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Journal articles on the topic 'Boolean'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 July 2024

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1

Chernoskutov, Yu Yu. "On the Syllogistic of G. Boole." Discourse 7, no. 2 (April 29, 2021): 5–15. http://dx.doi.org/10.32603/2412-8562-2021-7-2-5-15.

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Introduction. This article focuses on the investigation of Boole’s theory of categorical syllogism, exposed in his book “The Mathematical analysis of Logic”. That part of Boolean legacy has been neglected in the prevailed investigations on the history of logic; the latter provides the novelty of the work presented.Methodology and sources. The formal reconstruction of the methods of algebraic presentation of categorical syllogism, as it is exposed in the original work of Boole, is conducted. The character of Boolean methods is investigated in the interconnections with the principles of symbolic algebra on the one hand, and with the principles of signification, taken from R. Whately, on the other hand. The approaches to signification, grounding the syllogistic theories of Boole and Brentano, are analyzed in comparison, wherefrom we explain the reasons why the results of those theories are different so much.Results and discussion. It is demonstrated here that Boole has borrowed the principles of signification from the Whately’s book “The Elements of Logic”. The interpreting the content of the terms as classes, being combined with methods of symbolic algebra, has determined the core features of Boolean syllogism theory and its unexpected results. In contrast to Whately, Boole conduct the approach to ultimate ends, overcoming the restrictions imposed by Aristotelean doctrine. In particular, he neglects the distinction of subject and predicate among the terms of proposition, the order of premises, and provide the possibility to draw conclusions with negative terms. At the same time Boole missed that the forms of inference, parallel to Bramantip and Fresison, are legitimate forms in his system. In spite of the apparent affinities between the Boolean and Brentanian theories of judgment, the syllogistics of Boole appeared to be more flexible. The drawing of particular conclusion from universal premises is allowable in Boolean theory, but not in Brentanian one; besides, in his theory is allowable the drawing of conclusion from two negative premises, which is prohibited in Aristotelian syllogistic.Conclusion. Boole consistently interpreted signification of terms as classes; being combine with methods symbolic algebra it led to very flexible syllogism theory with rich results.

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Allart, Emilie, Joachim Niehren, and Cristian Versari. "Exact Boolean Abstraction of Linear Equation Systems." Computation 9, no. 11 (October 21, 2021): 113. http://dx.doi.org/10.3390/computation9110113.

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We study the problem of how to compute the boolean abstraction of the solution set of a linear equation system over the positive reals. We call a linear equation system ϕ exact for the boolean abstraction if the abstract interpretation of ϕ over the structure of booleans is equal to the boolean abstraction of the solution set of ϕ over the positive reals. Abstract interpretation over the booleans is thus complete for the boolean abstraction when restricted to exact linear equation systems, while it is not complete more generally. We present a new rewriting algorithm that makes linear equation systems exact for the boolean abstraction while preserving the solutions over the positive reals. The rewriting algorithm is based on the elementary modes of the linear equation system. The computation of the elementary modes may require exponential time in the worst case, but is often feasible in practice with freely available tools. For exact linear equation systems, we can compute the boolean abstraction by finite domain constraint programming. This yields a solution of the initial problem that is often feasible in practice. Our exact rewriting algorithm has two further applications. Firstly, it can be used to compute the sign abstraction of linear equation systems over the reals, as needed for analyzing function programs with linear arithmetics. Secondly, it can be applied to compute the difference abstraction of a linear equation system as used in change prediction algorithms for flux networks in systems biology.

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Huang, Jing Lian, Su Duo Li, Yong Liu, and Ke Yan Deng. "On Analysis and Judgment of Balance for Boolean Functions by E-Derivative." Applied Mechanics and Materials 643 (September 2014): 130–35. http://dx.doi.org/10.4028/www.scientific.net/amm.643.130.

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Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools and deeply into the internal structure of Boolean, we study the issues of the analysis and judgment of balance for Boolean functions. We get that the linear functions and the nonzero derivative of the product of two linear functions are balanced functions, and the product of two linear functions are not balanced functions. We also obtain the quadratic homogeneous Booleans are not all balanced function. Besides, we deduce the theorem which determine the sum of linear function and balanced function whether it is a balanced function. What is more, the features of balance of Boolean functions can be reflected easily and effectively by e-derivative.

