Filter theory of BL algebras

In this paper we consider fundamental properties of some types of filters (Boolean, positive implicative, implicative and fantastic filters) of BL algebras defined in Haveshki et al. (Soft Comput 10:657–664, 2006) and Turunen (Arch Math Logic 40:467–473, 2001). It is proved in Haveshki et al. (2006) that if F is a maximal and (positive) implicative filter then it is a Boolean filter. In that paper there is an open problem Under what condition are Boolean filters positive implicative filters? One of our results gives an answer to the problem, that is, we need no more conditions. Moreover, we give simple characterizations of those filters by an identity form ? x, y(t(x, y) ∈ F), where t(x, y) is a term containing x, y.      

Boolean filters and positive implicative filters of residuated lattices

The aim of this paper is to introduce the notions of Boolean filters and positive implicative filters in residuated lattices and to investigate their properties. Several characterizations of Boolean filters and positive implicative filters are derived. The extension theorems of implicative filters and positive implicative filters are obtained. The relations among Boolean filters, implicative filters and positive implicative filters are investigated and it is proved that Boolean filters are equivalent to implicative filters, and that every Boolean filter is a positive implicative filter, but the converse may not be true. Furthermore, the conditions under which a positive implicative filter is a Boolean filter are established.     

R0代数中的正蕴涵MP滤子和固执MP滤子

在R₀代数中,给出布尔MP滤子的几种性质特征。提出正蕴涵MP滤子和固执MP滤子的概念,讨论了它们的一些性质定理。证明正蕴涵MP滤子与布尔MP滤子等价,固执MP滤子与MP超滤等价,同时也讨论了正蕴涵MP滤子和固执MP滤子的关系。     

剩余格上的几类n-重滤子及其特征

在剩余格上引入了n-重蕴含滤子、n-重MV滤子和n-重布尔滤子,给出它们的一系列刻画定理。通过新的刻画定理得到了这些滤子之间的关系;并且证明了滤子是n-重布尔滤子当且仅当它是n-重蕴含MV滤子。     

R0代数中的正蕴涵MP滤子和固执MP滤子

在R0代数中,给出布尔MP滤子的几种性质特征。提出正蕴涵MP滤子和固执MP滤子的概念,讨论了它们的一些性质定理。证明正蕴涵MP滤子与布尔MP滤子等价,固执MP滤子与MP超滤等价,同时也讨论了正蕴涵MP滤子和固执MP滤子的关系。     

n-Fold obstinate filters in BL-algebras

In this paper, we introduced the notion of n-fold obstinate filter in BL-algebras and we stated and proved some theorems, which determine the relationship between this notion and other types of n-fold filters in a BL-algebra. We proved that if F is a 1-fold obstinate filter, then A/F is a Boolean algebra. Several characterizations of n-fold fantastic filters are given, and we show that A is a n-fold fantastic BL-algebra if A is a MV-algebra (n ≥ 1) and A is a 1-fold positive implicative BL-algebra if A is a Boolean algebra. Finally, we construct some algorithms for studying the structure of the finite BL-algebras and n-fold filters in finite BL-algebras.     

Redefined fuzzy implicative filters

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (θ, ψ)-fuzzy implicative filters where θ, ψ are any two of {∈, q, ∈ ∨ q, ∈ ∧ q} with θ ≠ ∈ ∧ q is introduced, and related properties are discussed. Relations between (∈ ∨ q, ∈ ∨ q)-fuzzy implicative filters and (∈, ∈ ∨ q)-fuzzy implicative filters are investigated, and conditions for an (∈, ∈ ∨ q)-fuzzy implicative filter to be an (∈, ∈)-fuzzy implicative filter are provided. Characterizations of (∈, ∈ ∨ q)-fuzzy implicative filters are given, and conditions for a fuzzy set to be a (q, ∈ ∨ q)-fuzzy implicative filter are provided.     

Generalized fuzzy filters of <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>-algebras

In this paper, we introduce the notions of interval valued -fuzzy filters and interval valued -fuzzy Boolean (implicative) filters in R ₀-algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that an interval valued fuzzy set F in R ₀-algebras is an interval valued -fuzzy Boolean filter if and only if it is an interval valued -fuzzy implicative filter.     

