On the number of prime implicants

Computers & Electrical Engineering, 1979

A procedure Is presented for constructing the Blake canonical form for a switching function (the sum of all its prime imphcants) conveniently and rapidly. This procedure combines Blake's method (iterated consensus) with the multiplying method of Samson and Mills

Artificial Intelligence, 1999

We present a new approach for selective enumeration of prime implicants of CNF formulae. The method uses a 0-1 programming schema, having feasible solutions corresponding to prime implicants. Prime implicants are generated one at a time, so that as many of them can be computed as needed by the specific application considered. Selective generation is also supported, whereby preferences on the structure of generated prime implicants can be specified. We present two algorithms for selective enumeration of prime implicants and discuss their properties. The former amounts to solving the basic 0-1 programming schema first, to obtain an implicant ψ (not necessarily a prime one), and then generating a prime implicant implied by ψ . The latter is based on adding a suitable minimization function to the basic 0-1 programming schema so that finding optimal solutions corresponds one-to-one to generating prime implicants of the original theory. We show that the latter algorithm has wider applicability but is less efficient than the former one. Finally we present experimental results, which confirm the effectiveness of our approach in computing prime implicants of CNF formulae. : S 0 0 0 4 -3 7 0 2 ( 9 9 ) 0 0 0 3 5 -1

ISAIM, 2012

In this short note we introduce a class of Boolean functions defined by a minimum length of its prime implicants. We show that given a DNF one can test in polynomial time whether it represents a function from this class. Moreover, in case that the answer is affirmative we present a polynomial time algorithm which outputs a shortest DNF representation of the given function. Therefore the defined class of functions is a new member of a relatively small family of classes for which the Boolean minimization problem can be solved in polynomial time. Finally, we present a generalization of the above class which is still recognizable in polynomial time, and for which the Boolean minimization problem can be approximated within a constatnt factor.

1998

Minimum-size prime implicants of Boolean functions find application in many areas of Computer Science including, among others, Electronic Design Automation and Artificial Intelligence. The main purpose of this paper is to describe and evaluate two fundamentally different modeling and algorithmic solutions for the computation of minimum-size prime implicants. One is based on explicit search methods, and uses Integer Linear Programming models and algorithms, whereas the other is based on implicit techniques, and so it uses Binary Decision Diagrams. For the explicit approach we propose new dedicated ILP algorithms, specifically target at solving these types of problems. As shown by the experimental results, other well-known ILP algorithms are in general impractical for computing minimumsize prime implicants. Moreover, we experimentally evaluate the two proposed algorithmic strategies. 1 Introduction Given a propositional formula ϕ in Conjunctive Normal Form (CNF), denoting a boolean function f, the problem of computing a minimum-size assignment (in the number of literals) that satisfies f is referred to as the minimum-size prime implicant problem. Minimum-size prime implicants find several applications in Artificial Intelligence and in Electronic Design Automation (EDA). For example, the computation of minimum-size test patterns in testing is tightly related with the computation of the minimum size prime implicant of a Boolean function [15]. In Artificial Intelligence, the identification of minimum-size prime implicants (i.e. minimum-size satisfying assignments for propositional formulas) is commonly encountered in Automated Reasoning and Non-Monotonic Reasoning [7, 13]. In this paper we describe and empirically compare two algorithms for computing minimum-size prime implicants. One is an explicit algorithm, based on Integer Linear Programming (ILP), whereas the other utilizes implicit techniques, and so it is based on Binary Decision Diagrams (BDDs). The explicit approach merges in a single algorithm, bsolo, two commonly used search paradigms, namely branch and bound search and backtrack search, applying search-pruning techniques from both paradigms. The implicit approach, min-bdd, extends the prime implicant implicit representation of [3, 4]. An experimental comparison of the two approaches is provided in Section 5, which also provides empirical evidence validating the proposed ILP algorithm against state of the art ILP solvers [2]. The paper is organized as follows. A few brief definitions are provided in Section 2. The ILP model and algorithm for computing minimum-size prime implicants are described in Section 3. Section 4 is dedicated to the implicit BDD-based approach. The two strategies are compared in Section 5, and the paper concludes in Section 6.

