Parallel Solution of Covering Problems Super-Linear Speedup on a Small Set of Cores

This paper aims at better possibilities to solveproblems of exponential complexity. Our special focus is thecombination of the computational power of four cores of astandard PC with better approaches in the application domain.As the main example we selected the unate covering problemwhich must be solved, among others, in the process of circuitsynthesis and for graph-covering (domination) problems.We introduce into the wide field of problems that can besolved using Boolean models. We explain the models and theclassic solutions, and discuss the results of a selected model byusing a benchmark set. Subsequently we study sources of parallelismin the application domain and explore improvementsgiven by the parallel utilization of the available four cores ofa PC. Starting with a uniform splitting of the problem, wesuggest improvements by means of an adaptive division andan intelligent master. Our experimental results confirm thatthe combination of improvements of the application modelsand of the algorithmic domain leads to a remarkable speedupand an overall improvement factor of more than 35 millionsin comparison with the improved basic approach

COMPACT XOR-BI-DECOMPOSITION FOR LATTICES OF BOOLEAN FUNCTIONS

Bi-Decomposition is a powerful approach for the synthesis of multi-level combinational circuits because it utilizes the properties of the given functions to find small circuits, with low power consumption and low delay. Compact bi-decompositions restrict the variables in the support of the decomposition functions as much as possible. Methods to find compact AND-, OR-, or XOR-bi-decompositions for a given completely specified function are well known.Lattices of Boolean Functions significantly increase the possibilities to synthesize a minimal circuit. However, so far only methods to find compact AND- or OR-bidecompositions for lattices of Boolean functions are known. This gap, i.e., a method to find a compact XOR-bi-decomposition for a lattice of Boolean functions, has been closed by the approach suggested in this paper

BOOLEAN DIFFERENTIAL EQUATIONS - A COMMON MODEL FOR CLASSES, LATTICES, AND ARBITRARY SETS OF BOOLEAN FUNCTIONS

The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean valuesor Boolean functions can be described. A Boolean Differential Equation (BDE)is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDE, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential Equations.In order to reach this aim, we give a short introduction into the BDC, emphasizethe general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDE that is restricted to all vectorial derivatives of f(x) and optionally the Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution.The basic operations of XBOOLE are sufficient to solve BDEs. We demonstratehow a XBOOLE-problem program (PRP) of the freely available XBOOLE-Monitorquickly solves some BDEs

Locally monotone Boolean and pseudo-Boolean functions

We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are shown to be tightly related to lattice counterparts of classical partial derivatives via the notion of permutable derivatives. More precisely, p-locally monotone functions are shown to have p-permutable lattice derivatives and, in the case of symmetric functions, these two notions coincide. We provide further results relating these two notions, and present a classification of p-locally monotone functions, as well as of functions having p-permutable derivatives, in terms of certain forbidden "sections", i.e., functions which can be obtained by substituting constants for variables. This description is made explicit in the special case when p=2

Logic functions and equations: examples and exercises

With a free, downloadable software package available to help solve the exercises, this book focuses on practical and relevant problems that arise in the field of binary logics, with its two main applications - digital circuit design, and propositional logics

Extremely Complex 4-Colored Rectangle-Free Grids: Solution of Open Multiple-Valued Problems

This paper aims at the rectangle-free coloring of grids using four colors. It has been proven in a well developed theory that there is an upper bound of rectangle-free 4-colorable grids as well as a lower bound of grids for which no rectangle-free color pattern of four colors exist. Between these tight bounds the grids of the size 17×17, 17×18, 18×17, and 18×18 are located for which it is not known until now whether a rectangle-free coloring by four colors exists. We present in this paper an approach that solves all these open problems. From another point of view this paper aims at the solution of a multiple-valued problem having an extremely high complexity. There are 1.16798 ∗ 10 195 different grids of four colors. It must be detected whether at least one of this hardly imaginable large number of patterns satisfies strong additional conditions. In order to solve this highly complex problem, several approaches were taken into account to find out properties of the problem which finally allowed us to calculate the solution

The Solution of SAT Problems Using Ternary Vectors and Parallel Processing

This paper will show a new approach to the solution of SAT-problems. It has been based on the isomorphism between the Boolean algebras of finite sets and the Boolean algebras of logic functions depending on a finite number of binary variables. Ternary vectors are the main data structure representing sets of Boolean vectors. The respective set operations (mainly the complement and the intersection) can be executed in a bit-parallel way (64 bits at present), but additionally also on different processors working in parallel. Even a hierarchy of processors, a small set of processor cores of a single CPU, and the huge number of cores of the GPU has been taken into consideration. There is no need for any search algorithms. The approach always finds all solutions of the problem without consideration of special cases (such us no solution, one solution, all solutions). It also allows to include problem-relevant knowledge into the problem-solving process at an early point of time. Very often it is possible to use ternary vectors directly for the modeling of a problem. Some examples are used to illustrate the efficiency of this approach (Sudoku, Queen's problems on the chessboard, node bases in graphs, graph-coloring problems, Hamiltonian and Eulerian paths etc.)