Boolean Algebra Research Articles

Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones

In this paper we explore the extent to which the algebraic structure of a monoid M M determines the topologies on M M that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or T 1 T_1 topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse monoids. If M M is a topological monoid such that every homomorphism from M M to a second countable topological monoid N N is continuous, then we say that M M has automatic continuity. We show that many well-known, and extensively studied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid N N \mathbb {N} ^\mathbb {N} ; the full binary relation monoid B N B_{\mathbb {N}} ; the partial transformation monoid P N P_{\mathbb {N}} ; the symmetric inverse monoid I N I_{\mathbb {N}} ; the monoid Inj ⁡ ( N ) \operatorname {Inj}(\mathbb {N}) consisting of the injective transformations of N \mathbb {N} ; and the monoid C ( 2 N ) C(2^{\mathbb {N}}) of continuous functions on the Cantor set 2 N 2^{\mathbb {N}} . The monoid N N \mathbb {N} ^\mathbb {N} can be equipped with the product topology, where the natural numbers N \mathbb {N} have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on N N \mathbb {N} ^\mathbb {N} , and its analogue on P N P_{\mathbb {N}} , is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on C ( 2 N ) C(2 ^\mathbb {N}) , and on the monoid C ( [ 0 , 1 ] N ) C([0, 1] ^\mathbb {N}) of continuous functions on the Hilbert cube [ 0 , 1 ] N [0, 1] ^\mathbb {N} . The symmetric inverse monoid I N I_{\mathbb {N}} has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid B N B_{\mathbb {N}} has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, Inj ⁡ ( N ) \operatorname {Inj}(\mathbb {N}) and the monoid Surj ⁡ ( N ) \operatorname {Surj}(\mathbb {N}) of all surjective transformations of N \mathbb {N} each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on N N \mathbb {N} ^\mathbb {N} , P N P_{\mathbb {N}} , and Inj ⁡ ( N ) \operatorname {Inj}(\mathbb {N}) coincide with the pointwise topology; and we characterise the Zariski topology on B N B_{\mathbb {N}} . Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in N N \mathbb {N}^{\mathbb {N}} and inverse monoids in I N I_{\mathbb {N}} . Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies.

Automorphism groups of the constituent graphs of integral distance graphs

In this paper, we consider the automorphism groups of Cayley graphs which are a basis of a complete Boolean algebra of strongly regular graphs, one of such graph is the integral distance graph Γ . The automorphism groups of the integral distance graphs Γ were determined by Kovács and Ruff in 2014 and by Kurz in 2009. Our point of interest will be restricted to the one determined by Kovács and Ruff. Here we consider the automorphism groups of subgraphs Γ 1 and Γ 2 , which inherit the properties of Γ . It will be shown that Aut Γ is isomorphic to Aut Γ 2 and that Aut Γ 1 is isomorphic to S F q 2 ≀ S [ 2 ] .

A Category of Ordered Algebras Equivalent to the Category of Multialgebras

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s. We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \(\textit{bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \(\textit{ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\). The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.

Phase transition induced threshold resistive switching in two-dimensional VTe2 nanosheets for Boolean logic operations

Vanadium chalcogenides have been extensively studied owing to the diverse crystallographic structures with various stoichiometric ratios. The metal-to-insulator transition (MIT) widely reported in vanadium chalcogenides is a rapid reversible phase transition that requires small energy, demonstrating potential applications in memory devices. In this work, two-dimensional (2D) vanadium telluride (VTe2) nanosheets are prepared by the chemical vapor deposition method. The synthesized VTe2 nanosheets exhibit volatile threshold switching (TS) behaviors due to the MIT phase transition, which can be further confirmed by the temperature dependent TS behaviors. The TS memristor demonstrates good stability and high reliability with up to 1000 continuous and repeatable writing/erasing operations. Furthermore, based on the TS behaviors, the fabricated memristor can be utilized to implement basic Boolean logic operations of “OR,” “AND,” and “NOT.” This study not only demonstrates the TS behaviors in the 2D VTe2 nanosheets owing to the MIT phase transition but also shows the potential applications of the TS devices in Boolean logic operations.

