119 research outputs found
A multi-modal logic for Galois connections
Advances in Modal Logic 2022
(Rennes, August 22-25
Tense logic based on finite orthomodular posets
It is widely accepted that the logic of quantum mechanics is based on
orthomodular posets. However, such a logic is not dynamic in the sense that it
does not incorporate time dimension. To fill this gap, we introduce certain
tense operators on such a logic in an inexact way, but still satisfying
requirements asked on tense operators in the classical logic based on Boolean
algebras or in various non-classical logics. Our construction of tense
operators works perfectly when the orthomodular poset in question is finite. We
investigate the behaviour of these tense operators, e.g. we show that some of
them form a dynamic pair. Moreover, we prove that if the tense operators
preserve one of the inexact connectives conjunction or implication as defined
by the authors recently in another paper, then they also preserve the other
one. Finally, we show how to construct the binary relation of time preference
on a given time set provided the tense operators are given, up to equivalence
induced by natural quasiorders
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Attribute Exploration of Gene Regulatory Processes
This thesis aims at the logical analysis of discrete processes, in particular
of such generated by gene regulatory networks. States, transitions and
operators from temporal logics are expressed in the language of Formal Concept
Analysis. By the attribute exploration algorithm, an expert or a computer
program is enabled to validate a minimal and complete set of implications, e.g.
by comparison of predictions derived from literature with observed data. Here,
these rules represent temporal dependencies within gene regulatory networks
including coexpression of genes, reachability of states, invariants or possible
causal relationships. This new approach is embedded into the theory of
universal coalgebras, particularly automata, Kripke structures and Labelled
Transition Systems. A comparison with the temporal expressivity of Description
Logics is made. The main theoretical results concern the integration of
background knowledge into the successive exploration of the defined data
structures (formal contexts). Applying the method a Boolean network from
literature modelling sporulation of Bacillus subtilis is examined. Finally, we
developed an asynchronous Boolean network for extracellular matrix formation
and destruction in the context of rheumatoid arthritis.Comment: 111 pages, 9 figures, file size 2.1 MB, PhD thesis University of
Jena, Germany, Faculty of Mathematics and Computer Science, 2011. Online
available at http://www.db-thueringen.de/servlets/DocumentServlet?id=1960
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