scholarly journals Minor complexity of discrete functions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Slavcho Shtrakov
Keyword(s):  
Data Structure ◽  
Combinatorial Problems ◽  
Complexity Measures ◽  
Essential Arity ◽  
Discrete Functions

In this paper we study a class of complexity measures, induced by a new data structure for representing k-valued functions (operations), called minor decision diagram. When assigning values to some variables in a function the resulting functions are called subfunctions, and when identifying some variables the resulting functions are called minors. The sets of essential variables in subfunctions of f are called separable in f.We examine the maximal separable subsets of variables and their conjugates, introduced in the paper, proving that each such set has at least one conjugate. The essential arity gap gap(f) of the function f is the minimal number of essential variables in f which become fictive when identifying distinct essential variables in f. We also investigate separable sets of variables in functions with non-trivial arity gap. This allows us to solve several important algebraic, computational and combinatorial problems about the finite-valued functions.

On the upper bounds for complexities of discrete functions

2022 ◽  
Author(s):  
Slavcho Shtrakov
Keyword(s):  
Data Structures ◽  
Upper Bounds ◽  
Complexity Measures ◽  
Discrete Functions

In this paper, we study two classes of complexity measures induced by new data structures (abstract reduction systems) for representing [Formula: see text]-valued functions (operations), namely subfunction and minor reductions. When assigning values to some variables in a function, the resulting functions are called subfunctions, and when identifying some variables, the resulting functions are called minors. The number of the distinct objects obtained under these reductions of a function [Formula: see text] is a well-defined measure of complexity denoted by [Formula: see text] and [Formula: see text], respectively. We examine the maximums of these complexities and construct functions which reach these upper bounds.

scholarly journals Compact-MDD: Efficiently Filtering (s)MDD Constraints with Reversible Sparse Bit-sets

2018 ◽  
Author(s):  
Hélène Verhaeghe ◽  
Christophe Lecoutre ◽  
Pierre Schaus
Keyword(s):  
Data Structure ◽  
Practical Interest ◽  
Combinatorial Problems ◽  
Experimental Results ◽  
Decision Diagrams ◽  
Central Layer ◽  
Filtering Algorithm ◽  
Related Data ◽  
Bitwise Operations

Multi-Valued Decision Diagrams (MDDs) are instrumental in modeling combinatorial problems with Constraint Programming.In this paper, we propose a related data structure called sMDD (semi-MDD) where the central layer of the diagrams is non-deterministic.We show that it is easy and efficient to transform any table (set of tuples) into an sMDD.We also introduce a new filtering algorithm, called Compact-MDD, which is based on bitwise operations, and can be applied to both MDDs and sMDDs.Our experimental results show the practical interest of our approach, both in terms of compression and filtering speed.

scholarly journals Genetic algorithm for binary and functional decision diagrams optimization

2018 ◽  
Vol 31 (2) ◽  
pp. 169-187
Author(s):  
Stojkovic Suzana ◽  
Velickovic Darko ◽  
Moraga Claudio
Keyword(s):  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Data Structure ◽  
Optimization Problems ◽  
Binary Decision Diagrams ◽  
Size Optimization ◽  
Decision Diagrams ◽  
Crossover Operator ◽  
Binary Decision ◽  
Discrete Functions

Decision diagrams (DD) are a widely used data structure for discrete functions representation. The major problem in DD-based applications is the DD size minimization (reduction of the number of nodes), because their size is dependent on the variables order. Genetic algorithms are often used in different optimization problems including the DD size optimization. In this paper, we apply the genetic algorithm to minimize the size of both Binary Decision Diagrams (BDDs) and Functional Decision Diagrams (FDDs). In both cases, in the proposed algorithm, a Bottom-Up Partially Matched Crossover (BU-PMX) is used as the crossover operator. In the case of BDDs, mutation is done in the standard way by variables exchanging. In the case of FDDs, the mutation by changing the polarity of variables is additionally used. Experimental results of optimization of the BDDs and FDDs of the set of benchmark functions are also presented.