泛逻辑学中UB代数系统的(∈,∈∨q)-fuzzy滤子

何华灿教授给出了理想状态下的泛逻辑学的形式演绎系统β,并证明了该系统的可靠性。并且提出了理想状态下的泛逻辑学对应的代数系统-UB代数,并讨论了它们的性质。在以上这些结果的基础上,引入UB代数的(∈,∈∨q)-fuzzy滤子和(∈,∈∨q)-fuzzy关联滤子的概念,获得了它们的若干等价刻画,证明了(∈,∈∨q)-fuzzy关联滤子的扩张定理。     

BL-代数上的几种直觉模糊滤子

结合直觉模糊集和滤子理论,对BL-代数上的直觉模糊滤子进行了研究。首先回顾了BL-代数和直觉模糊集的有关基础知识。然后引入BL-代数上的直觉模糊滤子、直觉模糊格滤子、直觉模糊布尔滤子和直觉模糊蕴涵滤子的概念,讨论了它们的一系列重要性质,证明了直觉模糊滤子与直觉模糊格滤子、直觉模糊布尔滤子和直觉模糊蕴涵滤子是等价的,并用实例进行了验证。最后探讨了直觉模糊滤子和模糊滤子的关系。     

Computing the Output Distribution and Selection Probabilities of a Stack Filter from the DNF of Its Positive Boolean Function

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function (PBF). Our algorithm avoids to enumerate the models of the PBF one by one, and thus is considerably more efficient than previous methods. The so called rank selection probabilities can be computed along the way.     

Fuzzy蕴涵代数的正则滤子

在Fuzzy蕴涵代数中引入正则滤子的概念并讨论其性质。获得了正则滤子的若干等价刻画。深入考察了正则滤子与其他类型的滤子之间的关系,证明了一个非空集合是正关联滤子当且仅当它既是关联滤子又是正则滤子的结论。     

Relationships between generalized Bosbach states and <Emphasis Type="Italic">L</Emphasis>-filters on residuated lattices

Generalized Bosbach states and filters on residuated lattices have been extensively studied in the literature. In this paper, relationships between generalized Bosbach states and residuated-lattice-valued filters, also called L-filters, on residuated lattices are investigated. Particularly, type I and type II L-filters and their subclasses are defined, and some their properties are obtained. Then relationships between special types of L-filters and the generalized Bosbach states are considered where generalized Bosbach states are characterized by some type I or type II L-filters with additional conditions. Associated with these relationships, new subclasses of generalized Bosbach states such as implicative type IV, V, VI states, fantastic type IV states and Boolean type IV states are introduced, and the relationships between various types of generalized Bosbach states are investigated in detail. In particular, the existence of several generalized Bosbach states is provided and, as application, some typical subclasses of residuated lattices such as Rl-monoids, Heyting algebras and Boolean algebras are characterized by these generalized Bosbach states.     

l2-l∞ filter design for discrete-time singular Markovian jump systems with time-varying delays

The problem of l₂-l_∞ filter design for discrete-time singular systems with Markovian jump and time-varying delays is investigated in this paper. By using the delay partitioning technique, a delay-dependent condition is established to guarantee the filtering error systems to be stochastically admissible and achieve a prescribed l₂-l_∞ performance index. Based on the derived condition, the full-order and reduced-order filters with mode-independent characterization are designed in a unified framework. The corresponding filter parameters can be obtained by solving a set of linear matrix inequalities. The reported results not only depend upon the delay, but also depend upon the partitioning, which aims at reducing the conservatism. Two numerical examples are provided to illustrate the effectiveness of the proposed methods.     

Notes on redefined fuzzy implicative filters of lattice implication algebras

The concept of -fuzzy implicative filters of lattice implication algebras is introduced and some of its related properties are investigated. In particular, the relationships among the ordinary fuzzy implicative filters, the (∈,∈∨q)-fuzzy implicative filters and the -fuzzy implicative filters of lattice implication algebras will be studied.     