Dissecting the Determinants of Domain Insertion Tolerance and Allostery in Proteins.

Domain insertion engineering is a promising approach to recombine the functions of evolutionarily unrelated proteins. Insertion of light-switchable receptor domains into a selected effector protein, for instance, can yield allosteric effectors with light-dependent activity. However, the parameters that determine domain insertion tolerance and allostery are poorly understood. Here, an unbiased screen is used to systematically assess the domain insertion permissibility of several evolutionary unrelated proteins. Training machine learning models on the resulting data allow to dissect features informative for domain insertion tolerance and revealed sequence conservation statistics as the strongest indicators of suitable insertion sites. Finally, extending the experimental pipeline toward the identification of switchable hybrids results in opto-chemogenetic derivatives of the transcription factor AraC that function as single-protein Boolean logic gates. The study reveals determinants of domain insertion tolerance and yielded multimodally switchable proteins with unique functional properties.

A Reduced Hardware SNG for Stochastic Computing

Stochastic Computing (SC) is an alternative way of computing with binary weighted words that can significantly reduce hardware resources. This technique relies on transforming information from a conventional binary system to the probability domain in order to perform mathematical operations based on probability theory, where smaller amounts of binary logic elements are required. Despite the advantage of computing with reduced circuitry, SC has a well known issue; the input interface known as stochastic number generator (SNG), is a hardware consuming module, which is disadvantageous for small digital circuits or circuits with several input data. Hence, in this work, efforts are dedicated to improving a classic weighted binary SNG (WBSNG). For this, one of the internal modules (weight generator) of the SNG was redesigned by detecting a pattern in the involved signals that helped to pose the problem in a different way, yielding equivalent results. This greatly reduced the number of logical elements used in its implementation. This pattern is interpreted with Boolean equations and transferred to a digital circuit that achieves the same behavior of a WBSNG but with less resources.

A parametrized axiomatization for a large number of restricted second-order logics

Abstract By limiting the range of the predicate variables in a second-order language, one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note, we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras. This article is dedicated to our friend John N. Crossley on the occasion of his 86th birthday.

Alternative Formulations of Decision Rule Learning from Neural Networks

This paper extends recent work on decision rule learning from neural networks for tabular data classification. We propose alternative formulations to trainable Boolean logic operators as neurons with continuous weights, including trainable NAND neurons. These alternative formulations provide uniform treatments to different trainable logic neurons so that they can be uniformly trained, which enables, for example, the direct application of existing sparsity-promoting neural net training techniques like reweighted L1 regularization to derive sparse networks that translate to simpler rules. In addition, we present an alternative network architecture based on trainable NAND neurons by applying De Morgan’s law to realize a NAND-NAND network instead of an AND-OR network, both of which can be readily mapped to decision rule sets. Our experimental results show that these alternative formulations can also generate accurate decision rule sets that achieve state-of-the-art performance in terms of accuracy in tabular learning applications.

GHOST-NOT and GHOST-YES: Two programs for generating high-speed biosensors with randomized oligonucleotide binding sites with NOT or YES Boolean logic functions based on experimentally validated algorithms.

High-speed allosteric hammerhead ribozymes can be engineered to distinguish well between a perfectly matching effector and the nucleic acid sequences with a few mismatches under physiologically relevant conditions. Such ribozymes can be designed to control the expression of exogenous mRNAs and can be used to develop new gene therapies, including anticancer treatments. The in vivo selection of such ribozymes is a complicated and lengthy procedure with no guarantee of success. Thus, in silico selection of high-speed ribozymes can be employed using secondary RNA structure computation based on the partition function of the RNA folding in combination with random search algorithms. This approach has already been proven very accurate in designing allosteric hammerhead ribozymes. Herein, we present two programs for the computational design of allosteric ribozymes sensing randomized oligonucleotides based on the extended version of the hammerhead ribozyme. A Generator for High-speed Oligonucleotide Sensing allosteric ribozymes with NOT logic function (GHOST-NOT) and a Generator for High-speed Oligonucleotide Sensing allosteric ribozymes with YES logic function (GHOST-YES) for computational design of high-speed allosteric ribozymes are described. The allosteric hammerhead ribozymes had a high self-cleavage rate of about 1.8 per minute and were very selective in sensing an effector sequence.