On the Computational Complexity of Finite Operations

2016 ◽  
Vol 27 (01) ◽  
pp. 15-38 ◽  
Author(s):  
Slavcho Shtrakov ◽  
Ivo Damyanov
Keyword(s):  
Computational Complexity ◽  
Combinatorial Problems ◽  
Affine Group ◽  
Equivalence Relations ◽  
Transformation Groups ◽  
Finite Function ◽  
Important Measure ◽  
Discrete Functions

The essential variables in a finite function f are defined as variables which occur in f and weigh with the values of that function. The number of essential variables is an important measure of complexity for discrete functions. When replacing some variables in a function with constants the resulting functions are called subfunctions, and when replacing all essential variables in a function with constants we obtain an implementation of this function. Such an implementation corresponds with a path in an ordered decision diagram (ODD) of the function which connects the root with a leaf of the diagram. The sets of essential variables in subfunctions of f are called separable in f. In this paper we study several properties of separable sets of variables in functions which directly affect the number of implementations and subfunctions in these functions. We define equivalence relations which classify the functions of k-valued logic into classes with the same number of: (i) implementations; (ii) subfunctions; and (iii) separable sets. These relations induce three transformation groups which are compared with the lattice of all subgroups of restricted affine group (RAG). This allows us to solve several important computational and combinatorial problems.

Complexity Measures of Brain Electrophysiological Activity

2010 ◽  
Vol 24 (2) ◽  
pp. 131-135 ◽  
Author(s):  
Włodzimierz Klonowski ◽  
Pawel Stepien ◽  
Robert Stepien
Keyword(s):  
Brain Activity ◽  
Deterministic Chaos ◽  
Epileptic Seizures ◽  
Eeg Signal ◽  
Eeg Signals ◽  
Complexity Measures ◽  
Electrophysiological Activity ◽  
Signal Complexity ◽  
Physiological Sleep ◽  
The Brain

Over 20 years ago, Watt and Hameroff (1987 ) suggested that consciousness may be described as a manifestation of deterministic chaos in the brain/mind. To analyze EEG-signal complexity, we used Higuchi’s fractal dimension in time domain and symbolic analysis methods. Our results of analysis of EEG-signals under anesthesia, during physiological sleep, and during epileptic seizures lead to a conclusion similar to that of Watt and Hameroff: Brain activity, measured by complexity of the EEG-signal, diminishes (becomes less chaotic) when consciousness is being “switched off”. So, consciousness may be described as a manifestation of deterministic chaos in the brain/mind.

Distributed Modeling Of Building Systems Through Cyber-Physical Integration

2019 ◽  
pp. 12-17
Keyword(s):  
Data Structure ◽  
Operating Conditions ◽  
Physical Models ◽  
Distributed Model ◽  
Physical Processes ◽  
Distributed Modeling ◽  
Distributed Models ◽  
Operating Modes ◽  
Building Systems ◽  
Important Practical Application

This article describes the proposed approaches to creating distributed models that can, with given accuracy under given restrictions, replace classical physical models for construction objects. The ability to implement the proposed approaches is a consequence of the cyber-physical integration of building systems. The principles of forming the data structure of designed objects and distributed models, which make it possible to uniquely identify the elements and increase the level of detail of such a model, are presented. The data structure diagram of distributed modeling includes, among other things, the level of formation and transmission of signals about physical processes inside cyber-physical building systems. An enlarged algorithm for creating the structure of the distributed model which describes the process of developing a data structure, formalizing requirements for the parameters of a design object and its operating modes (including normal operating conditions and extreme conditions, including natural disasters) and selecting objects for a complete group that provides distributed modeling is presented. The article formulates the main approaches to the implementation of an important practical application of the cyber-physical integration of building systems - the possibility of forming distributed physical models of designed construction objects and the directions of further research are outlined.

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