3,445 research outputs found
Fuzzy Galois connections on fuzzy sets
In fairly elementary terms this paper presents how the theory of preordered
fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy
sets, is established under the guidance of enriched category theory. Motivated
by several key results from the theory of quantaloid-enriched categories, this
paper develops all needed ingredients purely in order-theoretic languages for
the readership of fuzzy set theorists, with particular attention paid to fuzzy
Galois connections between preordered fuzzy sets.Comment: 30 pages, final versio
A Galois-Connection between Myers-Briggs' Type Indicators and Szondi's Personality Profiles
We propose a computable Galois-connection between Myers-Briggs' Type
Indicators (MBTIs), the most widely-used personality measure for
non-psychiatric populations (based on C.G. Jung's personality types), and
Szondi's personality profiles (SPPs), a less well-known but, as we show, finer
personality measure for psychiatric as well as non-psychiatric populations
(conceived as a unification of the depth psychology of S. Freud, C.G. Jung, and
A. Adler). The practical significance of our result is that our
Galois-connection provides a pair of computable, interpreting translations
between the two personality spaces of MBTIs and SPPs: one concrete from
MBTI-space to SPP-space (because SPPs are finer) and one abstract from
SPP-space to MBTI-space (because MBTIs are coarser). Thus Myers-Briggs' and
Szondi's personality-test results are mutually interpretable and
inter-translatable, even automatically by computers
Landauer's principle as a special case of Galois connection
It is demonstrated how to construct a Galois connection between two related
systems with entropy. The construction, called the Landauer's connection,
describes coupling between two systems with entropy. It is straightforward and
transfers changes in one system to the other one preserving ordering structure
induced by entropy. The Landauer's connection simplifies the description of the
classical Landauer's principle for computational systems. Categorification and
generalization of the Landauer's principle opens area of modelling of various
systems in presence of entropy in abstract terms.Comment: 24 pages, 3 figure
Comparing and evaluating extended Lambek calculi
Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was
innovative in many ways, notably as a precursor of linear logic. But it also
showed that we could treat our grammatical framework as a logic (as opposed to
a logical theory). However, though it was successful in giving at least a basic
treatment of many linguistic phenomena, it was also clear that a slightly more
expressive logical calculus was needed for many other cases. Therefore, many
extensions and variants of the Lambek calculus have been proposed, since the
eighties and up until the present day. As a result, there is now a large class
of calculi, each with its own empirical successes and theoretical results, but
also each with its own logical primitives. This raises the question: how do we
compare and evaluate these different logical formalisms? To answer this
question, I present two unifying frameworks for these extended Lambek calculi.
Both are proof net calculi with graph contraction criteria. The first calculus
is a very general system: you specify the structure of your sequents and it
gives you the connectives and contractions which correspond to it. The calculus
can be extended with structural rules, which translate directly into graph
rewrite rules. The second calculus is first-order (multiplicative
intuitionistic) linear logic, which turns out to have several other,
independently proposed extensions of the Lambek calculus as fragments. I will
illustrate the use of each calculus in building bridges between analyses
proposed in different frameworks, in highlighting differences and in helping to
identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona,
Spain. 201
Towards a generalisation of formal concept analysis for data mining purposes
In this paper we justify the need for a generalisation of Formal
Concept Analysis for the purpose of data mining and begin the
synthesis of such theory. For that purpose, we first review semirings and
semimodules over semirings as the appropriate objects to use in abstracting
the Boolean algebra and the notion of extents and intents, respectively.
We later bring to bear powerful theorems developed in the field
of linear algebra over idempotent semimodules to try to build a Fundamental
Theorem for K-Formal Concept Analysis, where K is a type of
idempotent semiring. Finally, we try to put Formal Concept Analysis in
new perspective by considering it as a concrete instance of the theory
developed
A Galois-Connection between Cattell's and Szondi's Personality Profiles
We propose a computable Galois-connection between, on the one hand, Cattell's
16-Personality-Factor (16PF) Profiles, one of the most comprehensive and
widely-used personality measures for non-psychiatric populations and their
containing PsychEval Personality Profiles (PPPs) for psychiatric populations,
and, on the other hand, Szondi's personality profiles (SPPs), a less well-known
but, as we show, finer personality measure for psychiatric as well as
non-psychiatric populations (conceived as a unification of the depth psychology
of S. Freud, C.G. Jung, and A. Adler). The practical significance of our result
is that our Galois-connection provides a pair of computable, interpreting
translations between the two personality spaces of PPPs (containing the 16PFs)
and SPPs: one concrete from PPP-space to SPP-space (because SPPs are finer than
PPPs) and one abstract from SPP-space to PPP-space (because PPPs are coarser
than SPPs). Thus Cattell's and Szondi's personality-test results are mutually
interpretable and inter-translatable, even automatically by computers.Comment: closely related to arXiv:1403.2000 as explained in the first
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