Omnidirectional edge detection

In this paper we propose a new method for extending 1-D step edge detection filters to two dimensions via complex-valued filtering. Complex-valued filtering allows us to obtain edge magnitude and direction simultaneously. Our method can be viewed either as an extension of n-directional complex filtering of Paplinski to infinite directions or as a variant of Canny’s gradient-based approach. In the second view, the real part of our filter computes the gradient in the x direction and the imaginary part computes the gradient in the y direction. Paplinski claimed that n-directional filtering is an improvement over the gradient-based method, which computes gradient only in two directions. We show that our omnidirectional and Canny’s gradient-based extensions of the 1-D DoG coincide. In contrast to Paplinski’s claim, this coincidence shows that both approaches suffer from being confined to the subspace of two 2-D filters, even though n-directional filtering hides these filters in a single complex-valued filter. Aside from these theoretical results, the omnidirectional method has practical advantages over both n-directional and gradient-based approaches. Our experiments on synthetic and real-world images show the superiority of omnidirectional and gradient-based methods over n-directional approach. In comparison with the gradient-based method, the advantage of omnidirectional method lies mostly in freeing the user from specifying the smoothing window and its parameter. Since the omnidirectional and Canny’s gradient-based extensions of the 1-D DoG coincide, we have based our experiments on extending the 1-D Demigny filter. This filter has been proposed by Demigny as the optimal edge detection filter in sampled images.     

BL-代数的n-重模糊蕴涵滤子

引入BL-代数的n-重模糊蕴涵滤子概念,讨论了n-重模糊蕴涵滤子的性质,给出若干等价刻画,给出了验证n-重模糊蕴涵滤子的一个简单条件。用n-重模糊蕴涵滤子对n-重蕴涵BL-代数进行了刻画,证明了一个模糊滤子是一个n-重模糊蕴涵滤子当且仅当对应的商代数是一个n-重蕴涵BL-代数。     

Fuzzy Aggregation with Artificial Color filters

Artificial Color filters are designed to attenuate some pixels and pass others. The pass/attenuate decision is made on the basis of the learned association of spectral components with user-defined concepts. In earlier work, it has been shown that there are various ways to design Artificial Color filters using multiple user-designated classes and those filters are subjected to useful manipulations such as image processing and Boolean Aggregation.The Artificial Color filtering has always been binary. Therefore, the Boolean logic was the only choice for aggregating filters. This paper shows how to fuzzify Artificial Color filters. Fuzzy logic subsumes Boolean logic and can do so in many ways. Several different fuzzy T-norms are applied to Artificial Color filters to illustrate the richness in aggregation. Margin Setting, a supervised statistical pattern recognition method to train the filters, is very conservative in what is definitely assigned to a class (μ=1) while allowing a useful gradation of membership (μ?1) for other cases. A parametric exploration of these effects for an image is presented.     

An alternative FIR filter for state estimation in discrete-time systems

To overcome the resulting problems of existing finite impulse response (FIR) structure filters, this paper proposes an alternative FIR filter for state estimation in discrete-time systems, which is derived from the well-known Kalman filter with recursive infinite impulse response (IIR) structure. The proposed FIR filter obtains a posteriori knowledge about the window initial condition from the most recent finite observations, while existing FIR filters handle this task arbitrarily or heuristically. The gain matrix for the proposed FIR filter incorporates a posteriori knowledge about the window initial condition during its design and is shown to be time-invariant. The proposed FIR filter is shown to have good inherent properties such as unbiasedness and deadbeat. Through extensive computer simulations, the proposed FIR filter can be shown to be comparable with the Kalman filter for the nominal system and better than that for the temporarily uncertain system.     

BR_0代数中的次极大滤子

首先在BR0代数M中引入滤子,然后又给出了M中次极大滤子的概念,并讨论了它的性质,得到了BR0代数中的每个滤子都可表示为一些次极大滤子的交的结果。特别地,在满足滤子降链条件的BR0代数中,每个滤子都可表示为有限个次极大滤子的交。