Logics of upsets of De Morgan lattices

AbstractWe study logics determined by matrices consisting of a De Morgan lattice with an upward closed set of designated values, such as the logic of non‐falsity preservation in a given finite Boolean algebra and Shramko's logic of non‐falsity preservation in the four‐element subdirectly irreducible De Morgan lattice. The key tool in the study of these logics is the lattice‐theoretic notion of an n‐filter. We study the logics of all (complete, consistent, and classical) n‐filters on De Morgan lattices, which are non‐adjunctive generalizations of the four‐valued logic of Belnap and Dunn (of the three‐valued logics of Priest and Kleene, and of classical logic). We then show how to find a finite Hilbert‐style axiomatization of any logic determined by a finite family of prime upsets of finite De Morgan lattices and a finite Gentzen‐style axiomatization of any logic determined by a finite family of filters on finite De Morgan lattices. As an application, we axiomatize Shramko's logic of anything but falsehood.

Methodical advances in reproducibility research: A proof of concept qualitative comparative analysis of reproducing animal data in humans

BackgroundWhile the term reproducibility crisis mainly reflects reproducibility of experiments between laboratories, reproducibility between species also remains problematic. We previously summarised the published reproducibility between animal and human studies; i.e. the translational success rates, which varied from 0% to 100%. Based on analyses of individual factors, we could not predict reproducibility.Several potential analyses can assess effect of combinations of predictors on an outcome. Regression analysis (RGA) is common, but not ideal to analyse multiple interactions and specific configurations (≈ combinations) of variables, which could be highly relevant to reproducibility.Qualitative comparative analysis (QCA) is based on set theory and Boolean algebra, and was successfully used in other fields. We reanalysed the data from our preceding review with QCA. ResultsThis QCA resulted in the following preliminary formula for successful translation:∼Old*∼Intervention*∼Large*MultSpec*QuantitativeWhich means that within the analysed dataset, the combination of relative recency (∼ means not; >1999), analyses at event or study level (not at intervention level), n < 75, inclusion of more than one species and quantitative (instead of binary) analyses always resulted in successful translation (>85%). Other combinations of factors showed less consistent or negative results. An RGA on the same data did not identify any of the included variables as significant contributors. ConclusionsWhile these data were not collected with the QCA in mind, they illustrate that the approach is viable and relevant for this research field. The QCA seems a highly promising approach to furthering our knowledge on between-species reproducibility.

Application of the Global Equilibrium Search Method for Solving Boolean Programming Problems

Introduction. The significance of methods and algorithms for solving discrete optimization problems in mathematical supporting computer technologies of diverse levels and objectives is increasing. Consequently, the efficacy of discrete optimization methods deserves particular attention, as it drives the advancement of techniques capable of solving complex real-world problems. This paper introduces the Global Equilibrium Search (GES) method as a highly effective approach for solving Boolean programming problems, thus contributing to the field's progress and applicability. Purpose. We describe the successful application of the approximate probabilistic GES method for effectively solving various Boolean programming problems. Results. This paper explores the application of sequential GES algorithms for solving Boolean linear, Boolean quadratic programming, and other related problems with their specific characteristics. In our study, we conducted a comparative analysis to assess the effectiveness of GES algorithms by evaluating them against state-of-the-art approaches. Additionally, to parallelize the optimization process for discrete programming problems, we introduced algorithm unions, specifically portfolios, and teams. The efficiency of GES algorithm portfolios and teams is investigated by solving the maximum weighted graph cut problem, with subsequent comparisons to identify distinctions between them. Conclusions. Based on the accumulated experience of applying GES algorithms and their modifications to solve discrete optimization problems, this study establishes the GES method as the leading approximate approach for Boolean programming. The results demonstrate the GES algorithm unions experience a significant boost in the optimization process speed, whereas algorithm teams demonstrate higher efficiency. Keywords: global equilibrium search method, Boolean programming problems, experimental studies, algorithm efficiency, algorithm unions (portfolios and teams).

