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The JK flip flop is a flexible type of bistable elements that has extensive uses in digital electronics and control circuits. It is usually described by its characteristic equation or next-state table (used for analysis purposes) and its excitation table (used for synthesis purposes). This paper explores a variety of novel characterizations of JK flip flops. First, equational and implicational descriptions are presented, and methods of logic deduction are utilized to produce complete statements of all propositions that are true for a general JK flip flop. Next, methods of Boolean-equation solving are employed to find all possible ways to express the excitations in terms of the present state and next state. The concept of Boolean quotient is used to impose certain requirements so as to find particularly useful expressions of the excitations. Relations of JK flip flops to other types of flip flops are also explored. This paper is expected to provide an immediate pedagogical benefit, and to help facilitate the analysis and synthesis of sequential digital circuits.

Given the contemporary importance of communications systems and infrastructures in general, networks need to be designed and operated in such a way that failures can be reduced, alleviated, or mitigated. Network components might for instance fail due to equipment failing, malicious attacks, planned or, unplanned activities, and so on. This paper presents a three-state system model (Good, Poor, and Fail). This model is demonstrated via two examples covering the proposed multistate model in detail. The first example studies a firewall system with two components in two operational modes, namely active-active and active-standby. The second example covers a simple private cloud setup with three layers, namely firewall, server, and service layer. The first example is solved by two methods (a) Tabular form of exhaustive enumeration; (b) Multi-state Decision Diagrams (MDD). The second Example is solved by (a) The exhaustive enumeration method, and (b) Dividing the system into 3 subsystems and integrating their results. The results of this study indicate that the firewall block modes of operation have a major impact on the studied systems. Firewalls in active-active mode significantly improve the probability of system’s Up state. The findings of this study should be of great help to designers to better design services to increase service availability.

This paper deals with the reliability of a multi-state delivery network with multiple suppliers, transfer stations and markets (depicted as vertices), connected by branches of multi-state capacities, delivering a certain commodity or service between their end vertices. We utilize a symbolic logic expression of the network success to satisfy the market demand within budget and production capacity limitations even when subject to deterioration. This system success is a two-valued function expressed in terms of multi-valued component successes, and it has been obtained in the literature in minimal form as the disjunction of particular prime implicants or minimal paths of the pertinent network. This paper provides a systematic procedure for converting this minimal expression into a probability-ready expression (PRE). We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain. This concept is important since it allows a direct transformation of a random logical expression, on a one-to-one one, to its statistical expectation form, simply by replacing all logic variables by their statistical expectations, and also substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing). The statistical expectation of the expression is its probability of being equal to 1, and is simply called reliability of the multi-state delivery network (MSDN).

An -bit comparator is a celebrated combinational circuit that compares two -bit inputs and and produces three orthonormal outputs: G (indicating that is strictly greater than ), E (indicating that and are equal or equivalent), and L (indicating that is strictly less than ). The symbols ‘G’, ‘E’, and ‘L’ are deliberately chosen to convey the notions of ‘Greater than,’ ‘Equal to,’ and ‘Less than,’ respectively. This paper analyzes an -bit comparator in the general case of arbitrary and visualizes the analysis for on a regular and modular version of the 8-variable Karnaugh-map. The cases 3, 2 and 1 appear as special cases on 6-variable, 4-variable, and 2-variable submaps of the original map. The analysis is a tutorial exposition of many important concepts in switching theory including those of implicants, prime implicants, essential prime implicants, irredundant disjunctive forms, minimal sums, the complete sum and disjoint sums of products (or probability-ready expressions).

A multi-state k-out-of-n: G system is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ? m ? j. This paper is devoted to the analysis of a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components. We express each instance of the multi-state system output as an explicit function of the multi-valued inputs of the system. The ultimate outcome of our analysis is a Multi-Valued Karnaugh Map (MVKM), which serves as a natural, unique, and complete representation of the multi-state system. To construct this MVKM, we use “binary” entities to relate each of the instances of the output to the multi-valued inputs. These binary entities are represented via an eight-variable Conventional Karnaugh Map (CKM) that is adapted to a map representing four variables that are four-valued each. Despite the relatively large size of the maps used, they are still very convenient, thanks to their regular structure. No attempt was made to draw loops on the maps or to seek minimal formulas. The maps just served as handy tools for combinatorial representation and for collectively implementing the operations of ANDing, ORing, and complementation. The MVKM obtained serves as a means for symbolic analysis yielding results that agree numerically with those obtained earlier. The map is a useful tool for visualizing many system properties, and is a valuable resource for computing a plethora of Importance Measures for the components of the system.