Proceedings Ninth IEEE International Conference on Tools with Artificial Intelligence, 1997

The computation of prime implicants has several and significant applications in different areas, including Automated Reasoning, Non-Monotonic Reasoning, Electronic Design Automation, among others. In this paper we describe a new model and algorithm for computing minimum-size prime implicants of propositional formulas. The proposed approach is based on creating an integer linear program (ILP) formulation for computing the minimumsize prime implicant, which simplifies existing formulations. In addition, we introduce two new algorithms for solving ILPs, both of which are built on top of an algorithm for propositional satisfiability (SAT). Given the organization of the proposed SAT algorithm, the resulting ILP procedures implement powerful search pruning techniques, including a non-chronological backtracking search strategy, clause recording procedures and identification of necessary assignments. Experimental results, obtained on several benchmark examples, indicate that the proposed model and algorithms are significantly more efficient than other existing solutions.

Theoretical Computer Science, 1999

Consider a uniform distribution of r-CNF formulae (in Conjunctive Normal Form) with cn clauses, each with r distinct literals, over a set of n variables. A prime implicant Y of a formula @ is a consistent conjunction of literals which implies @ but ceases to imply when deprived of any one literal. The normalized length of 9 is the ratio of the number of its literals to the number of variables occurring in @. We show that for any E > 0 and for some range of values of c depending on r, almost every r-CNF formula: _ either is satisfiable and any one of its prime implicants has a normalized length at least equal to (c&(c)/( 1e-"')) -F and at most equal to (&(c)/( 1e-l')) +E, a&(c) and c&(c) being well-defined as functions of c, or is unsatisfiable. A first practical consequence is when testing the satisfiability of r-CNF formulae by procedures such as the well-known Davis, Putnam and Loveland Procedure, for almost every r-CNF formula, when it is satisfiable, the proportion of variables which must be assigned a value by such procedures, in order to find a solution, is at least equal to (ccL(c)/(l -e-")) -E.

Intelligent Systems Design and Applications, 2018

The algorithm to compute theory prime implicates, a generalization of prime implicates, in propositional logic has been suggested in [16]. In this paper we have extended that algorithm to compute theory prime implicates of a knowledge base X with respect to another knowledge base Y using [2], where Y is a propositional knowledge base and X |= Y , in modal system T and we have also proved its correctness. We have also proved that it is an equivalence preserving knowledge compilation and the size of theory prime implicates of X with respect to Y is less than the size of the prime implicates of X ∪ Y. We have also extended the query answering algorithm in modal logic.

International Journal of Computer Applications, 2013

K Maps are generally and ideally , thought to be simplest form for obtaining solution of Boolean equations.Cubical Representation of Boolean equations is an alternate pick to incur a solution, otherwise to be meted out with Truth Tables, Boolean Laws and different traits of Karnaugh Maps. Largest possible k-cubes that exist for a given function are equivalent to its prime implicants. A technique of minimization of Logic functions is tried to be achieved through cubical methods. The main purpose is to make aware and utilise the advantages of cubical techniques in minimization of Logic functions. All this is done with an aim to achieve minimal cost solution.

University of Oxford, PRG technical monographs, 1990

The page numbers of this electronic version do NOT correspond exactly to the page numbers of the printed version. Please refer to sections or subsections.

1996

We present a new algorithm (called TPI /BDD) for computing the theory prime implicates compilation of a knowledge base X. In contrast to many compilation algorithms, TPI /BDD does not require the prime implicates of Z to be generated. Since their number can easily be exponential in the size of X, TPI/BDD can save a lot of computing. Thanks to TPI/BDD, we can now conceive of compiling knowledge bases impossible to before.