Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

Using Eight-Variable Karnaugh Maps to Unravel Hidden Technicalities in Qualitative Comparative Analysis

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.

Conventional and Improved Inclusion-Exclusion Derivations of Symbolic Expressions for the Reliability of a Multi-State Network

This paper deals with an emergent variant of the classical problem of computing the probability of the union of n events, or equivalently the expectation of the disjunction (ORing) of n indicator variables for these events, i.e., the probability of this disjunction being equal to one. The variant considered herein deals with multi-valued variables, in which the required probability stands for the reliability of a multi-state delivery network (MSDN), whose binary system success is a two-valued function expressed in terms of multi-valued component successes. The paper discusses a simple method for handling the afore-mentioned problem in terms of a standard example MSDN, whose success is known in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. This method utilizes the multi-state inclusion-exclusion (MS-IE) principle associated with a multi-state generalization of the idempotency property of the ANDing operation. The method discussed is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier. The example demonstrates the notorious shortcomings and the extreme inefficiency that the MS-IE method suffers, but, on the positive side, it reveals the way to alternative methods, in which such a shortcoming is (partially) mitigated. A prominent and well known example of these methods is the construction of a multi-state probability-ready expression (MS-PRE). Another candidate method would be to apply the MS-IE principle to the union of fewer (factored or composite) paths that is converted (at minimal cost) to PRE form. A third candidate method, employed herein, is a novel method for combining the MS-PRE and MS-IE concepts together. It confines the use of MS-PRE to ‘shellable’ disjointing of ORed terms, and then applies MS-IE to the resulting partially orthogonalized disjunctive form. This new method makes the most of both MS-PRE and MS-IE, and bypasses the troubles caused by either of them. The method is illustrated successfully in terms of the same real-case problem used with the conventional MS-IE.

Symbolic Derivation of a Probability-Ready Expression for the Reliability Analysis of a Multi-State Delivery Network

This paper deals with the reliability of a multi-state delivery network (MSDN) 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 prime implicants or minimal paths of the pertinent network. The main contribution of this paper is to provide 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 of paramount importance 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 network reliability. The proposed method is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier by an alternative means. This paper is a part of an ongoing activity to develop pedagogical material for various candidate techniques for assessing multi-state reliability.

ATEN: And/Or tree ensemble for inferring accurate Boolean network topology and dynamics

Motivation Inferring gene regulatory networks from gene expression time series data is important for gaining insights into the complex processes of cell life. A popular approach is to infer Boolean networks. However, it is still a pressing open problem to infer accurate Boolean networks from experimental data that are typically short and noisy. Results To address the problem, we propose a Boolean network inference algorithm which is able to infer accurate Boolean network topology and dynamics from short and noisy time series data. The main idea is that, for each target gene, we use an And/Or tree ensemble algorithm to select prime implicants of which each is a conjunction of a set of input genes. The selected prime implicants are important features for predicting the states of the target gene. Using these important features we then infer the Boolean function of the target gene. Finally, the Boolean functions of all target genes are combined as a Boolean network. Using the data generated from artificial and real-world gene regulatory networks, we show that our algorithm can infer more accurate Boolean network topology and dynamics from short and noisy time series data than other algorithms. Our algorithm enables us to gain better insights into complex regulatory mechanisms of cell life. Availability and implementation Package ATEN is freely available at https://github.com/ningshi/ATEN. Supplementary information Supplementary data are available at Bioinformatics online.

Logical Design of n-bit Comparators: Pedagogical Insight from Eight-Variable Karnaugh Maps

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, minimal sum, complete sum and disjoint sum of products (or probability-ready expressions).