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4

De Villiers, Michael D. "Teaching Modeling and Axiomatization with Boolean Algebra." Mathematics Teacher 80, no. 7 (October 1987): 528–32. http://dx.doi.org/10.5951/mt.80.7.0528.

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Traditionally, Boolean algebra is largely taught in connection with computer programming courses, logic, or set theory. Since Boolean algebra arose from George Boole's application of algebraic principles to the study of logic in 1854, this approach would seem natural.

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5

Chajda, I., and M. Kotrle. "Boolean semirings." Czechoslovak Mathematical Journal 44, no. 4 (1994): 763–67. http://dx.doi.org/10.21136/cmj.1994.128495.

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Ali M. Ali Rushdi, Ali M. Ali Rushdi. "Satisfiability in Big Boolean Algebras via Boolean-Equation Solving." journal of King Abdulaziz University Engineering Sciences 28, no. 1 (January 2, 2017): 3–18. http://dx.doi.org/10.4197/eng.28-1.1.

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This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM). In the classical method, the number of parameters used is minimized and compact solutions are obtained. However, the parameters belong to the underlying big Boolean algebra. By contrast, the novel method does not attempt to minimize the number of parameters used, as it uses independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom in the Boole-Shannon expansion of the formula ݃(ࢄ .(Though the method produces non-compact expressions, it is much quicker in generating particular solutions. The two methods are demonstrated via two detailed examples.

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7

Wehrung, Friedrich. "Boolean universes above Boolean models." Journal of Symbolic Logic 58, no. 4 (December 1993): 1219–50. http://dx.doi.org/10.2307/2275140.

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AbstractWe establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are “boundedly algebraically compact” in the language (+, −, ·, ∧, ∨, ≤), and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any first-order language. The proofs can be translated into “naive set theory” in a uniform way.

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8

Madsen, Magnus, Jaco van de Pol, and Troels Henriksen. "Fast and Efficient Boolean Unification for Hindley-Milner-Style Type and Effect Systems." Proceedings of the ACM on Programming Languages 7, OOPSLA2 (October 16, 2023): 516–43. http://dx.doi.org/10.1145/3622816.

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As type and effect systems become more expressive there is an increasing need for efficient type inference. We consider a polymorphic effect system based on Boolean formulas where inference requires Boolean unification. Since Boolean unification involves semantic equivalence, conventional syntax-driven unification is insufficient. At the same time, existing Boolean unification techniques are ill-suited for type inference. We propose a hybrid algorithm for solving Boolean unification queries based on Boole’s Successive Variable Elimination (SVE) algorithm. The proposed approach builds on several key observations regarding the Boolean unification queries encountered in practice, including: (i) most queries are simple, (ii) most queries involve a few flexible variables, (iii) queries are likely to repeat due similar programming patterns, and (iv) there is a long tail of complex queries. We exploit these observations to implement several strategies for formula minimization, including ones based on tabling and binary decision diagrams. We implement the new hybrid approach in the Flix programming language. Experimental results show that by reducing the overhead of Boolean unification, the compilation throughput increases from 8,580 lines/sec to 15,917 lines/sec corresponding to a 1.8x speed-up. Further, the overhead on type and effect inference time is only 16% which corresponds to an overhead of less than 7% on total compilation time. We study the hybrid approach and demonstrate that each design choice improves performance.

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Studenic, Paul, David Felson, Maarten de Wit, Farideh Alasti, Tanja A. Stamm, Josef S. Smolen, and Daniel Aletaha. "Testing different thresholds for patient global assessment in defining remission for rheumatoid arthritis: are the current ACR/EULAR Boolean criteria optimal?" Annals of the Rheumatic Diseases 79, no. 4 (February 5, 2020): 445–52. http://dx.doi.org/10.1136/annrheumdis-2019-216529.