This chapter reviews various types of techniques utilized for analyzing capacitated networks, with an eye on assessing the performance indexes of practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems. The techniques reviewed include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method. The chapter is supplemented with an appendix, which introduces the concept of a “probability-ready expression” for a Boolean-based coherent pseudo-Boolean function.

There is a definite need for representations of, and tools for, conditional probability that enhance understanding, simplify calculations, foster insight, and facilitate reasoning. Such representations and tools are desirable and useful in a wide variety of scientific disciplines, but their utility in medical contexts and applications are stressed herein, so as to address a clinical rather than a mathematical audience. We employ a plethora of time-tested pedagogical representations or tools of conditional probability including: (a) Visualization on Venn diagrams or Karnaugh maps, (b) Reformulation as natural frequencies, (c) Entity interrelations via Signal Flow Graphs, (d) Specification of certain problem formats such as the format of Trinomial Graphs, and (e) Use of a pair of complementary Karnaugh-map-like or eikosogram-like diagrams. Most of the new representations or tools have well known histories of pedagogical advantages, but are still to be tested in the specific realm of conditional probability. Further assessment of the novel representations or tools proposed herein is needed. Each of these is to be taught to a group of students, and a control group of students is to be instructed via the conventional representation. Detailed statistical analysis of the outcomes is warranted. Similar investigations were performed earlier, but they were on a limited scale. Therefore, the need arises for an explorative study that exhausts all proposed representations or tools.

This chapter attempts to enhance the interplay between the ecology and reliability fields by employing Boolean-based reliability language and techniques to quantify ecological metrics related to connectivity and redundancy. We emphasize the question of connectivity in models of probabilistic networks as a common area of interest for both fields. The chapter borrows techniques from mainstream reliability theory to treat a prominent problem of ecology, namely that of survivability (of a species), defined here as the probability of successful migration of a certain organism escaping from critical source habitat patches and seeking refuge in specific destination habitat patches via heterogeneous deletable ecological corridors, possibly with uninhabitable stepping stones en route. This problem might be reformulated in contexts other than that of migration, including those of (a) dynamics of metapopulations, colonization, or invasion, (b) gene flow, (c) spread of infectious diseases, epidemics, or pandemics, and (d) energy transfer within food webs. Indicators of network connectivity in classical reliability theory are probabilities that might be designated according to the set of source nodes and the set of destination nodes as one to one, one to many, many to many, or all to all. Our present notion of survivability (of a species) is also a probability of connectivity, now measured from any node (among many nodes) to any node (among many nodes). We explore methods for computing the survivability (of a species) by adapting switching-algebraic techniques that are usually employed in the reliability field. In addition to this survivability metric, we comment on some other connectivity indicators that are currently used in ecology. We stress two recent contributions to the ecology literature, one employing analogy with electric circuit theory, and another concerning the most reliable (or minimum-lag) dispersal paths.

A multi-state k-out-of-n: G system is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components. We express each instance of the multi-state system output as an explicit function of the multi-valued inputs of the system. The ultimate outcome of our analysis is a Multi-Valued Karnaugh Map (MVKM), which serves as a natural, unique, and complete representation of the multi-state system. To construct this MVKM, we use “binary” entities to relate each of the instances of the output to the multi-valued inputs. These binary entities are represented via an eight-variable Conventional Karnaugh Map (CKM) that is adapted to a map representing four variables that are four-valued each. Despite the relatively large size of the maps used, they are still very convenient, thanks to their regular structure. No attempt was made to draw loops on the maps or to seek minimal formulas. The maps just served as handy tools for combinatorial representation and for collectively implementing the operations of ANDing, ORing, and complementation. The MVKM obtained serves as a means for symbolic analysis yielding results that agree numerically with those obtained earlier. The map is a useful tool for visualizing many system properties, and is a valuable resource for computing a plethora of Importance Measures for the components of the system.

