Extremal boolean functions with long prime implicants

Most methods for the determination of prime implicants of a Boolean function depend on minterms of the function. Deviating from this philosophy, this paper presents a method which depends on maxterms ( minterms of the complement of the function) for this purpose. Normally maxterms are used to get prime implicates and not prime implicants. It is shown that all prime implicants of a Boolean function can be obtained by expanding and simplifying any product of sums form of the function appropriately. No special form of product of sums is required. More generally prime implicants can be generated from any form of the function by converting it into a POS using well known techniques. The prime implicants of a product of Boolean functions can be obtained from the prime implicants of individual Boolean functions. This allows us to handle big functions by breaking them into product of smaller functions. A simple method is presented to obtain one minimal set of prime implicants from all prime implicants without using minterms. Similar statements hold for prime implicates also . In particular all prime implicates can be obtained from any sum of products form. Twelve variable examples are solved to illustrate the methods.

One of the major areas in switching theory research has been concerned with obtaining suitable algorithrns for the minimization of Boolean functions in connection with the general problem of their economic realization. A solution of the minimization problem, in general, involves consideration of two distinct phases. In the first phase all the prime implicants of the function are found, while in the second phase, from this set of all the prime implicants, a minimal subset (according to some criterion of minimality) of prime implicants is selected such that their disjunction is equivalent to the function and from which none of the prime implicants can be dropped without sacrificing equivalence. Many different algorithms exist for solving both the first and the second phase of this minimization problem. In a recent paper,' Slagle et al. describe a new algorithm for the generation of all the prime implicants of a Boolean function. As claimed by the authors, this algorithm is different from those previously given in the literature. The algorithm is efficient, does not generate the same prime implicant more than once (though the algorithm sometimes generates some non-prime implicants), and does not need large capacity of memory for implementation on a digital computer. The algorithm works equally well with either the conjunctive or the disjunctive (both canonical and noncanonical) form of the function.

In practice, users may only be interested in subsets of associations containing attributes satisfying given Boolean conditions also called Boolean constraints. This chapter, which builds upon the previous chapter, deals with solving the problem of mining for association rules in the presence of such constraints. Taxonomies may be present and constraints may contain both terminal and nonterminal attributes. A set of Boolean constraints can be identified with a Boolean function. In the first section, we define the syntax and semantics of Boolean functions. In the second section we review the notion of prime implicant. The prime implicants are the basic building blocks of Boolean functions. Any Boolean function can be identified with the set of its prime implicants (often, only identified with a subset of it, since the set of prime implicants is, in general, redundant). Each prime implicant defines a sublattice in 2 A . In the last section, we take advantage of the sublattices attached to the prime implicants to devise a sequential and a parallel algorithm solving the problem of mining for association rules under Boolean constraints. The algorithms derive from the ones developed in Chapter 4. Cas enumeration takes advantage of the sublattices to discard all those cass that do not meet the given constraints or cannot be expected to lead to cass meeting these constraints.

The construction of a decimal grouping table and its use to determine essential and nonessential prime implicants for the minimisation of an n-variable Boolean function are presented in the paper. In existing tabular methods, e.g. the Quine-McCluskey technique, each fundamental product is represented by a row of binary 1s and 0s and the finding of a set of prime implicants necessitates the formation of successive tables of binary characters, and only after an exhaustive search in the tables can one discover any prime implicants. Dealing with binary characters is rather tedious, and searching through several tables to establish a prime implicant is time consuming. The proposed grouping table offers the convenience of using decimal minterm numbers and the advantage of using one table in the search for prime implicants. In the grouping table the decimal equivalent of function terms appear in a column and the entries to a row corresponding to a function term N are the decimal equivalent of product terms related to N by one change of variable. From these entries, only those terms which appear in the function under investigation are selected and only these need to be considered for the minimisation of the problem. Thus, unlike other tabular methods, the grouping table provides all possible combinational terms for each fundamental product term as its row terms and also the facility of at-a-glance comparison of all function terms by referring to the same table. In the paper, a method of minimising Boolean functions with the aid of grouping table is illustrated with examples.

The performances of almost all available fault tree analysis tools are limited by the performance of their prime implicant computation procedure. All these procedures manipulate the prime implicants of the fault trees in extension, so that the analysis costs are directly related to the number of prime implicants to be generated, which in practice makes these tools difficult to apply on fault trees with more than 20 000 prime implicants. This paper introduces an analysis method of coherent as well as noncoherent fault trees that overcomes this limitation because its computational cost is related to neither the number of basic events, nor the number of gates, nor the number of prime implicants of these trees. The authors present the concepts underlying the prototype tool MetaPrime, and the experimental results obtained with this tool on real fault trees. These results show that these concepts provide complete analysis in seconds on fault trees that no previously available technique could ever even partially analyze, for instance noncoherent fault trees with more than 10/sup 20/ prime implicants. These concepts can also be used to analyze event trees because such trees denote Boolean functions on which these concepts can be applied. Prime implicant computation is also critical in many other domains, in particular in expert system applications such as reasoning maintenance and multiple fault diagnosis. The application of the concepts underlying MetaPrime to the resolution of these problems is under study.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Quine–McCluskey method of minimizing a Boolean function gives all the prime implicants, from which the essential terms are selected by one or more cover tables known as the prime implicant tables. This note describes a tabular method where the essential prime implicants are selected during the process of forming the combination tables, and other essential terms are selected from what have been described in the note as chains of selective prime implicants. Consequently, the need for successive prime implicant tables is eliminated.