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ObjectivesThis study aimed to evaluate different patient global assessment (PGA) cut-offs required in the American College of Rheumatology/European League Against Rheumatism (ACR/EULAR) Boolean remission definition for their utility in rheumatoid arthritis (RA).MethodsWe used data from six randomised controlled trials in early and established RA. We increased the threshold for the 0–10 score for PGA gradually from 1 to 3 in steps of 0.5 (Boolean1.5 to Boolean3.0) and omitted PGA completely (BooleanX) at 6 and 12 months. Agreement with the index-based (Simplified Disease Activity Index (SDAI)) remission definition was analysed using kappa, recursive partitioning (classification and regression tree (CART)) and receiver operating characteristics. The impact of achieving each definition on functional and radiographic outcomes after 1 year was explored.ResultsData from 1680 patients with early RA and 920 patients with established RA were included. The proportion of patients achieving Boolean remission increased with higher thresholds for PGA from 12.4% to 19.7% in early and 5.9% to 12.3% in established RA at 6 months. Best agreement with SDAI remission occurred at PGA cut-offs of 1.5 and 2.0, while agreement decreased with higher PGA (CART: optimal agreement at PGA≤1.6 cm; sensitivity of PGA≤1.5 95%). Changing PGA thresholds at 6 months did not affect radiographic progression at 12 months (mean ꙙsmTSS for Boolean, 1.5, 2.0, 2.5, 3.0, BooleanX: 0.35±5.4, 0.38±5.14, 0.41±5.1, 0.37±4.9, 0.34±4.9, 0.27±4.7). However, the proportion attaining HAQ≤0.5 was 90.2%, 87.9%, 85.2%, 81.1%, 80.7% and 73.1% for the respective Boolean definitions.ConclusionIncreasing the PGA cut-off to 1.5 cm would provide high consistency between Boolean with the index-based remission; the integer cut-off of 2.0 cm performed similarly.

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10

Stempel, Rachel. "BOOLEAN." Minnesota review 2022, no. 98 (May 1, 2022): 17. http://dx.doi.org/10.1215/00265667-9563639.

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11

Ricci, Gabriele. "Boolean matrices ... neither Boolean nor matrices." Discussiones Mathematicae - General Algebra and Applications 20, no. 1 (2000): 141. http://dx.doi.org/10.7151/dmgaa.1012.

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12

Reith, Steffen, and Klaus W. Wagner. "On boolean lowness and boolean highness." Theoretical Computer Science 261, no. 2 (June 2001): 305–21. http://dx.doi.org/10.1016/s0304-3975(00)00146-8.

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13

Khuzam, Hazar Abu, and Adil Yaqub. "Generalized Boolean and Boolean-like rings." International Journal of Algebra 7 (2013): 429–38. http://dx.doi.org/10.12988/ija.2013.2894.

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14

Sachwanowicz, Wojcech. "Boolean powers over incomplete boolean algebras." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36, no. 5 (1990): 431–40. http://dx.doi.org/10.1002/malq.19900360508.

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15

Cherchi, Gianmarco, Fabio Pellacini, Marco Attene, and Marco Livesu. "Interactive and Robust Mesh Booleans." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–14. http://dx.doi.org/10.1145/3550454.3555460.

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Boolean operations are among the most used paradigms to create and edit digital shapes. Despite being conceptually simple, the computation of mesh Booleans is notoriously challenging. Main issues come from numerical approximations that make the detection and processing of intersection points inconsistent and unreliable, exposing implementations based on floating point arithmetic to many kinds of degeneracy and failure. Numerical methods based on rational numbers or exact geometric predicates have the needed robustness guarantees, that are achieved at the cost of increased computation times that, as of today, has always restricted the use of robust mesh Booleans to offline applications. We introduce an algorithm for Boolean operations with robustness guarantees that is capable of operating at interactive frame rates on meshes with up to 200K triangles. We evaluate our tool thoroughly, considering not only interactive applications but also batch processing of large collections of meshes, processing of huge meshes containing millions of elements and variadic Booleans of hundreds of shapes altogether. In all these experiments, we consistently outperform prior robust floating point methods by at least one order of magnitude.

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16

Espitia, Helbert, José Soriano, Iván Machón, and Hilario López. "Compact Fuzzy Systems Based on Boolean Relations." Applied Sciences 11, no. 4 (February 18, 2021): 1793. http://dx.doi.org/10.3390/app11041793.

