scholarly journals Study of T-norms and Quantum Logic Functions on BL-algebra and Their Relationships to the Classical Probability Structures

2020 ◽  
pp. 591-599
Author(s):  
Ahmed AL-Adilee ◽  
Habeeb Kareem Abdullah ◽  
Hawraa A. AL-Challabi
Keyword(s):  
Quantum Logic ◽  
Probability Space ◽  
Binary Operation ◽  
The Other ◽  
Symmetric Difference ◽  
Logic Function ◽  
Classical Probability ◽  
Binary Operations ◽  
Basic Probability ◽  
Logic Functions

This paper is concerned with the study of the T-norms and the quantum logic functions on BL-algebra, respectively, along with their association with the classical probability space. The proposed constructions depend on demonstrating each type of the T-norms with respect to the basic probability of binary operation. On the other hand, we showed each quantum logic function with respect to some binary operations in probability space, such as intersection, union, and symmetric difference. Finally, we demonstrated the main results that explain the relationships among the T-norms and quantum logic functions. In order to show those relations and their related properties, different examples were built.

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

scholarly journals Card-Based Cryptographic Logical Computations Using Private Operations

2020 ◽  
Author(s):  
Hibiki Ono ◽  
Yoshifumi Manabe
Keyword(s):  
Cryptographic Protocols ◽  
The Other ◽  
General Logic ◽  
Private Values ◽  
Minimum Number ◽  
Logic Functions ◽  
Logical Functions

Abstract This paper proposes new card-based cryptographic protocols to calculate logic functions with the minimum number of cards using private operations under the semi-honest model. Though various card-based cryptographic protocols were shown, the minimum number of cards used in the protocol has not been achieved yet for many problems. Operations executed by a player where the other players cannot see are called private operations. Private operations have been introduced in some protocols to solve a particular problem or to input private values. However, the effectiveness of introducing private operations to the calculation of general logic functions has not been considered. This paper introduces three new private operations: private random bisection cuts, private reverse cuts, and private reveals. With these three new operations, we show that all of AND, XOR, and copy protocols are achieved with the minimum number of cards by simple three-round protocols. This paper then shows a protocol to calculate any logical functions using these private operations. Next, we consider protocols with malicious players.

The branching property in generalized information theory

1978 ◽  
Vol 10 (04) ◽  
pp. 788-802
Author(s):  
Bruce Ebanks
Keyword(s):  
Information Theory ◽  
Probability Space ◽  
Binary Operation ◽  
Information Measure ◽  
Additive Functions ◽  
Expected Information ◽  
Generalized Information ◽  
Generalized Information Theory ◽  
Branching Property

It is shown that every measure of expected information which has the branching property is of the form where J is a given information measure which is compositive under a regular binary operation and the Ψ n are antisymmetric, bi-additive functions. In a probability space, such measures (entropies) take the form

DESIGN AND APPLICATION OF AN INTERACTIVE WHEELCHAIR TRAINING SYSTEM

2008 ◽  
Vol 20 (06) ◽  
pp. 377-385 ◽  
Author(s):  
Chern-Sheng Lin ◽  
Chia-Chang Chang ◽  
Wei-Lung Chen
Keyword(s):  
Control Function ◽  
Training System ◽  
Digital Logic ◽  
3D Simulation ◽  
Initial Time ◽  
Logic Function ◽  
Wheelchair Users ◽  
Rehabilitation Training ◽  
Direction Control ◽  
Logic Functions

In this paper we constructed an interactive wheelchair rehabilitation training platform. The roller wheel on the platform is driven mainly by turning the wheelchair, and then the relative position of wheelchair on the screen can be adjusted based on the rotation speed of left and right wheels on the platform. Comparing the digital logic function when two wheels rotate at the same time and judging the variance in digital logic, the steering direction of wheels can be known and be controlled forward or backward. Additionally, the standard digital logic function could be individually judged when left wheel rotates and vice versa, so as to control the steering. Through judging three digital logic functions, the initial time of left wheel, next signal selecting time of left wheel, initial time of right wheel, and next signal selecting time of right wheel could be obtained, then the system can achieve the required direction control function through the judgment formula. The direction control function is indicated by standard digital logic function, so that the user can operate smoothly in the interactive situation software and make an interaction with the computer 3D simulation scene, the patient would have rehabilitation training through various 3D simulation real exteriors. This study not only provides basic trainings but also records the service behavior of wheelchair users, so that the rehabilitation consultant would have reference for the future diagnosis.

3D Visualization Model and Sample Implementation of 0-1 Function on Variant Logic

2013 ◽  
Vol 718-720 ◽  
pp. 480-483
Author(s):  
Huan Wang ◽  
Jie Ao Zhu ◽  
Xue Liu ◽  
Jeffrey Zheng
Keyword(s):  
Data Visualization ◽  
3D Visualization ◽  
Random Sequence ◽  
Relevant Logic ◽  
Complex Data ◽  
Paper Sample ◽  
Logic Function ◽  
Visual Model ◽  
Logic Functions ◽  
Visual Results

Random sequences generated by different logic functions play an important role in cryptography. The structure and the special properties of the logic function has been one of the most active areas of research. In order to study the random sequence and its related logic functions, many models have been established, and different advanced tools are applied to make complex data visualization. In this paper, sample logic functions are transferred into variant logic expressions to form a set of measurements. Using selected measurements, a 3D visual model is proposed. Selected 3D visual results are shown their intrinsic 3D spatial characteristics of relevant logic functions respectively.