Application of Hypergraphs in the Prime Implicants Selection Process

The Quine-McCluskey method is an algorithm to minimize Boolean functions. Although the method can be programmed on computers, it takes a long time to return the set of prime implicants, thus slowing the analysis and design of digital logic circuits. As a result, it slows down the dynamic reconfiguration process of programmable logic devices. In this paper, we first propose a data representation for storing implicants in memory to reduce the cache misses of the program. We then propose an algorithm to find all prime implicants of a Boolean function. The algorithm aims to reuse the data available on cache, thus decreasing cache misses. After that, we propose an algorithm for step 2 of the Quine-McCluskey method to select the minimal number of essential prime implicants. The evaluation shows that our proposals achieve much higher performance than the original Quine-McCluskey method. The number of essential prime implicants is a low percentage, less than 50%, of the total prime implicants generated in step 1 of the method.

We consider the problem Enum·IP of enumerating prime implicants of Boolean functions represented by decision decomposable negation normal form (dec-DNNF) circuits. We study Enum·IP from dec-DNNF within the framework of enumeration complexity and prove that it is in OutputP, the class of output polynomial enumeration problems, and more precisely in IncP, the class of polynomial incremental time enumeration problems. We then focus on two closely related, but seemingly harder, enumeration problems where further restrictions are put on the prime implicants to be generated. In the first problem, one is only interested in prime implicants representing subset-minimal abductive explanations, a notion much investigated in AI for more than thirty years. In the second problem, the target is prime implicants representing sufficient reasons, a recent yet important notion in the emerging field of eXplainable AI, since they aim to explain predictions achieved by machine learning classifiers. We provide evidence showing that enumerating specific prime implicants corresponding to subset-minimal abductive explanations or to sufficient reasons is not in OutputP.

This paper describes an algorithm for minimizing an arbitrary Boolean function. The approach differs from most previous procedures in which first all prime implicants are found and then a minimal set is then determined. This procedure imposes a set of conditions on the selection of the next prime implicant in order to obtain a near minimal sum-of-products realization. Extension to the multiple output and incompletely specified function cases is given. An important characteristic of the proposed procedure is the relatively small amount of computer time spent to solve a problem, as compared to other procedures. The MINI algorithm may give better results for a large number of inputs and outputs if relatively few product terms are needed. This procedure is also well suited to find a solution for programmable logic arrays (PLA's) which internally implement large Boolean functions as a sum-of-products.

A new method for computing the prime implicants of a Boolean function from an arbitrary sum-of-products form is given. It depends on the observation that the prime implicants of a Boolean function can be obtained from the prime implicants of its subfunctions with respect to a fixed but arbitrary variable. The problem of obtaining all irredundant sums from the list of all prime implicants and an arbitrary list of implicants representing the function is solved. The irredundant sums are in one-to-one relation to the prime implicants of a positive Boolean function associated with these lists. The known formulas of Petrick, Ghazala, Tison, Mott, and Chang are obtained as special cases and incompletely specified functions can also be handled. We give a complete and simple method for finding the positive Boolean function mentioned above. The paper is self-contained and examples are included.

Horn functions are Boolean functions where each of the prime implicants contains at most one negative literal. A class of Boolean functions is considered in this letter where a single term containing two negative literals is added by logical-or operation to a Horn function. We show that the function does not have any prime implicant containing three negative literals. We also show that if two terms containing two negative literals are added to a Horn function, then it may have many prime implicants all of which contain three negative literals. We show that it is P-complete to determine whether a given Boolean function in disjunctive normal form of the considered class is a tautology.

A procedure for finding out the prime implicants and hence the minimal sum(s) of Boolean functions containing essential prime implicants, cyclic type prime implicants or functions with optional terms has been suggested in this paper. The conception of the present procedure is based on the geometrical representation of Boolean functions. The procedure is also applicable for obtaining the prime implicants of a multiple output function.

Recent advances in nanotechnology have led to the emergence of energy efficient circuit technologies like spintronics and domain wall magnets that use Majority gates as their primary logic elements. Logic synthesis that exploits these technologies demand an understanding of the mathematics of n-input Majority terms. One of the problems that turn up in such a study is the checking of equivalence of two n-input Majority terms on the same set of variables. We provide an algorithm based on prime implicants as a solution to this problem. In this quest, we extend the concept of implicants to two cases - 1-implicants and prime 1-implicants that imply a function evaluates to '1', and 0-implicants and prime 0-implicants that imply that it evaluates to '1'. We exploit the properties of Majority to create an efficient algorithm to generate the sums of all prime 1-implicants and all prime 0-implicants of an n-input Majority term, both being canonical representations of Boolean functions. As Majority and Threshold functions have been shown to be logically equivalent, our algorithms are applicable to Threshold functions as well. Also, being based on prime implicants, our algorithms improve on the known algorithm for equivalence checking for threshold logic terms.

A systematic procedure of finding out all the irredundant forms of a Boolean function is presented in this paper. It has been shown that when a particular prime implicant or a group of prime implicants is considered for covering a term or a group of terms then certain restrictions in the form of compulsory occurrence or non-occurrence are imposed on the remaining prime implicants. These restrictions are found from the prime implicant matrix. The knowledge and use of these restrictions at proper stages makes the problem of determining the irredundant forms of a Boolean function a rapidly converging one.