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This document presents some considerations and procedures to design a compact fuzzy system based on Boolean relations. In the design process, a Boolean codification of two elements is extended to a Kleene’s of three elements to perform simplifications for obtaining a compact fuzzy system. The design methodology employed a set of considerations producing equivalent expressions when using Boole and Kleene algebras establishing cases where simplification can be carried out, thus obtaining compact forms. In addition, the development of two compact fuzzy systems based on Boolean relations is shown, presenting its application for the identification of a nonlinear plant and the control of a hydraulic system where it can be seen that compact structures describes satisfactory performance for both identification and control when using algorithms for optimizing the parameters of the compact fuzzy systems. Finally, the applications where compact fuzzy systems are based on Boolean relationships are discussed allowing the observation of other scenarios where these structures can be used.

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17

Rushdi, Ali Muhammad Ali, and Waleed Ahmad. "Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras." International Journal of Mathematical, Engineering and Management Sciences 3, no. 4 (December 1, 2018): 404–28. http://dx.doi.org/10.33889/ijmems.2018.3.4-029.

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A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design.

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18

Takeuti, Gaisi. "Boolean Simple Groups and Boolean Simple Rings." Journal of Symbolic Logic 53, no. 1 (March 1988): 160. http://dx.doi.org/10.2307/2274435.

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19

Avilés, Antonio. "Boolean Metric Spaces and Boolean Algebraic Varieties." Communications in Algebra 32, no. 5 (December 31, 2004): 1805–22. http://dx.doi.org/10.1081/agb-120029903.

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20

MOVSISYAN, YU M., and V. A. ASLANYAN. "SUPER-BOOLEAN FUNCTIONS AND FREE BOOLEAN QUASILATTICES." Discrete Mathematics, Algorithms and Applications 06, no. 02 (March 19, 2014): 1450024. http://dx.doi.org/10.1142/s1793830914500244.

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A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables.

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21

Gudder, Stan, and Frédéric Latrémolière. "Boolean inner-product spaces and Boolean matrices." Linear Algebra and its Applications 431, no. 1-2 (July 2009): 274–96. http://dx.doi.org/10.1016/j.laa.2009.02.028.

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22

Takeuti, Gaisi. "Boolean simple groups and boolean simple rings." Journal of Symbolic Logic 53, no. 1 (March 1988): 160–73. http://dx.doi.org/10.1017/s0022481200029005.

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Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V() of set theory. If we observe G outside V(), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V(). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

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23

TAKAHASHI, Makoto. "Completeness of Boolean powers of Boolean algebras." Journal of the Mathematical Society of Japan 40, no. 3 (July 1988): 445–56. http://dx.doi.org/10.2969/jmsj/04030445.

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Couceiro, Miguel, Jean-Luc Marichal, and Tamás Waldhauser. "Locally monotone Boolean and pseudo-Boolean functions." Discrete Applied Mathematics 160, no. 12 (August 2012): 1651–60. http://dx.doi.org/10.1016/j.dam.2012.03.006.

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25

Buszkowski, Wojciech. "Embedding Boolean Structures into Atomic Boolean Structures." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32, no. 13-16 (1986): 227–28. http://dx.doi.org/10.1002/malq.19860321307.

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26

Pushpalatha, K., and V. M.L.Hima Bindu. "A Note on Boolean Like Algebras." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 1015. http://dx.doi.org/10.14419/ijet.v7i4.10.26660.

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In this paper we develop on abstract system: viz Boolean-like algebra and prove that every Boolean algebra is a Boolean-like algebra. A necessary and sufficient condition for a Boolean-like algebra to be a Boolean algebra has been obtained. As in the case of Boolean ring and Boolean algebra, it is established that under suitable binary operations the Boolean-like ring and Boolean-like algebra are equivalent abstract structures.

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Pushpalatha, K., and . "Some Contributions to Boolean like near Rings." International Journal of Engineering & Technology 7, no. 3.34 (September 1, 2018): 670. http://dx.doi.org/10.14419/ijet.v7i3.34.19413.

