Implementation of Boolean minimization in an abductive framework

Dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. This article presents some transformations and analytics for some example algorithms and Boolean domain problems. Our focus is on the methodology of complexity evaluation and integration of several types of postulations concerning special hypercube structures. Our primary goal is to demonstrate the usual formulas and analytics in this area, giving the necessary set of common formulas often used for complexity estimations and approximations. The basic example under considered is the Boolean minimization problem, in terms of the average complexity of the so-called reduced disjunctive normal form (also referred to as complete, prime irredundant, or Blake canonical form). In fact, combinatorial counterparts of the disjunctive normal form complexities are investigated in terms of sets of their maximal intervals. The results obtained compose the basis of logical separation classification algorithmic technology of pattern recognition. In fact, these considerations are not only general tools of minimization investigations of Boolean functions, but they also prove useful structures, models, and analytics for constraint logic programming, machine learning, decision policy optimization and other domains of computer science.

In this article, we investigate the complexity of abduction, a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining the world's behaviour, it aims at finding an explanation for some observed manifestation. In this article, we consider propositional abduction, where the knowledge base and the manifestation are represented by propositional formulae. The problem of deciding whether there exists an explanation has been shown to be Σ2p-complete in general. We focus on formulae in which the allowed connectives are taken from certain sets of Boolean functions. We consider different variants of the abduction problem in restricting both the manifestations and the hypotheses. For all these variants, we obtain a complexity classification for all possible sets of Boolean functions. In this way, we identify easier cases, namely NP-complete, coNP-complete and polynomial cases. Thus, we get a detailed picture of the complexity of the propositional abduction problem, hence highlighting the sources of intractability. Further, we address the problem of counting the full explanations and prove a trichotomous classification theorem.

The reported study has established the possibility of improving the productivity of an algorithm for the minimization of Boolean functions using a method of the optimal combination of the sequence of logical operations applying different techniques for gluing the variables ‒ simple gluing and super-gluing. The correspondence of intervals I (α, β) in the Boolean space ẞ n has been established, given by a pair of Boolean vectors α and β, such that α≼β, with a complete combinatorial system with the repeated 2-( n , b )-designs. The internal components of the interval I (α, β) correspond to the complete 2-( n , b )-design system while external ones are determined by calculating the number of zeros or unities in the columns of the truth table of the assigned logical function. This makes it possible to use the theory of I (α, β) intervals in the mathematical apparatus of 2-( n , b )-design combinatorial systems to minimize Boolean functions by the method of equivalent figurative transformations, in particular, to perform automated search for the 2-( n , b )-design systems in the structure of a truth table. Experimental study has confirmed that the combinatorial 2-( n , b )-design system and the consistent alternation of logical operations of super-gluing the variables (if such an operation is possible) and the simple gluing of variables in the first truth table improves the efficiency of the process and the reliability of results from minimizing the Boolean functions. This simplifies algorithmizing the search for the 2-( n , b )-design system in the structure of a truth table of the assigned logical function, which would provide for the tool to further automate the search for the 2-( n , b )-design system. In comparison with analogs, it enables increasing the productivity of the Boolean function minimization process by 100–200 % by using an optimum alternation of operations of super gluing and simple gluing of variables by the method of equivalent figurative transformations. There are reasons to argue about the possibility to improve the productivity of the Boolean function minimization process by the optimal combination of the sequences of the logical operations of super-gluing the variables and the simple gluing of variables by the method of equivalent figurative transformations

We use a regular and modular eight-variable Karnaugh map to reveal some technical details of Boolean minimization usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, that looks better than the published solution in two aspects, namely: (a) it is based on a smaller minimal set of supporting variables, and (b) it provides a more compact Boolean formula. However, we refrain from labelling our solution as a better one, but instead we stress that it is simply un-comparable with the published solution. The comparison between any two rival solutions should be context-specific and not tool-specific. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as is in QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. Moreover, each of them is unate as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.

The article introduces advanced mathematical modeling methods for the automated reproduction of discrete functions of majority and Boolean logic using multiplexer systems. In this work, structural configuration is not understood as the development of specialized software implemented using microprocessors, but, on the contrary, as the transformation of technological operations for setting up and changing the configurations of large micro- and nanocircuits in such a way as to synthesize a functionally complete logical base on a structural-logical basis. The latest technology of nanoscale cellular automata (QCA or QCA-quantum-dots cellular automata) is the most promising direction of single-electron electronics, which also confirms Moore's law. Unlike traditional microtechnologies, for example, based on complementary MOS transistors, in QCA binary information is encoded by polarizations of two electrons, and not through logical voltage levels. The results of computer programming of nanodevices obtained in the article con-firmed their advantages over microelectronic equivalents in terms of increased speed and minimal energy consumption. The modern CAD QCA Designer was used for the analysis and synthesis of Boolean, majority and hybrid logic functions using micro- and nanocircuits in space applications at cryogenic temperatures to predict their trouble-free operation. The results obtained and their analysis confirm the possibility of improving the method-ology for configuring modern micro- and single-electron nanodevices.