Boolean Logic Computing Based on Neuromorphic Transistor

AbstractGeneral‐purpose computers usually use logic gate computing units based on complementary metal oxide semiconductors (CMOS). Due to the separate memory and computing units in Von Neumann architecture, data transmission requires great energy and time consumption. Developing novel neuromorphic devices and comprehensively investigating their logical computing mode are crucial to achieve high‐performance and low‐power neuromorphic computation. Here, a systematic summary of Boolean logic computing based on emerging neuromorphic transistors is presented. This summary encompasses logical operation modes, materials, device structures, and working mechanisms. The input mode of Boolean logic operation is classified into electrical input, optical input, and synergistic optical/electrical input. Besides, additional modulation strategies to construct programmable logic functions by electrical, optical, and thermal signals are also summarized. These strategies hold great significance as they enable dynamic reconfiguration of logic operations and provide neuromorphic devices with decision‐making capabilities. Finally, the application prospects and current challenges to Boolean logic computing based on dendritic integration are discussed from the aspects of device integration, synergistic input/modulation modes, auxiliary peripheral circuit, software/hardware system, etc. It is believed that comprehensive investigations on neuromorphic Boolean logic operations are crucial to push forward the development of future neuromorphic computing toward high efficiency and high integration density.

Some simple theories from a Boolean algebra point of view

We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories Tm reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories Tn,k, which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters “by hand” to satisfy certain model theoretically meaningful chain conditions. This may be seen as advancing a line of work going back through Kunen's construction of good ultrafilters in ZFC using families of independent functions. We conclude with a theorem on flexible ultrafilters, and open questions.

On Hausdorff operators in ZF$\mathsf {ZF}$

AbstractA Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.

CONNEXIVE IMPLICATIONS IN SUBSTRUCTURAL LOGICS

Abstract This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we provide some philosophical upshots like, e.g., a discussion on the relevance of the above operation in relationship with G. Polya’s logic of plausible inference, and some characterization results on weak and strong connexivity.

Energy Efficient All-Electric-Field-Controlled Multiferroic Magnetic Domain-Wall Logic.

Magnetic domain wall (DW)-based logic devices offer numerous opportunities for emerging electronics applications allowing superior performance characteristics such as fast motion, high density, and nonvolatility to process information. However, these devices rely on an external magnetic field, which limits their implementation; this is particularly problematic in large-scale applications. Multiferroic systems consisting of a piezoelectric substrate coupled with ferromagnets provide a potential solution that provides the possibility of controlling magnetization through an electric field via magnetoelastic coupling. Strain-induced magnetization anisotropy tilting can influence the DW motion in a controllable way. We demonstrate a method to perform all-electrical logic operations using such a system. Ferromagnetic coupling between neighboring magnetic domains induced by the electric-field-controlled strain has been exploited to promote noncollinear spin alignment, which is used for realizing essential building blocks, including DW generation, propagation, and pinning, in all implementations of Boolean logic, which will pave the way for scalable memory-in-logic applications.

Counterfactuals as modal conditionals, and their probability

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.

A generalization of de Vries duality to closed relations between compact Hausdorff spaces

Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category DeVS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then gives that KHausR is equivalent to DeVS, thus resolving a problem recently raised in the literature.The equivalence between KHausR and DeVS further restricts to an equivalence between the category KHaus of compact Hausdorff spaces and continuous functions and the wide subcategory DeVF of DeVS whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition.

Lateral order on complex vector lattices and narrow operators

AbstractIn this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification of a real vector lattice E and introduce the lateral order on . Our first main result asserts that the set of all fragments of an element of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator from the complexification of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator from to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from to a complex Banach lattice is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.