Multi-State systems are systems whose outputs are multi-valued (due to multiple levels of capacity or performance) and (possibly) whose inputs are also multi-valued (due to multiple performance levels or multiple modes of failure). These systems are a generalization of binary or dichotomous systems that have binary or two-valued outputs and inputs. The multi-state reliability model generalizes and adapts many of the concepts and techniques of the binary reliability model, and naturally ends up with sophisticated concepts and techniques of its own. This paper explores the possibility of simply analyzing a multi-state system by reformulating or encoding its inputs in terms of binary inputs and evaluating each of its multiple output levels as an individual binary output of these alternative inputs. This means that we dispense with multiple-valued logic in the analysis of a multi-state system, since this system is now analyzed solely via switching algebra (two-valued Boolean algebra). The wealth of tools and techniques of switching algebra are now used (without any modification or adaptation) in the analysis of the multi-state system (at the cost of an expanded input domain). The paper makes its point though the analysis of a standard commodity-supply system, whose multi-valued inputs are expressed in terms of physically-meaningfully binary inputs. The analysis is made possible through the use of advanced techniques for deriving probability–ready expressions together with the employment of large-size Karnaugh maps and utilization of multiplication tables, symmetric switching functions, and Boolean quotients. Though the system studied involves twelve binary input variables, its manual analysis is completed successfully herein, yielding results that exactly agree with those obtained earlier via automated methods, and are possibly less prone to the notorious effects of round-off errors.

This paper advocate and demonstrates the utility of the Karnaugh map, as a pictorial manual tool of Boolean algebra, in the exploration of medical problems as exemplified herein by the problem of Fetal Malnutrition (FM). The paper briefly introduces the FM problem, and specifies four metrics or tests used frequently in its study. Clinical data collected about these metrics (as continuous variables or dichotomized versions thereof) are conventionally treated via statistical methods. The Karnaugh map serves as a convenient way for aggregating the set of clinical data available into a pseudo-Boolean function. The map can be used to produce a two-by-two contingency matrix (confusion matrix or frequency matrix) that relates an assessed test or metric to a reference or standard one. Each of these two metrics can be any of the map variables or a function of some or all of these variables. While the map serves in this capacity as a supplement or aid to statistical methods, it is also useful for certain non-statistical methods (specifically Boolean ones). The paper shows how the map entries can be dichotomized via an appropriate threshold for use in Boolean Analysis (BA), which can be conducted despite the lack of a gold standard. The map also implements Qualitative Comparative Analysis (QCA) for the given clinical data. The map variable-handling capability does not pose as a shortcoming for either BA or QCA, since the number of variables involved (not only herein but in other typical medical problems as well) is relatively small. The concepts and methods introduced herein are demonstrated through application to the aforementioned set of clinical data for the FM problem, and can be extended to a wide variety of medical problems.

Modern digital circuits, especially those based on large-scale integration devices employ DE flip flops, which are an extension of the D type with the capacity to store an input value only upon request or enabling. The DE flip flop could possibly be described algebraically by its characteristic equation or tabularly by its next-state table (used for analysis purposes) and its excitation table (used for synthesis purposes). This paper explores two novel characterizations of DE flip flops. First, equational and implicational descriptions are presented, and the Modern Syllogistic Method is utilized to produce complete statements of all propositions that are true for a general DE flip flop. Next, methods of Boolean-equation solving are employed to find all possible ways to express the excitations in terms of the present state and next state. The concept of Boolean quotient plays a crucial role in exposing the pertinent concepts and implementing the various desired derivations. This paper is expected to be of an immediate pedagogical benefit, and to facilitate the analysis and synthesis of contemporary sequential digital circuits.

The paper utilizes the Karnaugh map in a pedagogical calculation of the Shapley-Shubik index of voting power, which is perhaps the most prominent such index for managerial, political, and engineering voting systems. The Karnaugh map has features that allow it to readily implement this calculation, namely: (a) it can conveniently represent pseudo-switching functions of the form g= B_2^n →R, (b) it can immediately detect the number of orderings in going from one of its configurations (cells) to another, and (c) it visually identifies a pair of configurations that is critical w. r. t. a certain voter, which are a primitive winning coalition of which this voter is a member and a corresponding primitive losing coalition that results when this voter defects. The paper offers a detailed description of these three features and of the associated calculations using the celebrated system of the European Economic Community as a running example.