As was seen in Chapter 12 of BFE, several Boolean-based algebraic structures have been studied, that is, algebras having as support a Boolean algebra and whose operations are Boolean functions; homomorphisms connecting such algebras and expressed by Boolean functions have also been investigated. Chapter 13 of BFE deals with Boolean arithmetic and Boolean geometry. The former means the study of divisibility between Boolean functions. In Boolean geometry the role of the space is played by a Boolean algebra and one looks for analogues of the basic concepts of geometry; for instance, a “good” analogue of the distance function is the symmeric difference d(x, y) = x + y. As was seen in BFE, Chapter 14, Boolean analysis replaces the real line by a Boolean algebra, while the functions dealt with are Boolean functions.

Boolean algebra is the primary mathematical tool used in designing the computing machines that have come into widespread use in the latter part of this century. Switching functions, described using two-valued Boolean algebra, are implemented using transistor circuits [4]. These transistors act as on-off switches, which is a natural implementation for representing the primitive operations in a Boolean algebra. Once the primitive operations of a Boolean algebra are implemented using transistor circuits, any Boolean function can be implemented using these primitive operations. Operations AND, OR and NOT form the primitive operations of Boolean algebra. Other variations of these primitive operations such and NAND and NOT can also be used as the set of primitive operations.

This chapter addresses Boolean algebra, which is based on Boolean logic. In the social sciences, Boolean algebra comes under different labels. It is often used in set-theoretic and qualitative comparative analysis to assess complex causation that leads to particular outcomes involving different combinations of conditions. The basic features of Boolean algebra are the use of binary data, combinatorial logic, and Boolean minimization to reduce the expressions of causal complexity. By calculating the intersection between the final Boolean equation and the hypotheses formulated in Boolean terms, three subsets of causal combinations emerge: hypothesized and empirically confirmed; hypothesized, but not detected within the empirical evidence; and causal configurations found empirically, but not hypothesized. This approach is both holistic and analytic because it examines cases as a whole and in parts.

Chapter 12 - Boolean Algebra and Combinatorial Circuits

Undetected failures in critical control systems can be catastrophic, including the loss of human lives or huge amounts of money. The safety assessment process aims to minimize such problems. In a previous work, we showed a strategy based on the process algebra CSP to obtain failure logic (a kind of Boolean structure expression or function representing part of a static fault tree) from a set of sequences of faults (or failure traces) which leads to unwanted outputs. Following common practices in industry, we used Simulink diagrams as an input of our strategy. In this work we define a representation of failure traces in terms of a lists-based bounded lattice. These failure traces are now able to describe temporal failures (with a notion of fault propagation). We show that the proposed lattice is indeed a Boolean algebra. This means we benefit from Boolean algebra's properties, laws and existing reduction techniques. We illustrate our work on a simple but real case study supplied by our industrial partner EMBRAER.

Stimuli-responsive multistep switching phenomena of a luminescent bimetallic Ru(II) complex are employed herein to fabricate multiple configurable logic devices. The complex exhibits "off-on" and "on-off" emission switching upon alternative treatment with visible and UV light. Additionally, remarkable augmentation of the rate as well as quantum yield of photoisomerization was achieved via the use of a chemical oxidant (Ce4+) as well as a reductant (metallic sodium). Upon exploiting the emission spectral response of the complex, several advanced Boolean logic functions, including IMPLICATION as well as 2-input 2-output and 3-input 2-output complex combinational logic gates, are successfully implemented. Additionally, by utilizing the vast efficacy of Python, a novel "logic_circuit" model is devised that is capable of making accurate decisions under the influence of various input combinations. This model transcends traditional Boolean logic gates, offering flexibility and intuition to design logical functions tailored to specific chemical contexts. By integrating principles of logic circuits with chemical processes, this innovative approach enables structure determination of the chemical states based on input conditions, thereby unlocking avenues for exploring intricate interactions and reactions beyond conventional Boolean logic paradigms.

