14,045 research outputs found
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
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
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
The Budget-Constrained Functional Dependency
Armstrong's axioms of functional dependency form a well-known logical system
that captures properties of functional dependencies between sets of database
attributes. This article assumes that there are costs associated with
attributes and proposes an extension of Armstrong's system for reasoning about
budget-constrained functional dependencies in such a setting.
The main technical result of this article is the completeness theorem for the
proposed logical system. Although the proposed axioms are obtained by just
adding cost subscript to the original Armstrong's axioms, the proof of the
completeness for the proposed system is significantly more complicated than
that for the Armstrong's system
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
Metalogic and the Overgeneration Argument
A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic
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
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