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In this paper we extend Foster’s Boolean-like ring to Near-rings. We introduce the concept of a Boolean like near-ring. A near-ring N is said to be a Boolean-like near-ring if the following conditions hold: (i) a+a = 0 for all aÎ N , (ii) ab(a+b+ab) = ba for all a, b Î N and (iii) abc = acb for all a,b, c Î N (right weak commutative law). We have proved that every Boolean ring is a Boolean like near-ring. An example is given to show that the converse is not true. We prove that if N is a Boolean near-ring then conditions (i) and (ii) of the above definition are equivalent. We also proved that a Boolean near-ring with condition (iii) is a Boolean ring. As a consequence we show that a Boolean –like near-ring N is a Boolean ring if and only if it is a Boolean near-ring. Obviously, every Boolean like ring is a Boolean like near-ring. We show that if N is a Boolean-like near-ring with identity, then N is a Boolean-like ring. In addition we prove several interesting properties of Boolean-like near-rings. We prove that the set of all nilpotent elements of a Boolean –like near-ring N forms an ideal and the quotient near-ring N/I is a Boolean ring. Every homomorphic image of a Boolean like near ring is a Boolean like near ring. We further prove that every Boolean-like near-ring is a Boolean-like semiring As example is given to show that the converse of this result is not true.

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28

Frické, Martin. "Boolean Logic." KNOWLEDGE ORGANIZATION 48, no. 2 (2021): 177–91. http://dx.doi.org/10.5771/0943-7444-2021-2-177.

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The article describes and explains Boolean logic (or Boolean algebra) in its two principal forms: that of truth-values and the Boolean connectives and, or, and not, and that of set membership and the set operations of intersection, union and complement. The main application areas of Boolean logic to knowledge organization, namely post-coordinate indexing and search, are introduced and discussed. Some wider application areas are briefly mentioned, such as: propositional logic, the Shannon-style approach to electrical switching and logic gates, computer programming languages, probability theory, and database queries. An analysis is offered of shortcomings that Boolean logic has in terms of potential uses in knowledge organization.

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29

Vardi, Moshe Y. "Boolean satisfiability." Communications of the ACM 57, no. 3 (March 2014): 5. http://dx.doi.org/10.1145/2578043.

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Reddy, J. K. "Boolean Algebra." IETE Journal of Education 28, no. 4 (October 1987): 153–58. http://dx.doi.org/10.1080/09747338.1987.11436173.

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Brinn, L. W. "Boolean algebra." International Journal of Mathematical Education in Science and Technology 20, no. 6 (November 1989): 799–807. http://dx.doi.org/10.1080/0020739890200602.

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Hodgkin, J. "Boolean yeast?" Trends in Genetics 14, no. 2 (February 1998): 53. http://dx.doi.org/10.1016/s0168-9525(98)01411-5.

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McAllester, David, and Ramin Zabih. "Boolean classes." ACM SIGPLAN Notices 21, no. 11 (November 1986): 417–23. http://dx.doi.org/10.1145/960112.28740.

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Vechtomov, E. M. "Boolean rings." Mathematical Notes of the Academy of Sciences of the USSR 39, no. 2 (February 1986): 101–3. http://dx.doi.org/10.1007/bf01159890.

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Wang, Chi. "Boolean minors." Discrete Mathematics 141, no. 1-3 (June 1995): 237–58. http://dx.doi.org/10.1016/0012-365x(93)e0191-6.

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Boros, E., P. L. Hammer, and J. N. Hooker. "Boolean regression." Annals of Operations Research 58, no. 3 (May 1995): 201–26. http://dx.doi.org/10.1007/bf02032132.

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Okhotin, Alexander. "Boolean grammars." Information and Computation 194, no. 1 (October 2004): 19–48. http://dx.doi.org/10.1016/j.ic.2004.03.006.

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Bishop, Barbara A. "Beware Boolean:." College & Undergraduate Libraries 1, no. 2 (December 9, 1994): 23–24. http://dx.doi.org/10.1300/j106v01n02_03.

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Min, Yinghua, Zhongcheng Li, and Zhuxing Zhao. "Boolean process." Science in China Series E: Technological Sciences 40, no. 3 (June 1997): 250–57. http://dx.doi.org/10.1007/bf02916600.

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40

Banković, Dragić. "Boolean inequations." Discrete Mathematics 307, no. 6 (March 2007): 750–55. http://dx.doi.org/10.1016/j.disc.2006.07.006.

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Crama, Yves, and Peter L. Hammer. "Boolean Functions." Discrete Applied Mathematics 161, no. 1-2 (January 2013): 315. http://dx.doi.org/10.1016/j.dam.2012.07.021.

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Bubeck, Uwe, and Hans Kleine Büning. "Encoding Nested Boolean Functions as Quantified Boolean Formulas." Journal on Satisfiability, Boolean Modeling and Computation 8, no. 1-2 (January 1, 2012): 101–16. http://dx.doi.org/10.3233/sat190092.