Aim/Purpose: The purpose of this study was to examine whether the authoring of computer games in a mainstream primary school setting can support the learning of game design and programming concepts. Background: Despite the benefits for students when they learn how to program and the significant body of research regarding this matter, these benefits are still under debate, and the teaching of programming has a relatively undeveloped pedagogy. With this in mind, a project was designed and implemented, having constructionism as its theoretical framework. Also, Microsoft’s Kodu Game Lab was used for the development of students’ games. Methodology: The project lasted for almost a school year (fifty two-hour sessions) and the target group was fifth-grade students (ages 10-11). A total of 138 students participated, coming from five schools in Athens, Greece. Students were divided into three groups. While all groups worked in pairs, to the first there was no teachers’ guidance, to the second there was limited teachers’ guidance, and to the third, a combination of teacher-led and pair work was used. Each group developed three games of escalating complexity and a total of 207 games were analyzed. Data were collected by analyzing students’ games and with a short questionnaire. Contribution: The study contributes to the debate surrounding the pedagogy of computer game authoring as a tool for teaching programming. That is because few studies have examined the above in mainstream settings, having as a target group primary school students. Furthermore, the present study is in contrast to most studies which involved intensive research projects, since it lasted for almost a school year. Findings: It was found that the most commonly used programming concept was conditions, followed by variables and loops, while Boolean logic and functions were the least used ones. The most problematic concepts proved to be Boolean logic and loops, closely followed by functions. The least problematic concepts were conditions and variables. Also, the number of programming concepts that were used was increasing in each game, while the errors were decreasing. All in all, students’ final games fall into the relational level according to a modified version of the SOLO taxonomy. While the findings indicated that, as well as learning some basic programming concepts, students enjoyed the activity and demonstrated positive attitudes to learning programming by developing games; it was also found that the teaching method did not have any effect on the learning outcomes nor in their views for game authoring. Recommendations for Practitioners: Extended projects can be considered for teaching programming to primary school students, using visual programming tools that allow the development of games. The classes’ teachers can undertake the task to teach programming if they are properly trained. The SOLO taxonomy can be used for assessing students’ games. Future Research: Future studies can examine a variety of game-like programming environments and the target group can be older or younger students. The assessment of students’ games is also an interesting topic. Finally, research can be conducted by using other devices and compare the results.

Neural networks have been applied in various domain including science, commerce, medicine, and industry. However, The knowledge learned by a trained neural network is difficult to understand. This paper proposes a Boolean algebra based algorithm to extract comprehensible Boolean rules from supervised feed-forward neural networks to uncover the black-boxed knowledge. This algorithm is called the BAB-BB rule extraction algorithm, which stands for a Boolean algebra based rule extraction algorithm for neural networks with binary and bipolar inputs. Decomposition techniques and interval arithmetic are used in the algorithm. First, each neuron associated with its inputs is analyzed and a Boolean function, describing the activation rule from its inputs to the neuron, is derived. These Boolean functions are merged into an aggregated Boolean rule according to the network topology. The Boolean rule is then further simplified by Boolean algebra operations. During the rule extraction procedure, redundant hidden neurons can be detected and removed without affecting the original function of the neural network. Examples of unipolar and bipolar inputs are presented to demonstrate the use of our algorithm. Finally, the Exclusive OR problem is presented and solved by our algorithm. Results show that our BAB-BB algorithm is practicable and of high efficiency.

After while the “Human Genome Project” proposes, the people complete measures the foreword plan after the multi-gene genome team, also starts to change to these genes and their reciprocity function understanding research. Varieties of gene regulation Boolean networks algorithms have been proposed of the gene expression profiles, however, the problem of noise could always be found in creating a Boolean network. Due to gene expression data are always noisy. In this paper, it show that after the Boolean networks logic function are learned from noisy data, some noise in the Boolean function could be restructure Karnaugh Maps. It could find logic relationships between protein and protein and restructure protein logic networks. It find logic relationship among proteins as well as COGs (clusters of orthogous groups) and build the logic network of protein.

Boolean algebra operating on the binary numbers 0 and 1 was first developed by George Boolean in 19th century [1]. A Boolean function is an expression using binary variables. We review basic Boolean algebra laws, duals of these laws, functions and a visual method called Karnaugh-maps to simplify a Boolean expression in the first part of this chapter. We then review combinational circuits in the second part and describe simple structures called logic gates to build a logic circuit to represent a Booelan function. We conclude this chapter by arithmetic circuits to add binary numbers.