29,311 research outputs found
Hidden assumptions in the derivation of the Theorem of Bell
John Bell's inequalities have already been considered by Boole in 1862. Boole
established a one-to-one correspondence between experimental outcomes and
mathematical abstractions of his probability theory. His abstractions are
two-valued functions that permit the logical operations AND, OR and NOT and are
the elements of an algebra. Violation of the inequalities indicated to Boole an
inconsistency of definition of the abstractions and/or the necessity to revise
the algebra. It is demonstrated in this paper, that a violation of Bell's
inequality by Einstein-Podolsky-Rosen type of experiments can be explained by
Boole's ideas. Violations of Bell's inequality also call for a revision of the
mathematical abstractions and corresponding algebra. It will be shown that this
particular view of Bell's inequalities points toward an incompleteness of
quantum mechanics, rather than to any superluminal propagation or influences at
a distance
Integrity Constraints Revisited: From Exact to Approximate Implication
Integrity constraints such as functional dependencies (FD), and multi-valued
dependencies (MVD) are fundamental in database schema design. Likewise,
probabilistic conditional independences (CI) are crucial for reasoning about
multivariate probability distributions. The implication problem studies whether
a set of constraints (antecedents) implies another constraint (consequent), and
has been investigated in both the database and the AI literature, under the
assumption that all constraints hold exactly. However, many applications today
consider constraints that hold only approximately. In this paper we define an
approximate implication as a linear inequality between the degree of
satisfaction of the antecedents and consequent, and we study the relaxation
problem: when does an exact implication relax to an approximate implication? We
use information theory to define the degree of satisfaction, and prove several
results. First, we show that any implication from a set of data dependencies
(MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most
quadratic in the number of variables; when the consequent is an FD, the factor
can be reduced to 1. Second, we prove that there exists an implication between
CIs that does not admit any relaxation; however, we prove that every
implication between CIs relaxes "in the limit". Finally, we show that the
implication problem for differential constraints in market basket analysis also
admits a relaxation with a factor equal to 1. Our results recover, and
sometimes extend, several previously known results about the implication
problem: implication of MVDs can be checked by considering only 2-tuple
relations, and the implication of differential constraints for frequent item
sets can be checked by considering only databases containing a single
transaction
Lattices with non-Shannon Inequalities
We study the existence or absence of non-Shannon inequalities for variables
that are related by functional dependencies. Although the power-set on four
variables is the smallest Boolean lattice with non-Shannon inequalities there
exist lattices with many more variables without non-Shannon inequalities. We
search for conditions that ensures that no non-Shannon inequalities exist. It
is demonstrated that 3-dimensional distributive lattices cannot have
non-Shannon inequalities and planar modular lattices cannot have non-Shannon
inequalities. The existence of non-Shannon inequalities is related to the
question of whether a lattice is isomorphic to a lattice of subgroups of a
group.Comment: Ten pages. Submitted to ISIT 2015. The appendix will not appear in
the proceeding
Querying Schemas With Access Restrictions
We study verification of systems whose transitions consist of accesses to a
Web-based data-source. An access is a lookup on a relation within a relational
database, fixing values for a set of positions in the relation. For example, a
transition can represent access to a Web form, where the user is restricted to
filling in values for a particular set of fields. We look at verifying
properties of a schema describing the possible accesses of such a system. We
present a language where one can describe the properties of an access path, and
also specify additional restrictions on accesses that are enforced by the
schema. Our main property language, AccLTL, is based on a first-order extension
of linear-time temporal logic, interpreting access paths as sequences of
relational structures. We also present a lower-level automaton model,
Aautomata, which AccLTL specifications can compile into. We show that AccLTL
and A-automata can express static analysis problems related to "querying with
limited access patterns" that have been studied in the database literature in
the past, such as whether an access is relevant to answering a query, and
whether two queries are equivalent in the accessible data they can return. We
prove decidability and complexity results for several restrictions and variants
of AccLTL, and explain which properties of paths can be expressed in each
restriction.Comment: VLDB201
Integrity Constraints Revisited: From Exact to Approximate Implication
Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked by considering only databases containing a single transaction
Uncertainty in Crowd Data Sourcing under Structural Constraints
Applications extracting data from crowdsourcing platforms must deal with the
uncertainty of crowd answers in two different ways: first, by deriving
estimates of the correct value from the answers; second, by choosing crowd
questions whose answers are expected to minimize this uncertainty relative to
the overall data collection goal. Such problems are already challenging when we
assume that questions are unrelated and answers are independent, but they are
even more complicated when we assume that the unknown values follow hard
structural constraints (such as monotonicity).
In this vision paper, we examine how to formally address this issue with an
approach inspired by [Amsterdamer et al., 2013]. We describe a generalized
setting where we model constraints as linear inequalities, and use them to
guide the choice of crowd questions and the processing of answers. We present
the main challenges arising in this setting, and propose directions to solve
them.Comment: 8 pages, vision paper. To appear at UnCrowd 201
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