Persona: Rivieccio, Umberto
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Rivieccio
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Publicación Residuated bilattices(Springer, 2012) Jansana, Ramón; Rivieccio, UmbertoWe introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lattice factors. Finally, we show how to employ our product construction to define first-order definable classes of bilattices corresponding to any first-order definable subclass of residuated lattices.Publicación Priestley duality for bilattices(Springer, 2012) Jung, Achim; Rivieccio, UmbertoWe develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.Publicación An infinity of super-Belnap logics(Taylor & Francis, 2012) Rivieccio, UmbertoWe look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable.Publicación Characterizing finite-valuedness(Elsevier, 2018-08-15) Caleiro, Carlos; Marcelino, Sérgio; Rivieccio, UmbertoWe introduce properties of consequence relations that provide abstract counterparts of different notions of finite-valuedness in logic. In particular, we obtain characterizations of logics that are determined (i) by a single finite matrix, (ii) by a finite set of finite matrices, and (iii) by a set of n-generated matrices for some natural number n. A crucial role is played in our proofs by two closely related notions, local tabularity and local finiteness.Publicación An algebraic view of super-Belnap logics. Studia Logica(Springer Nature, 2017-07-28) Albuquerque, Hugo; Přenosil, Adam; Rivieccio, UmbertoThe Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.Publicación Bilattice Logic Properly Displayed(Elsevier, 2019-05-15) Greco, Giuseppe; Fei Liang,; Palmigiano,Alessandra; Rivieccio, UmbertoWe introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.Publicación Compatibly involutive residuated lattices and the Nelson identity(Springer Nature, 2018-11-03) Matthew Spinks; Rivieccio, Umberto; Nascimento, ThiagoNelson’s constructive logic with strong negation N3 can be presented (to within definitional equivalence) as the axiomatic extension NInFL ew of the involutive full Lambek calculus with exchange and weakening by the Nelson axiom[Figure not available: see fulltext.] The algebraic counterpart of NInFL ew is the recently introduced class of Nelson residuated lattices. These are commutative integral bounded residuated lattices ⟨ A; ∧ , ∨ , ∗ , ⇒ , 0 , 1 ⟩ that: (i) are compatibly involutive in the sense that ∼ ∼ a= a for all a∈ A, where ∼ a: = a⇒ 0 , and (ii) satisfy the Nelson identity, namely the algebraic analogue of (Nelson ⊢ ), viz.(x⇒(x⇒y))∧(∼y⇒(∼y⇒∼x))≈x⇒y.The present paper focuses on the role played by the Nelson identity in the context of compatibly involutive commutative integral bounded residuated lattices. We present several characterisations of the identity (Nelson) in this setting, which variously permit us to comprehend its model-theoretic content from order-theoretic, syntactic, and congruence-theoretic perspectives. Notably, we show that a compatibly involutive commutative integral bounded residuated lattice A is a Nelson residuated lattice iff for all a, b∈ A, the congruence condition ΘA(0,a)=ΘA(0,b)andΘA(1,a)=ΘA(1,b)impliesa=bholds. This observation, together with others of the main results, opens the door to studying the characteristic property of Nelson residuated lattices (and hence Nelson’s constructive logic with strong negation) from a purely abstract perspective.Publicación The logic of distributive bilattices(Oxford University Press, 2011) Bou, Félix; Rivieccio, UmbertoBilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú (1995, Algebraizable Gentzen Systems and the Deduction of Theorem for Gentzen Systems) (even if the logic itself is not algebraizable). We also prove some purely algebraic results concerning bilattices, for instance that the variety of (unbounded) distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron (1996, Math. Struct. Comput. Sci., 6, 287–299) stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices (of which distributive bilattices are a proper subclass).Publicación Nothing but the truth(Springer, 2013) Pietz, Andreas; Rivieccio, UmbertoA curious feature of Belnap’s “useful four-valued logic”, also known as first-degree entailment (FDE), is that the overdetermined value B (both true and false) is treated as a designated value. Although there are good theoretical reasons for this, it seems prima facie more plausible to have only one of the four values designated, namely T (exactly true). This paper follows this route and investigates the resulting logic, which we call Exactly True Logic.Publicación Representation of interlaced trilattices(Elsevier, 2013) Rivieccio, UmbertoTrilattices are algebraic structures introduced ten years ago into logic with the aim to provide a uniform framework for the notions of constructive truth and constructive falsity. In more recent years, trilattices have been used to introduce a number of many-valued systems that generalize the Belnap–Dunn logic of first-degree entailment, proposed as logics of how several computers connected together in a network should think in order to deal with incomplete and possibly contradictory information. The aim of the present work is to develop a first purely algebraic study of trilattices, focusing in particular on the problem of representing certain subclasses of trilattices as special products of bilattices. This approach allows to extend the known representation results for interlaced bilattices to the setting of trilattices and to reduce many algebraic problems concerning these new structures to the better-known framework of lattice theory.