1,306,842 research outputs found
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
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
The lattice of varieties of implication semigroups
In 2012, the second author introduced and examined a new type of algebras as
a generalization of De Morgan algebras. These algebras are of type (2,0) with
one binary and one nullary operation satisfying two certain specific
identities. Such algebras are called implication zroupoids. They invesigated in
a number of articles by the second author and J.M.Cornejo. In these articles
several varieties of implication zroupoids satisfying the associative law
appeared. Implication zroupoids satisfying the associative law are called
implication semigroups. Here we completely describe the lattice of all
varieties of implication semigroups. It turns out that this lattice is
non-modular and consists of 16 elements.Comment: Compared with the previous version, we rewrite Section 3 and add
Appendixes A and
A cognitive view of relevant implication
Relevant logics provide an alternative to classical implication
that is capable of accounting for the relationship between the antecedent
and the consequence of a valid implication. Relevant implication is usually
explained in terms of information required to assess a proposition.
By doing so, relevant implication introduces a number of cognitively relevant
aspects in the denition of logical operators. In this paper, we
aim to take a closer look at the cognitive feature of relevant implication.
For this purpose, we develop a cognitively-oriented interpretation of the
semantics of relevant logics. In particular, we provide an interpretation
of Routley-Meyer semantics in terms of conceptual spaces and we show
that it meets the constraints of the algebraic semantics of relevant logic
On Independence Atoms and Keys
Uniqueness and independence are two fundamental properties of data. Their
enforcement in database systems can lead to higher quality data, faster data
service response time, better data-driven decision making and knowledge
discovery from data. The applications can be effectively unlocked by providing
efficient solutions to the underlying implication problems of keys and
independence atoms. Indeed, for the sole class of keys and the sole class of
independence atoms the associated finite and general implication problems
coincide and enjoy simple axiomatizations. However, the situation changes
drastically when keys and independence atoms are combined. We show that the
finite and the general implication problems are already different for keys and
unary independence atoms. Furthermore, we establish a finite axiomatization for
the general implication problem, and show that the finite implication problem
does not enjoy a k-ary axiomatization for any k
On the Usability of Probably Approximately Correct Implication Bases
We revisit the notion of probably approximately correct implication bases
from the literature and present a first formulation in the language of formal
concept analysis, with the goal to investigate whether such bases represent a
suitable substitute for exact implication bases in practical use-cases. To this
end, we quantitatively examine the behavior of probably approximately correct
implication bases on artificial and real-world data sets and compare their
precision and recall with respect to their corresponding exact implication
bases. Using a small example, we also provide qualitative insight that
implications from probably approximately correct bases can still represent
meaningful knowledge from a given data set.Comment: 17 pages, 8 figures; typos added, corrected x-label on graph
Predicativity and parametric polymorphism of Brouwerian implication
A common objection to the definition of intuitionistic implication in the
Proof Interpretation is that it is impredicative. I discuss the history of that
objection, argue that in Brouwer's writings predicativity of implication is
ensured through parametric polymorphism of functions on species, and compare
this construal with the alternative approaches to predicative implication of
Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
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