This paper deals with one of the most prominent problems in industrial prognostics, namely the estimation of the Remaining Useful Life (RUL) of the most popular industrial battery, viz., the lithium-ion battery. The paper presents a state-space model of the battery, and then estimates the dynamic behavior of seven of its process variables and two of its sensor variables. The estimation is achieved via two well known estimators, the Unscented Kalman Filter (UKF) and the Particle Filter (PF) when noise of various levels and types is injected. Numerical and chart comparisons of these two computing estimators are reported and discussed.

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.

Internet Service Providers (ISPs) are expected to continuously provide services without interruptions. However, planning for this activity requires a number of tasks such as forecasting clients' demand that needs to be considered in fulfilling their objectives particularly in face of calamity and traffic load variations. We present a Software-Defined Networking (SDN) load distribution technique for a typical ISP. We have implemented an optimization method employing linear programming to achieve this goal. A multi-objective model has been developed for this purpose. The multiple objectives are maximization of link utilization, maximization of link availability and minimization of average round-trip time for the top visited websites by users. Furthermore, an automated load distribution system has been developed for carrying out these optimization related activities. In addition, an example is presented to illustrate traffic load distribution under different operational modes (normal, critical, and catastrophic). The findings of this study should be of great help to the service provider to better plan for load distribution between international links, and help in approving down times and maintenance windows requested by cable providers.

Natural and designed ecological corridors are key elements for the survival of a species, as they allow the species to avoid local extinction by migrating to more suitable habitat patches. This paper studies various reliability metrics for the process of migration in a metapopulation landscape network from a critical habitat patch to destination habitat patches via perfect stepping stones and imperfect (deletable) corridors. The work presented herein generalizes earlier work on the application of reliability theory in ecology by allowing corridors to be heterogeneous (of non-identical unreliabilities). The paper is a tutorial exposition of modern reliability techniques, which formulate a problem in the Boolean domain, manipulate formulas to achieve disjointness of logically added subexpressions and retain statistical independence of logically multiplied ones, and finally reach a probability-ready expression that is directly transformed back to the probability domain. Several metrics are covered including system unreliability, life expectancy (MTTF), and component importance measures. An interesting finding is that the life expectancy of a classical landscape network is more than double that of a single corridor. Extensions to quantification of uncertainty in the above metrics and to evaluation of more sophisticated metrics of landscape connectivity are also pointed out.

An important class for Boolean reasoning problems involves interdependence among the members of a set T of Boolean functions. Two notable problems among this class are (a) to establish whether a given subset of T is dependent, and (b) to produce economical representations for the complementary families of all dependent subsets and independent subsets of T. This paper solves these two problems via a powerful manual pictorial tool, namely, the variableentered Karnaugh map (VEKM). The VEKM is utilized in executing a Label-and-Eliminate procedure for producing certain prime implicants or consequents used in tackling the two aforementioned problems. The VEKM procedure is a time-saving short cut indeed, since it efficiently handles the three basic tasks demanded by the solution procedure, which are: (a) To combine several Boolean relations into a single one, (b) to compute conjunctive eliminants of a Boolean function, and (c) to derive the complete sum (CS) of a Boolean function. The VEKM procedure significantly reduces the complexities of these tasks by introducing useful shortcuts and allowing simultaneous processing. The VEKM procedure is described in detail, and then demonstrated via two illustrative examples, which previously had only black-box computer solutions as they were thought to be not amenable to manual solution. The first example deals with switching or bivalent functions while the second handles 'big' Boolean functions. Both examples indicate that the VEKM procedure proposed herein enjoys the merits of insightfulness, simplicity and efficiency

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premisesallthat can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology.

