Irredundant Disjunctive Form Research Articles

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.

Further Improved Variable-Entered Karnaugh Map Procedures for Obtaining the Irredundant Forms of an Incompletely-Specified Switching Function

Out of the potpourri of methods available for traditional minimization of switching functions, the method of the Karnaugh map is distinguished as a quick manual method that provides the user with pictorial insight. An advanced version of this map, viz., the variable- entered Karnaugh map (VEKM) doubles the variable-handling capa- bility of the map and allows its use for big Boolean algebras. The present paper offers a novel exposition of the essential features and properties of the VEKM, many of which are published for the first time. It also presents a simple and further improved VEKM procedure that obtains one of the irredundant disjunctive forms (IDFs) of an in- completely specified switching function (ISSF). Duality concepts are used to convert the present procedure into a dual one that obtains an irredundant conjunctive form for an ISSF. These procedures differ from their predecessors in two respects. First, the present procedures are rather advanced ones equipped with an explicitly stated set of rules that are clearer, though more powerful, than those of the preced- ing procedures. Second, the present procedures are more precise in handling the contributions of a map entered term, or alterm, and hence are more likely to capture minor details in the intrinsic structure of the ISSF under consideration. Therefore, the present procedures, if fol- lowed strictly, are more likely to achieve exact minimality, and even if not, the resulting expressions from them are always guaranteed to be almost minimal. Many detailed examples are given to demonstrate the essential features and properties of the map and to illustrate the rules and steps of the new procedures.

A Boolean minimization procedure using the variable-entered Karnaugh map and the generalized consensus concept

Both the conventional Karnaugh map (CKM) and the variable-entered Karnaugh map (VEKM) are powerful manual tools that have many pictorial and pedagogical advantages. However, while the CKM is suitable only for small-sized problems, the VEKM can handle more variables and hence can be used for a variety of mediumsized applications. This paper presents an advanced and further improved VEKM procedure that aims at obtaining most, if not all, of the irredundant disjunctive forms of an incompletely-specified switching function (ISSF). This procedure differs from previous ones in two respects. First, it is a rather elaborate one employing the concept of the generalized consensus of a set of terms. Second, it uses a more systematic and more precise two-pass strategy that is more likely to capture minor details in the intrinsic structure of the ISSF under consideration. This procedure is intended as a supplement rather than a replacement of more powerful algorithms that have computer implementations but are too complicated to have some graphical insight for the user. Two detailed examples are given to illustrate the rules and steps of the new procedure and to demonstrate its efficiency and power.

P-functions-ternary logic functions capable of correcting input failures and suitable for treating ambiguities

The authors consider a new class of special ternary logic functions: the P-functions capable of correcting input failures, based on the regular ternary logic. There are three major results. First, the P-functions are a special subset of the regular functions with no information loss. Second a ternary logic function defined by the all-prime-implicant disjoint of any Boolean connective is a P-function. Third, a P-function may be identified from the irredundant disjunctive form. The relationship between P-functions and other meaningful ternary logic functions is discussed. >

Comments on "Generalization of Consensus Theory and Application to the Minimization of Boolean Functions"

An algorithm for finding a minimal irredundant disjunctive form representation for an incompletely specified multiple-output switching function presented by P. Tison1is shown to be incorrect by counterexample and corrected.