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Beasley, Leroy B., Kyung-Tae Kang, and Seok-Zun Song. "CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES." Pure and Applied Mathematics 21, no. 2 (May 31, 2014): 121–28. http://dx.doi.org/10.7468/jksmeb.2014.21.2.121.

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Vechtomov, E. M. "Annihilator characterizations of Boolean rings and Boolean lattices." Mathematical Notes 53, no. 2 (February 1993): 124–29. http://dx.doi.org/10.1007/bf01208314.

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Kleine Büning, Hans, K. Subramani, and Xishun Zhao. "Boolean Functions as Models for Quantified Boolean Formulas." Journal of Automated Reasoning 39, no. 1 (April 14, 2007): 49–75. http://dx.doi.org/10.1007/s10817-007-9067-0.

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Leech, Jonathan, and Matthew Spinks. "Skew Boolean algebras derived from generalized Boolean algebras." Algebra universalis 58, no. 3 (May 14, 2008): 287–302. http://dx.doi.org/10.1007/s00012-008-2069-x.

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Ramesh, S., and V. Venkatesawara Rao. "An Efficient Security Measure Extended Boolean Retrieval." International Journal for Research in Applied Science and Engineering Technology 10, no. 2 (February 28, 2022): 968–72. http://dx.doi.org/10.22214/ijraset.2022.40362.

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Abstract: The impact of Extended Boolean retrieval (EBR) has drastic changes over the decades and despite their significant advantages compared to either ranked keyword or pure Boolean retrieval. In particular, EBR models produce meaningful rankings; their query model allows the representation of complex concepts in an and-or format; and they are scriptable, in that the score assigned to a document depends solely on the content of that document, unaffected by any collection statistics or other external factors. Security is the major concern to make the queries retrieval and the entire data mining process. However, EBR is much more computationally expensive than the alternatives. We consider the implementation of the p-norm approach to EBR, and demonstrate that ideas used in the max-score and wand exact optimization techniques for ranked keyword retrieval can be adapted to allow selective by pass of documents via a low-cost screening process for this and similar retrieval models. We also propose term independent bounds that are able to further reduce the number of score calculations for short, simple queries under the extended Boolean retrieval model. Overall saving from 50 to 80 percent of the evaluation cost on test queries drawn from biomedical search. (Stefan Pohl, VOL. 24, NO. 6, JUNE 2012) Keywords: extended, boolen, model, EBR

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Awwad, Mohamad. "FROM BOOLE’S LOGIC TO BOOLEAN APPLICATIONS IN COMPUTER SCIENCE." Educational Discourse: collection of scientific papers, no. 32(4) (May 5, 2021): 18–25. http://dx.doi.org/10.33930/ed.2019.5007.32(4)-2.

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The developments of an algebraic logical language of thoughts by G. Boole are considered using historical and theoretical perspectives. The technical implementations of Boolean logic in combinational circuits and in modern cryptography show strong influences of a 19th century logic on the latest technologies of computing.

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Steinbach, Bernd, and Christian Posthoff. "Boolean differential equations: A common model for classes, lattices, and arbitrary sets of Boolean functions." Facta universitatis - series: Electronics and Energetics 28, no. 1 (2015): 51–76. http://dx.doi.org/10.2298/fuee1501051s.

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The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean values or Boolean functions can be described. A Boolean Differential Equation (BDe) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDe, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential equations. In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDe that is restricted to all vectorial derivatives of f (x) and optionally contains Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution. The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBooLe-problem program (PRP) of the freely available XBooLe-Monitor quickly solves some BDes.

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B, S., A. Kumar P H, M. Gopalakrishnan, and M. K. Garg. "AB0208 COMPARISON OF PERFORMANCE OF ACR EULAR 2022 REMISSION CRITERIA IN PATIENTS WITH RHEUMATOID ARTHRITIS FROM INDIA." Annals of the Rheumatic Diseases 82, Suppl 1 (May 30, 2023): 1287.2–1288. http://dx.doi.org/10.1136/annrheumdis-2023-eular.4372.

