Abstract
In this paper, we mainly investigate the relation theory and topological structures of rough neutrosophic sets which constructed by combining rough sets and neutrosophic sets. Firstly, we ...introduce the concept of algebra which is operations and identity relation. Then, discuss the properties of inverse, reflexive, symmetric as well as transitive relation of rough neutrosophic set. Further, regular sets form a Boolean algebra is introduced from the interior and closure of interior in rough neutrosophic sets. The results establish can help to gain more insight in rough neutrosophic sets.
Abstract
In this paper, the concept of (λ, μ)-hesitant fuzzy subalgebras is introduced in Boolean algebra. Some properties of (λ, μ)-hesitant fuzzy subalgebras are discussed. Finally, we proved that ...the intersection and direct product of two (λ, μ)-hesitant fuzzy subalgebras are also (λ, μ)-hesitant fuzzy subalgebras in Boolean algebra.
Abstract
In this paper, the concepts of multiset Boolean filter, multiset Boolean pseudofilter, anti-multiset Boolean filter, and anti-multiset Boolean pseudofilter are introduced. Also, the relation ...between multiset Boolean filter(pseudofilter) and some other types of multiset filters(pseudofilters) are given.
This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in that context. In particular we introduce a construction which ...defines a (finite) Boolean algebra of conditionals from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.
Complex-valued neural networks have many advantages over their real-valued counterparts. Conventional digital electronic computing platforms are incapable of executing truly complex-valued ...representations and operations. In contrast, optical computing platforms that encode information in both phase and magnitude can execute complex arithmetic by optical interference, offering significantly enhanced computational speed and energy efficiency. However, to date, most demonstrations of optical neural networks still only utilize conventional real-valued frameworks that are designed for digital computers, forfeiting many of the advantages of optical computing such as efficient complex-valued operations. In this article, we highlight an optical neural chip (ONC) that implements truly complex-valued neural networks. We benchmark the performance of our complex-valued ONC in four settings: simple Boolean tasks, species classification of an Iris dataset, classifying nonlinear datasets (Circle and Spiral), and handwriting recognition. Strong learning capabilities (i.e., high accuracy, fast convergence and the capability to construct nonlinear decision boundaries) are achieved by our complex-valued ONC compared to its real-valued counterpart.
The Grover quantum search algorithm is a hallmark application of a quantum computer with a well-known speedup over classical searches of an unsorted database. Here, we report results for a complete ...three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state marking scheme required to perform a classical search. We also report the deterministic implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.
A sequent calculus
$ \mathfrak {S} $
S
for the variety
$ \mathsf {tqBa} $
tqBa
of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the ...finite model property of
$ \mathfrak {S} $
S
is shown, and thus the decidability of
$ \mathfrak {S} $
S
is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety
$ \mathsf {TDM5} $
TDM
5
of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus
$ \mathfrak {S}_5 $
S
5
and prove that the cut elimination holds for it. Consequently the decidability of
$ \mathfrak {S}_5 $
S
5
is established. Furthermore, this proof-theoretic method is applied to some subvarieties of
$ \mathsf {TDM5} $
TDM
5
.
In the article "Toward Human-Understandable, Explainable AI," which appeared in the September 2018 issue of Computer, Figure 2(a) appears where Figure 2(b) should be, and vice versa. The graph in ...Figure 2(a) represents Type-1 fuzzy sets, while the graph in Figure 2(b) represents Boolean sets. We regret the error and any confusion that may have resulted.
A Boolean generator for a large number of standard complementary QAM sequences of length 2K is proposed. This Boolean generator is derived from the authors' earlier paraunitary generator, which is ...based on matrix multiplications. Both generators are based on unitary matrices. In contrast to previous Boolean QAM algorithms which represent complementary sequences as a weighted sum, our algorithm has a multiplicative form. Any element of a sequence can be generated efficiently by indexing the entries of unitary matrices with the binary representation of the discrete time index (which is easily implemented as a binary counter). Our 1Qum (based on one QAM unitary matrix) and 2Qum (based on two QAM unitary matrices) algorithms generate generalized Case I-III sequences and generalized Case IV and V sequences, respectively, as specified by Liu et al. in 2013, in addition to many new 2Qum sequences. The ratio of the numbers of sequences that are generated by our new construction and the previous construction increases with the constellation size. For example, for a 1024-QAM sequence of length 1024, this ratio is 4.4. However, if we compare only 2Qum sequences to Case IV and V sequences, this ratio is 267.
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