322,836 research outputs found
Dualities for Płonka Sums
No full textPłonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Płonka sum representation in terms of dualisable algebras
The Płonka product of topological spaces
No full textWe introduce a topological counterpart to the Płonka sums of algebraic structures: the Płonka product of topological spaces. This leads to a duality when considering spaces that are dually equivalent to the algebras used in the construction of the Płonka sum
Embeddings of metric Boolean algebras in RN
No full textA Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A,dm), where dm(a,b)=m((a∧¬b)∨(¬a∧b)), for any a,b∈A, sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in RN (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At(A),dm) embeds isometrically in RN (with the Euclidean metric
The rhythm of quantum algorithms
No full textQuantum algorithms can be generally represented as the dynamical evolution of an input quantum register, with the action of each logical gate, as well as of any transmission channel, defined by some quantum propagator. From a global viewpoint, this unitary dynamics is ruled by the flow of a continuous time, and the possible splitting into shorter logical sub-units is nothing but a harmless, though useful, zooming process. On the other hand, understanding how elementary units of the quantum register, namely single qubits, are actually hauled along the algorithm, is a more complex matter, as it involves the dynamical entanglement generation entailed in the action of two-qubit gates. In this work, we first review how the essential elements of quantum algorithms can be described in terms of dynamical processes, and then analyze the corresponding non-unitary dynamics of single qubits, by referring to the formalism adopted in the study of open quantum systems. We show that single qubits evolution cannot be split into intervals shorter than the typical time needed by two-qubit gates for accomplishing their task, which somehow gives a rhythmical structure to the algorithm itself. We further point out that the local evolution entails a memory, in that the way each qubit takes an infinitesimally small step forward in time, is set by its previous history, back to the instant when it entered the last two-qubit gate. This memory originates from quantum correlations, and it is suggested to play an essential role in quantum information processing. As a concluding remark, we just touch on the idea that a similar analysis could be put forward for getting a clue on how we extract meaningful contents out of complex informational input
Probability over Płonka sums of Boolean algebras: States, metrics and topology
No full textThe paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric
Residuated relational systems
No full textThe aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation. By enriching such binary relation with additional properties, we get interesting properties of residuated relational systems which are analogical to those of residuated posets and lattices
A Note on Orthomodular Lattices
No full textWe introduce a new identity equivalent to the orthomodular law in every ortholattice
Containment Logics: Algebraic Completeness and Axiomatization
No full textThe paper studies the containment companion (or, right variable inclusion companion) of a logic ⊢. This consists of the consequence relation ⊢ r which satisfies all the inferences of ⊢ , where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the Płonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization
On the structure of Bochvar algebras
No full textBochvar algebras consist of the quasivariety BCA playing the role of
equivalent algebraic semantics for Bochvar (external) logic, a logical
formalism introduced by Bochvar in the realm of (weak) Kleene logics. In this
paper, we provide an algebraic investigation of the structure of Bochvar
algebras. In particular, we prove a representation theorem based on Plonka sums
and investigate the lattice of subquasivarieties, showing that Bochvar
(external) logic has only one proper extension (apart from classical logic),
algebraized by the subquasivariety NBCA of BCA. Finally, we prove that both BCA
and NBCA enjoy the Amalgamation Property (AP).Comment: Revised versio
Counterfactuals as modal conditionals, and their probability
No full textIn this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update
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