19 research outputs found
An algebraic study of exactness in partial contexts
DMF@?s are the natural algebraic tool for modelling reasoning with Korner@?s partial predicates. We provide two representation theorems for DMF@?s which give rise to two adjunctions, the first between DMF and the category of sets and the second between DMF and the category of distributive lattices with minimum. Then we propose a logic L"{"1"} for dealing with exactness in partial contexts, which belongs neither to the Leibniz, nor to the Frege hierarchies, and carry on its study with techniques of abstract algebraic logic. Finally a fully adequate and algebraizable Gentzen system for L"{"1"} is given
An Algebraic Approach to Valued Constraint Satisfaction
[EN]We study the complexity of the valued CSP (VCSP, for short) over arbitrary templates, taking
the general framework of integral bounded linearly order monoids as valuation structures. The
class of problems considered here subsumes and generalizes the most common one in VCSP
literature, since both monoidal and lattice conjunction operations are allowed in the formulation
of constraints. Restricting to locally finite monoids, we introduce a notion of polymorphism that
captures the pp-definability in the style of Geigerâs result. As a consequence, sufficient conditions
for tractability of the classical CSP, related to the existence of certain polymorphisms, are shown
to serve also for the valued case. Finally, we establish the dichotomy conjecture for the VCSP,
modulo the dichotomy for classical CSP.The work was partly supported by the grant No. GA17-04630S of the Czech Science Foundation and partly by the long-term strategic development financing of the Institute of Computer Science
(RVO:67985807).Peer reviewe
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Strutture, logiche e pensiero
L'articolo si pone l'obiettivo di studiare un metodo per categorizzare la realtà che ci circonda e analizzarne le relazioni con il linguaggio in cui intendiamo parlarne. Vengono poi discussi i problemi filosofici che la formalizzazione comporta e le loro relazioni con il nostro processo di chiarificazione e conoscenza del mondo. Dopo aver mostrato come tutto ci si risolve nello studio dei rapporti che intercorrono tra la nozione di realtà , quella di linguaggio e quella di logica vengono presentate alcune possibili interpretazioni filosofiche, alcune confutate altre accettate, di questi rapporti. Infine si cerca di caratterizzare brevemente la relazione che il nostro pensiero deve instaurare con il linguaggio formale nel corso dei suoi processi deduttivi cercando cosÏ di legare questa nozione a quelle precedentemente introdotte
Degrees of the finite model property: the antidichotomy theorem
A classic result in modal logic, known as the Blok Dichotomy Theorem, states
that the degree of incompleteness of a normal extension of the basic modal
logic is or . It is a long-standing open problem
whether Blok Dichotomy holds for normal extensions of other prominent modal
logics (such as or ) or for extensions of the intuitionistic
propositional calculus . In this paper, we introduce the notion
of the degree of finite model property (fmp), which is a natural variation of
the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem
that the degree of fmp of a normal extension of remains or
. In contrast, our main result establishes the following
Antidichotomy Theorem for the degree of fmp for extensions of :
each nonzero cardinal such that or is realized as the degree of fmp of some extension of
. We then use the Blok-Esakia theorem to establish the same
Antidichotomy Theorem for normal extensions of and
Positive Modal Logic Beyond Distributivity
We develop a duality for (modal) lattices that need not be distributive, and
use it to study positive (modal) logic beyond distributivity, which we call
weak positive (modal) logic. This duality builds on the Hofmann, Mislove and
Stralka duality for meet-semilattices. We introduce the notion of
-persistence and show that every weak positive modal logic is
-persistent. This approach leads to a new relational semantics for weak
positive modal logic, for which we prove an analogue of Sahlqvist
correspondence result
The algebraic significance of weak excluded middle laws
Please read abstract in the article.National Research Foundation of South Africa;
Ministry of Science and Innovation of Spain;
AgĂšncia de GestiĂł d'Ajuts Universitaris i de Recerca.https://onlinelibrary.wiley.com/journal/15213870hj2023Mathematics and Applied Mathematic
Varieties of De Morgan monoids : covers of atoms
The variety DMM of De Morgan monoids has just four minimal
subvarieties. The join-irreducible covers of these atoms in the subvariety
lattice of DMM are investigated. One of the two atoms consisting
of idempotent algebras has no such cover; the other has just one. The
remaining two atoms lack nontrivial idempotent members. They are generated,
respectively, by 4{element De Morgan monoids C4 and D4, where
C4 is the only nontrivial 0{generated algebra onto which nitely subdirectly
irreducible De Morgan monoids may be mapped by non-injective
homomorphisms. The homomorphic pre-images of C4 within DMM (together
with the trivial De Morgan monoids) constitute a proper quasivariety,
which is shown to have a largest subvariety U. The covers of the
variety V(C4) within U are revealed here. There are just ten of them
(all nitely generated). In exactly six of these ten varieties, all nontrivial
members have C4 as a retract. In the varietal join of those six classes,
every subquasivariety is a variety|in fact, every nite subdirectly irreducible
algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)]
within DMM are discriminator varieties. Of these, we identify in nitely
many that are nitely generated, and some that are not. We also prove
that there are just 68 minimal quasivarieties of De Morgan monoids.The European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, RVO 67985807 and by the CAS-ICS postdoctoral fellowship, the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.https://www.cambridge.org/core/journals/review-of-symbolic-logic2021-06-01am2021Mathematics and Applied Mathematic
Singly generated quasivarieties and residuated structures
Please read abstract in the article.H2020 Marie SkĆodowska-Curie Actions;
DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa and National Research Foundation of South Africa.https://onlinelibrary.wiley.com/journal/15213870hj2021Mathematics and Applied Mathematic
Epimorphisms, definability and cardinalities
We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m+s+â”0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable âbridge theoremsâ, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic âą with suitable infinitary definability properties of âą, while not making the standard but awkward assumption that âą comes furnished with a proper class of variables.The European Unionâs Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project âSyntax Meets Semantics: Methods, Interactions, and Connections in Substructural logicsâ). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAÄR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.http://link.springer.com/journal/112252020-02-07hj2019Mathematics and Applied Mathematic