Any system of ‘big’ Boolean equations can be reduced to a single Boolean equation . We propose a novel method for producing a general parametric solution for such a Boolean equation without attempting to minimize the number of parameters used, but instead using independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom that appears in the discriminants of the function . We sacrifice minimality of parameters and algebraic expressions for ease, compactness and efficiency in listing all particular solutions. These solutions are given by additive formulas expressing a weighted sum of the asserted atoms of , with the weight of every atom (called its contribution) having a number of alternative possible values equal to the number of appearances of the atom in the discriminants of . This allows listing a huge number of particular solutions within a very small space and the possibility of constructing solutions of desirable features. The new method is demonstrated via three examples over the ‘big’ Boolean algebras, , , and , respectively. The examples demonstrate a variety of pertinent issues such as complementation, algebra collapse, incremental solution, and handling of equations separately or jointly.

A prominent reliability model is that of the partially-redundant (k-out-of-n) system. We use algebraic as well as signal-flow-graph methods to explore and expose the AR algorithm for computing k-out-of-n reliability. We demonstrate that the AR algorithm is, in fact, both a recursive and an iterative implementation of the strategy of Reduced Ordered Binary Decision Diagrams (ROBDDs). The underlying ROBDD for the AR recursive algorithm is represented by a compact Signal Flow Graph (SFG) that is used to deduce AR iterative algorithms of quadratic temporal complexity and linear spatial complexity. Extensions of the AR algorithm for (single or scalar) threshold, double-threshold, vector-threshold, and k-to-l-out-of-n systems have similar ROBDD interpretations.

There is an established equivalence between relatio nal database Functional Dependencies (FDs) and a fragment of switching algebra that is built typical ly of Horn clauses. This equivalence pertains to bo th concepts and procedures of the FD relational databa se domain and the switching algebraic domain. This study is an exposition of the use of switching -algebraic tools in solving problems typically encountered in the analysis and design of relationa l databases. The switching-algebraic tools utilized include purely-algebraic techniques, purely-visual techniques employing the Karnaugh map and intermediary techniques employing the variable-ente red Karnaugh map. The problems handled include; (a) the derivation of the closure of a Dep endency Set (DS), (b) the derivation of the closure of a set of attributes, (c) the determination of all c andidate keys and (d) the derivation of irredundant dependency sets equivalent to a given DS and conseq uently the determination of the minimal cover of such a set. A relatively large example illustrates the switching-algebraic approach and demonstrates i ts pedagogical and computational merits over the tradi tional approach.

Boolean-equation solving permeates many diverse areas of modern science. To solve a system of Boolean equations, one usually combines them into an equiva lent single Boolean equation {f (X) 0} = � whose set of solutions is exactly the same as that of the origin al system of equations. One of the general classes of solutions for Boolean equations is the subsumptive general solution, in which each variable is express ed as an interval decided by a double inequality in terms of the succeeding variables. The solution validity depends on the satisfaction of a required consisten cy condition. In this study, we introduce a novel m ethod (henceforth called the CS method) for producing sub sumptive Boolean-equation solutions based on deriving the complete sum (CS(f (X)) �� of the pertinent Boolean function f (X) � . The complete sum CS(f (X)) � is a disjunction of all prime implicants of f (X) �� and nothing else. It explicitly shows all informat ion about f (X) � in the most compact form. We demonstrate the proposed CS solutions in terms of four examples, covering Boolean algebras of different sizes and using two p rominent methods for deriving CS(f (X)) � . Occasionally, the consistency condition results in a collapse of the underlying Boolean algebra into a smaller subal gebra. We also illustrate how an expansion tree (typically reduced to an acyclic graph) can be used to deduce a complete list of all particular solutions from the subsumptive solution. The present CS method yields correct solutions, since it fits into the frame of the most general subsumptive solution. Among competing subsumptive methods, the CS method strikes a reasonable tradeoff between the conflicting requirements of less computational cost and more compact form for t he solution obtained. In fact, it is the second bes t known method from both criteria of efficiency and c ompactness of solution.