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BackgroundRemission is a state, not change or transition with absence of disease activity in rheumatoid arthritis (RA) [1]. Remission at 6 months has been shown to predict future good outcomes such as arresting radiographic progression and better functional status at 12 and 24 months [2]. ACR EULAR have endorsed Boolean2.0 in the revised remission criteria in 2022 [3]. None of these participants were recruited from India. In this ongoing study, we compare the various remission criteria in patients with RA in our setting.ObjectivesTo compare the performance of various remission criteria (Disease Activity Score 28 (DAS28 ESR), Disease activity Score 28 CRP (DAS28 CRP), Simple Disease Activity Index (SDAI), Clinical Disease Activity Index (CDAI), Boolean, Boolean2.0, and BooleanX) in patients with RA presenting to our rheumatology clinic, at Jodhpur, India.MethodsA longitudinal study was initiated in July 2021 after Ethics Committee approval. Demographic and treatment details were collected after written informed consent. Remission criteria were calculated at 6 months follow up. We analysed agreement between the Boolean- and index-based criteria SDAI, CDAI, DAS28 ESR and DAS28 CRP. The study is an ongoing study, with a calculated sample size of 165.ResultsFollow-up data of 85 participants of which 85.9% female with median age of 47 (12) years with RA diagnosed by ACR EULAR 2010 were collected. At enrolment, 10.6% (n = 9) were on biologicals, 35.3% (n = 30) on methotrexate alone and 54.1% (n = 46) on combination Disease-modifying Antirheumatic Drugs (DMARDs) therapy and at 6 months, 5.9% (n = 5) were on biologicals, 61.17% (n = 52) on methotrexate alone and 32.9% (n = 28) on combination DMARDs therapy. The proportion of patients in remission at 6 months using DAS28 ESR was 65.9% (n = 56), DAS28 CRP was 54.1% (n = 46), SDAI was 38.8% (n = 33), CDAI was 32.9% (n = 28), Boolean was 10.6% (n = 9), Boolean2.0 was 40% (n = 34) and BooleanX was 47.1% (n = 40) (Figure 1). DAS28 CRP showed substantial agreement with Boolean2.0 (kappa 0.63) at enrolment and moderate agreement (kappa 0.49) at 6 months, however, SDAI had only slight agreement with modified Boolean based indices (Table 1).Table 1.Agreement between modified Boolean indices and disease activityKappaSDAI at enrolment (CI)SDAI at 6 months (CI)DAS28 CRP at enrolment (CI)DAS28 CRP at 6 months (CI)Boolean0.23 (0.00-0.08)0.09 (0.20-0.38)0.10 (0.09-0.23)0.15 (0.00-0.03)Boolean2.00.13 (0.13-0.29)0.08 (0.27-0.45)0.63 (0.00-0.03)0.49 (0.00-0.03)BooleanX0.03 (0.97-1.00)0.06 (0.39-0.59)0.71(0.00-0.03)0.39 (0.00-0.03)ConclusionThis study provides evidence of external validation of the newly proposed modification of the Boolean ACR/EULAR remission criteria; however, we found that the Boolean2.0 criteria did not exhibit agreement with index-based remission definitions in the Indian population from our centre. Our results replicate the findings that a Boolean definition using 2 cm as threshold for patient global assessment of disease activity (Boolean2.0) yields better agreement than Boolean [2]. With the validation Boolean2.0 in our settings, our data support the revised ACR/EULAR remission criteria as a target in clinical practice.References[1]Felson DT, Anderson JJ, Boers M, Bombardier C, Chernoff M, Fried B, et al. The American College of Rheumatology preliminary core set of disease activity measures for rheumatoid arthritis clinical trials. The Committee on Outcome Measures in Rheumatoid Arthritis Clinical Trials. Arthritis Rheum. 1993 Jun;36(6):729–40.[2]Studenic P, Felson D, Wit M de, Alasti F, Stamm TA, Smolen JS, et al. Testing different thresholds for patient global assessment in defining remission for rheumatoid arthritis: are the current ACR/EULAR Boolean criteria optimal? Ann Rheum Dis. 2020 Apr 1;79(4):445–52.[3]Studenic P, Aletaha D, de Wit M, Stamm TA, Alasti F, Lacaille D, et al. American College of Rheumatology/EULAR Remission Criteria for Rheumatoid Arthritis: 2022 Revision. Arthritis Rheumatol. 2023;75(1):15–22.Acknowledgements:NIL.Disclosure of InterestsNone Declared.

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