scholarly journals On the generalization of Conway algebra

2018 ◽  
Vol 27 (02) ◽  
pp. 1850014 ◽  
Author(s):  
Seongjeong Kim
Keyword(s):  
Algebraic Structure ◽  
The Other ◽  
Second Step ◽  
Polynomial Invariant ◽  
Binary Operations ◽  
Other Hand ◽  
Skein Relation

In [Przytyski and Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1989) 115–139], Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in [Kauffman and Lambropoulou, New invariants of links and their state sum models, arXiv:1703.03655v2 [math.GT] 15 Mar 2017], Kauffman and Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step, we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply another skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra [Formula: see text] with two binary operations and we construct an invariant valued in [Formula: see text] by applying those two binary operations to mixed crossings and pure crossing, respectively. The 4-variable invariant of Kauffman and Lambropoulou with a specific condition is derived from the invariant valued in [Formula: see text]. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies nonlinear skein relations.

Dempster Combination Rule for Signed Belief Functions

1998 ◽  
Vol 06 (01) ◽  
pp. 79-102 ◽  
Author(s):  
Ivan Kramosil
Keyword(s):  
Large Class ◽  
Probability Space ◽  
Binary Operation ◽  
Random Variables ◽  
Belief Function ◽  
Measurable Space ◽  
Point Of View ◽  
Belief Functions ◽  
Combination Rule ◽  
Real Numbers

A possibility to define a binary operation over the space of pairs of belief functions, inverse or dual to the well-known Dempster combination rule in the same sense in which substraction is dual with respect to the addition operation in the space of real numbers, can be taken as an important problem for the purely algebraic as well as from the application point of view. Or, it offers a way how to eliminate the modification of a belief function obtained when combining this original belief function with other pieces of information, later proved not to be reliable. In the space of classical belief functions definable by set-valued (generalized) random variables defined on a probability space, the invertibility problem for belief functions, resulting from the above mentioned problem of "dual" combination rule, can be proved to be unsolvable up to trivial cases. However, when generalizing the notion of belief functions in such a way that probability space is replaced by more general measurable space with signed measure, inverse belief functions can be defined for a large class of belief functions generalized in the corresponding way. "Dual" combination rule is then defined by the application of the Dempster rule to the inverse belief functions.

scholarly journals ORTHOMODULAR-VALUED MODELS FOR QUANTUM SET THEORY

2017 ◽  
Vol 10 (4) ◽  
pp. 782-807
Author(s):  
MASANAO OZAWA
Keyword(s):  
Hilbert Space ◽  
Set Theory ◽  
Binary Operation ◽  
Boolean Logic ◽  
Transfer Principle ◽  
Orthomodular Lattices ◽  
Flexible Approach ◽  
Binary Operations ◽  
The Axiom Of Choice ◽  
Models Of Set Theory

AbstractIn 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.

Automorphism groups with some finiteness conditions

2017 ◽  
Vol 67 (4) ◽  
Author(s):  
Konrad Pióro
Keyword(s):  
Lower Bound ◽  
Binary Operation ◽  
Automorphism Groups ◽  
The Other ◽  
Finiteness Conditions ◽  
Other Hand ◽  
The Right

AbstractAll considered groups are torsion or do not contain infinitely generated subgroups. If such a groupNext, we show that ifThe Birkhoff’s construction can be slightly modified so as to obtain a smaller set of operations. In fact, it is enough to take the right multiplications by generators. Moreover, we show that this is the best possible lower bound for the number of unary operations in the case of groups considered here. If we admit non-unary operations, then for finite and countable groups we can reduce the number of operations to one binary operation. On the other hand, if

scholarly journals Foundations for Applications of Gibbs Derivatives in Logic Design and VLSI

VLSI Design ◽  
2002 ◽  
Vol 14 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Radomir S. Stanković ◽  
Milena Stanković ◽  
Reiner Creutzburg
Keyword(s):  
New Technologies ◽  
Differential Operators ◽  
Rate Of Change ◽  
Logic Design ◽  
Logic Function ◽  
Spectral Techniques ◽  
Cad Systems ◽  
Vlsi Cad ◽  
Efficient Calculation ◽  
Logic Functions

New technologies and increased requirements for performances of digital systems require new mathematical theories and tools as a basis for future VLSI CAD systems. New or alternative mathematical approaches and concepts must be suitable to solve some concrete problems in VLSI and efficient algorithms for their efficient application should be provided. This paper is an attempt in this direction and relates with the recently renewed interest in arithmetic expressions for switching functions, instead representations in Boolean structures, and spectral techniques and differential operators in switching theory and applications. Logic derivatives are efficiently used in solving different tasks in logic design, as for example, fault detection, functional decomposition, detection of symmetries and co-symmetries of logic functions, etc. Their application is based on the property that by differential operators, we can measure the rate of change of a logic function. However, by logic derivatives, we can hardly distinguish the direction of the change of the function, since they are defined in finite algebraic structures. Gibbs derivatives are a class of differential operators on groups, which applied to logic functions, permit to overcome this disadvantage of logic derivatives. Therefore, they may be useful in logic design in the same areas where the logic derivatives have been already using. For such applications, it is important to provide fast algorithms for calculation of Gibbs derivatives on finite groups efficiently in terms of space and time. In this paper, we discuss the methods for efficient calculation of Gibbs derivatives. These methods should represent a basis for further applications of these and related operators in VLSI CAD systems.

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