A total description of a synchronous Boolean networ k is typically achieved by a matrix recurrence relation. A simpler alternative is to use a scalar equation which is a possibly nonlinear equation tha t involves two or more instances of a single scalar variable and some Bool ean operator(s). Further simplification is possible in terms of a linear reduced scalar equation which is the simplest two-term scalar equa tion that includes no Boolean operators and equates the value of a scalar variabl e at a latter instance t 2 to its value at an earlier instance t 1. This equation remains valid when the times t 1 and t2 are both augmented by any integral multiple of the underlying time period. In other words, there are i nfinitely many versions of a reduced scalar equatio n, any of which is useful for deducing information about the cyclic behavior of the network. However, to obtain correct information about the transient behavior of the network, one must find the true red uced scalar equation for which instances t1 and t 2 are minimal. This study investigates the nature, d erivation and utilization of reduced scalar equations. It relies on Boolean-algebraic ma nipulations for the derivation of such equations an d suggests that this derivation can be facilitated by seeking certain orthogonality relations among certain successive (albeit not necessarily consecutive) instances of t he same scalar variable. We demonstrate, contrary t o previously published assumptions or assertions, tha t there is typically no common reduced scalar equat ion for all the scalar variables. Each variable usually sat isfies its own distinct reduced scalar equation. We also demonstrate that the derivation of a reduced scalar equation is achieved not only by proving it but also by disproving an immediately preceding version of it when such a version might exist. We also demonstrate that, despite the useful insight supplied by the reduced scalar equations, they do not provide a total solut ion like the one offered by matrix methods and therefore they ne ed be supported by other techniques of mathematical reasoning. We present three classical examples to i llustrate our techniques. Two examples are tutorial s on the necessary Boolean-algebraic techniques. They present corrections of previously published results and r efute purported claims of discrepancies between scalar an d matrix methods. The third example illustrates how the reduced scalar equations can be supplemented by tec hniques of number theory, Diophantine equations and Boolean equations in making subtle inferences about Boolean networks. We achieved a better understanding of the nature of reduced scalar equations, demonstr ated Boolean-algebraic techniques for deriving them, presented other mathematical tools for utilizing th em and finally reconciled them with the more encompassing but more complex and less insightful matrix methods.

An important class of square binary matrices over t he simplest finite or Galois Field GF(2) is the cla ss of involutory or self-inverse (SI) matrices. These mat rices are of significant utility in prominent engin eering applications such as the study of the Preparata Tra nsformation or the analysis of synchronous Boolean Networks. Therefore, it is essential to devise appr opriate methods, not only for understanding the pro perties of these matrices, but also for characterizing and constructing them. We survey square binary matrices of orders 1, 2 and 3 to identify primitive SI matrices among them. Larger SI matrices are constructed as (a) the direct sum, or (b) the Kronecker product, of smalle r ones. Illustrative examples are given to demonstr ate the construction and properties of binary SI matrices. The intersection of the sets of SI and permutation binary matrices is studied. We also study higher-order SI binary matrices and describe them via recursive rel ations or Kronecker products. Our work culminates in an exposition of the two most common representations of Boolean functions via two types of Boolean SI matri ces. A better understanding of the properties and methods of constructing SI binary matrices over GF (2) is achieved. A clearer picture is attained abou t the utility of binary matrices in the representation of Boolean functions.

This chapter deals with the paradigm of handling system reliability analysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain. This paradigm is well-established within the academic circles of reliability theory and engineering, albeit virtually unknown outside these circles. The chapter lists and explains arguments in favor of this paradigm for systems described by verbal statements, fault trees, block diagrams, and network graphs. This is followed by a detailed exposition of the pertinent concept of the Real or Probability Transform of a switching (two-valued Boolean) function, and that of a Probability-Ready Expression (PRE). Some of the important rules used in generating a PRE are presented, occasionally along with succinct proofs. These include rules to achieve disjointness (orthogonality) of ORed formulas, and to preserve statistical independence, as much as possible, among ANDed formulas. Recursive relations in the Boolean domain are also discussed, with an application to the four versions of the AR algorithm for evaluating the reliability and unreliability of the k-out-of-n:G and the k-out-of-n:F systems. These four versions of the algorithm are explained in terms of signal flow graphs that are compact, regular, and acyclic, in addition to being isomorphic to the Reduced Ordered Binary Decision Diagram (ROBDD). An appendix explains some important properties of the concept of Boolean quotient, whose expectation in Boolean-based probability is the counterpart of conditional probability in event-based probability.