444 research outputs found
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
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
A Characterization Result for Non-Distributive Logics
Recent published work has addressed the Shalqvist correspondence problem for
non-distributive logics. The natural question that arises is to identify the
fragment of first-order logic that corresponds to logics without distribution,
lifting van Benthem's characterization result for modal logic to this new
setting. Carrying out this project is the contribution of the present article.
The article is intended as a demonstration and application of a project of
reduction of non-distributive logics to (sorted) residuated modal logics. The
reduction is an application of recent representation results by this author for
normal lattice expansions and a generalization of a canonical and fully
abstract translation of the language of substructural logics into the language
of their companion sorted, residuated modal logics. The reduction of
non-distributive logics to sorted modal logics makes the proof of a van Benthem
characterization of non-distributive logics nearly effortless, by adapting and
reusing existing results, demonstrating the usefulness and suitability of this
approach in studying logics that may lack distribution
Reconciliation of Approaches to the Semantics of Logics without Distribution
This article contributes in that it clarifies and indeed completes an
approach (initiated by Dunn and this author several years ago and again pursued
by the present author over the last three years or so) to the relational
semantics of logics that may lack distribution (Dunn's non-distributive
gaggles). The approach uses sorted frames with an incidence relation on sorts
(polarities), equipped with additional sorted relations, but, in the spirit of
Occam's razor principle, it drops the extra assumptions made in the generalized
Kripke frames approach, initiated by Gehrke, that the frames be separated and
reduced (RS-frames). We show in this article that, despite rejecting the
additional frame restrictions, all the main ideas and results of the RS-frames
approach relating to the semantics of non-distributive logics are captured in
this simpler framework. This contributes in unifying the research field, and,
in an important sense, it complements and completes Dunn's gaggle theory
project for the particular case of logics that may drop distribution
Two-sorted Modal Logic for Formal and Rough Concepts
In this paper, we propose two-sorted modal logics for the representation and
reasoning of concepts arising from rough set theory (RST) and formal concept
analysis (FCA). These logics are interpreted in two-sorted bidirectional
frames, which are essentially formal contexts with converse relations. On one
hand, the logic contains ordinary necessity and possibility
modalities and can represent rough set-based concepts. On the other hand, the
logic has window modality that can represent formal concepts. We
study the relationship between \textbf{KB} and \textbf{KF} by proving a
correspondence theorem. It is then shown that, using the formulae with modal
operators in \textbf{KB} and \textbf{KF}, we can capture formal concepts based
on RST and FCA and their lattice structures
Dual characterizations for finite lattices via correspondence theory for monotone modal logic
International audienceWe establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation's characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic
Relational lattices via duality
The natural join and the inner union combine in different ways tables of a
relational database. Tropashko [18] observed that these two operations are the
meet and join in a class of lattices-called the relational lattices- and
proposed lattice theory as an alternative algebraic approach to databases.
Aiming at query optimization, Litak et al. [12] initiated the study of the
equational theory of these lattices. We carry on with this project, making use
of the duality theory developed in [16]. The contributions of this paper are as
follows. Let A be a set of column's names and D be a set of cell values; we
characterize the dual space of the relational lattice R(D, A) by means of a
generalized ultrametric space, whose elements are the functions from A to D,
with the P (A)-valued distance being the Hamming one but lifted to subsets of
A. We use the dual space to present an equational axiomatization of these
lattices that reflects the combinatorial properties of these generalized
ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that
these equations correspond to combinatorial properties of the dual spaces of
lattices, in a technical sense analogous of correspondence theory in modal
logic. In particular, this leads to an exact characterization of the finite
lattices satisfying these equations.Comment: Coalgebraic Methods in Computer Science 2016, Apr 2016, Eindhoven,
Netherland
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