fresh elements. Due to the close relationship between
generalized quantifiers and oracles, we also investigate generalized
quantifiers in team semantics. We show that monotone quantifiers of type (1)
can be canonically eliminated from quantifier extensions of first-order logic
by introducing corresponding generalized dependence atoms
Weak MSO: Automata and Expressiveness Modulo Bisimilarity
We prove that the bisimulation-invariant fragment of weak monadic
second-order logic (WMSO) is equivalent to the fragment of the modal
-calculus where the application of the least fixpoint operator is restricted to formulas that are continuous in . Our
proof is automata-theoretic in nature; in particular, we introduce a class of
automata characterizing the expressive power of WMSO over tree models of
arbitrary branching degree. The transition map of these automata is defined in
terms of a logic that is the extension of first-order
logic with a generalized quantifier , where means that there are infinitely many objects satisfying . An
important part of our work consists of a model-theoretic analysis of
.Comment: Technical Report, 57 page
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
A logic with temporally accessible iteration
Deficiency in expressive power of the first-order logic has led to developing
its numerous extensions by fixed point operators, such as Least Fixed-Point
(LFP), inflationary fixed-point (IFP), partial fixed-point (PFP), etc. These
logics have been extensively studied in finite model theory, database theory,
descriptive complexity. In this paper we introduce unifying framework, the
logic with iteration operator, in which iteration steps may be accessed by
temporal logic formulae. We show that proposed logic FO+TAI subsumes all
mentioned fixed point extensions as well as many other fixed point logics as
natural fragments. On the other hand we show that over finite structures FO+TAI
is no more expressive than FO+PFP. Further we show that adding the same
machinery to the logic of monotone inductions (FO+LFP) does not increase its
expressive power either
Uniform Interpolation for Coalgebraic Fixpoint Logic
We use the connection between automata and logic to prove that a wide class
of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first
we generalize one of the central results in coalgebraic automata theory, namely
closure under projection, which is known to hold for weak-pullback preserving
functors, to a more general class of functors, i.e.; functors with
quasi-functorial lax extensions. Then we will show that closure under
projection implies definability of the bisimulation quantifier in the language
of coalgebraic fixpoint logic, and finally we prove the uniform interpolation
theorem
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