The Operators of Vector Logic

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type I and Type II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. Mathematics Subject Classification: 03B05, 03B50.     

A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication A → B in terms of &, , and ¬ (e.g., S-implications that are based on the formula B ¬ A), and R-implications that are based on an implicit representation of implication A → B as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as b^a) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, , and ¬ is described by (implicit) axioms.     

完备格Brouwer上伪t-模与蕴涵算子的注记(II)(英文)

进一步研究完备Brouwer格上伪t-模与蕴涵算子,给出同二元算子生成的伪t-模与蕴涵算子的计算公式。     

Modelling Inference in Argumentation through Labelled Deduction: Formalization and Logical Properties

Artificial Intelligence (AI) has long dealt with the issue of finding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a confluence point for many alternative logical frameworks. Different formalisms have been developed, most of them sharing the common notions of argument and warrant. In defeasible argumentation, an argument is a tentative (defeasible) proof for reaching a conclusion. An argument is warranted when it ultimately prevails over other conflicting arguments. In this context, defeasible consequence relationships for modelling argument and warrant as well as their logical properties have gained particular attention. This article analyzes two non-monotonic inference operators C_arg and C_war intended for modelling argument construction and dialectical analysis (warrant), respectively. As a basis for such analysis we will use the LDS_ar framework, a unifying approach to computational models of argument using Labelled Deductive Systems (LDS). In the context of this logical framework, we show how labels can be used to represent arguments as well as argument trees, facilitating the definition and study of non-monotonic inference operators, whose associated logical properties are studied and contrasted. We contend that this analysis provides useful comparison criteria that can be extended and applied to other argumentation frameworks. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B42.     

完备格Brouwer上伪t-模与蕴涵算子的注记(Ⅱ)

进一步研究完备Brouwer格上伪t-模与蕴涵算子,给出同二元算子生成的伪t-模与蕴涵算子的计算公式。     

Fuzzy implication operators for difference operations for fuzzy sets and cardinality-based measures of comparison

In this paper, we determine by means of fuzzy implication operators, two classes of difference operations for fuzzy sets and two classes of symmetric difference operations for fuzzy sets which preserve properties of the classical difference operation for crisp sets and the classical symmetric difference operation for crisp sets respectively. The obtained operations allow us to construct as in [B. De Baets, H. De Meyer, Transitivity-preserving fuzzification schemes for cardinality-based similarity measures, European Journal of Operational Research 160 (2005) 726–740], cardinality-based similarity measures which are reflexive, symmetric and transitive fuzzy relations and, to propose two classes of distances (metrics) which are fuzzy versions of the well-known distance of cardinality of the symmetric difference of crisp sets.     

The influence of some fuzzy implication operators on the accuracy of a fuzzy model-part II

The influence of fuzzy implication operators and the connective Also on the accuracy of a fuzzy model of a d.c. series motor is considered. Some typical fuzzy implication operators are applied to the construction of a fuzzy model of a d.c. series motor. A root-mean-square error is used as the criterion of the fuzzy model's adequacy to the real system. A number of mathematical operations necessary for the implementation of the fuzzy model are used as the criterion by which the fuzzy model's applicability if estimated from the point of view of computing techniques. The best types of fuzzy relations, representing fuzzy models of a real system, are chosen in order to secure the least root-mean-square error with minimal number of mathematical operations necessary for computer implementation.     

Precovers,Modalities and Universal Closure Operators in a Topos

In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set. Mathematics Subject Classification: 03G30, 03B20, 03F55.     

A dynamical approach to logical decisions

We propose a first approximation to the modeling of cognitive decisions based on the theory of associative memories. The basic structure are the matrix memories of Anderson-Kohonen adapted to include the modulation of associations via tensorial preprocessing of inputs. This view admits an easy implementation of logical operations as matrix operators. From this “vectorial logic” springs a variety of models for fuzzy decision processes. Degrees of fuzziness are introduced at two different levels: (a) by the operators of modal logic, and (b) by using logical variables as vectors with projections inside the interval [0, 1]. The outcomes of this vectorial logic can be projected onto unit vectors yielding scalar difference equations. As examples we study the dynamics of contradictory self-referential systems and processes leading to competition between options. These models exhibit a variety of dynamical patterns that include stable steady states, oscillations, and deterministic chaos. © 1997 John Wiley & Sons, Inc.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

Rectangularity and paramonotonicity of maximally monotone operators

Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has a useful range of properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were not known. Our aim is to provide new results and examples concerning these notions. It is shown that rectangularity and paramonotonicity are actually independent. Moreover, for linear relations, rectangularity implies paramonotonicity but the converse implication requires additional assumptions. We also consider continuous linear monotone operators, and we point out that in the Hilbert space both notions are automatic for certain displacement mappings.     

General Logic-Systems and Finite Consequence Operators

In this paper, the significance of using general logic-systems and finite consequence operators defined on non-organized languages is discussed. Results are established that show how properties of finite consequence operators are independent from language organization and that, in some cases, they depend only upon one simple language characteristic. For example, it is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language L that cannot be generated from any finite logic-system. On the other hand, it is shown that for any nonempty language L, a set map is a finite consequence operator if and only if it is defined by a general logic-system. Simple logic-system examples that determine specific consequence operator properties are given. Mathematics Subject Classification (2000): Primary 03B22, Secondary 03B65     

Associativity, Commutativity and Symmetry in Residuated Structures

In this paper we develop the study of extended-order algebras, recently introduced by C. Guido and P. Toto, which are implicative algebras that generalize all the widely considered integral residuated structures. Particular care is devoted to the requirement of completeness that can be obtained by the MacNeille completion process. Associativity, commutativity and symmetry assumptions are characterized and their role is discussed toward the structure of the algebra and of its completion. As an application, further operations corresponding to the logical connectives of conjunction negation and disjunction are considered and their properties are investigated, either assuming or excluding the additional conditions of associativity, commutativity and symmetry. An overlook is also devoted to the relationship with other similar structures already considered such as implication algebras (in particular Heyting algebras), BCK algebras, quantales, residuated lattices and closed categories.     

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds. The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010.     

The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients

The present paper discusses relations between regularity, Dirichlet, and Neumann problems. We investigate the boundary problems for block operators and prove, in particular, that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implication was established in 1993 by C. Kenig, J. Pipher [Invent. Math. 113 (3) (1993) 447-509] and since then has served as an integral part of many results.     

Polymodal Lattices and Polymodal Logic

A polymodal lattice is a distributive lattice carrying an n-place operator preserving top elements and certain finite meets. After exploring some of the basic properties of such structures, we investigate their freely generated instances and apply the results to the corresponding logical systems — polymodal logics — which constitute natural generalizations of the usual systems of modal logic familiar from the literature. We conclude by formulating an extension of Kripke semantics to classical polymodal logic and proving soundness and completeness theorems. Mathematics Subject Classification: 03G10, 06D99, 03B45.     

Equivalence after extension and Schur coupling coincide for inessential operators

In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE $?$ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators.     

Progressive and merging-proof taxation

We investigate the implications and logical relations between progressivity (a principle of distributive justice) and merging-proofness (a strategic principle) in taxation. By means of two characterization results, we show that these two principles are intimately related, despite their different nature. In particular, we show that, in the presence of continuity and consistency (a widely accepted framework for taxation) progressivity implies merging-proofness and that the converse implication holds if we add an additional strategic principle extending the scope of merging-proofness to a multilateral setting. By considering operators on the space of taxation rules, we also show that progressivity is slightly more robust than merging-proofness.     

Singular Measures and Convolution Operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.