Essential Arity Gap of Boolean Functions

In this paper we investigate the Boolean functions with maximum essential arity gap. Additionally we propose a simpler proof of an important theorem proved by M. Couceiro and E. Lehtonen in [3]. They use Zhegalkin’s polynomials as normal forms for Boolean functions and describe the functions with essential arity gap equals 2. We use to instead Full Conjunctive Normal Forms of these polynomials which allows us to simplify the proofs and to obtain several combinatorial results concerning the Boolean functions with a given arity gap. The Full Conjunctive Normal Forms are also sum of conjunctions, in which all variables occur.

Decomposition of Boolean Functions – Recognizing a Good Solution by Traces

The problem of sequent two-block decomposition of a Boolean function is regarded in case when a good solution does exist. The problem consists mainly in finding an appropriate weak partition on the set of arguments of the considered Boolean function, which should be decomposable at that partition. A new fast heuristic combinatorial algorithm is offered for solving this task. At first the randomized search for traces of such a partition is fulfilled. The recognized traces are represented by some "triads" - the simplest weak partitions corresponding to non-trivial decompositions. After that the whole sought-for partition is restored from the discovered trace by building a track initialized by the trace and leading to the solution. The results of computer experiments testify the high practical efficiency of the algorithm.

Implementation of a Heuristic Method of Decomposition of Partial Boolean Functions

An original heuristic algorithm of sequential two-block decomposition of partial Boolean functions is researched. The key combinatorial task is considered: finding of suitable partition on the set of arguments, i. e. such one, on which the function is separable. The search for suitable partition is essentially accelerated by preliminary detection of its traces. Within the framework of the experimental system the efficiency of the algorithm is evaluated, the boundaries of its practical application are determined.

On some Properties of Boolean Functions and Their Binary Decision Diagrams

Иво Й. Дамянов - Манипулирането на булеви функции е основнo за теоретичната информатика, в това число логическата оптимизация, валидирането и синтеза на схеми. В тази статия се разглеждат някои първоначални резултати относно връзката между граф-базираното представяне на булевите функции и свойствата на техните променливи.

The Boundary Descriptors of the n-dimensional Unit Cube Subset Partitioning

* The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be

Logic Based Pattern Recognition - Ontology Content (2)

Logic based Pattern Recognition extends the well known similarity models, where the distance measure is the base instrument for recognition. Initial part (1) of current publication in iTECH-06 reduces the logic based recognition models to the reduced disjunctive normal forms of partially defined Boolean functions. This step appears as a way to alternative pattern recognition instruments through combining metric and logic hypotheses and features, leading to studies of logic forms, hypotheses, hierarchies of hypotheses and effective algorithmic solutions. Current part (2) provides probabilistic conclusions on effective recognition by logic means in a model environment of binary attributes.

Logic Based Pattern Recognition - Ontology Content (1)

* The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be

The Knowledge: Its Presentation and Role in Recognition Systems

The concept of knowledge is the central one used when solving the various problems of data mining and pattern recognition in finite spaces of Boolean or multi-valued attributes. A special form of knowledge representation, called implicative regularities, is proposed for applying in two powerful tools of modern logic: the inductive inference and the deductive inference. The first one is used for extracting the knowledge from the data. The second is applied when the knowledge is used for calculation of the goal attribute values. A set of efficient algorithms was developed for that, dealing with Boolean functions and finite predicates represented by logical vectors and matrices.

Using SAT for Combinational Implementation Checking

The problem of checking whether a system of incompletely specified Boolean functions is implemented by the given combinational circuit is considered. The task is reduced to testing out if two given logical descriptions are equivalent on the domain of one of them having functional indeterminacy. We present a novel SAT-based verification method that is used for testing whether the given circuit satisfies all the conditions represented by the system of incompletely specified Boolean functions.

Generating More Boundary Elements of Subset Projections

Composition problem is considered for partition constrained vertex subsets of n dimensional unit cube E^n . Generating numerical characteristics of E^n subsets partitions is considered by means of the same characteristics in 1 − n dimensional unit cube, and construction of corresponding subsets is given for a special particular case. Using pairs of lower layer characteristic vectors for E^(1-n) more characteristic vectors for E^n are composed which are boundary from one side, and which take part in practical recognition of validness of a given candidate vector of partitions.

Finding the Relationship between a Search Algorithm and a Class of Functions on Discrete Space by Exhaustive Search

* The work is supported by RFBR, grant 04-01-00858-a.

On some Generalizations of a Class of Discrete Functions

In this paper we examine discrete functions that depend on their variables in a particular way, namely the H-functions. The results obtained in this work make the “construction” of these functions possible. H-functions are generalized, as well as their matrix representation by Latin hypercubes.

A Recession Notion for a Class of Monotone Bivariate Functions

Using monotone bifunctions, we introduce a recession concept for general equilibrium problems relying on a variational convergence notion. The interesting purpose is to extend some results of P. L. Lions on variational problems. In the process we generalize some results by H. Brezis and H. Attouch relative to the convergence of the resolvents associated with maximal monotone operators.

Alternative Characterization of the Class k-UCV and Related Classes of Univalent Functions

In this paper an alternative characterization of the class of functions called k -uniformly convex is found. Various relations concerning connections with other classes of univalent functions are given. Moreover a new class of univalent functions, analogous to the ’Mocanu class’ of functions, is introduced. Some results concerning this class are derived.

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35