Method for analytical description and modeling of the working space of a manipulation robot

This paper reports a method, built in the form of a logic function, for describing the working spaces of manipulation robots analytically. A working space is defined as a work area or reachable area by a manipulation robot. An example of describing the working space of a manipulation robot with seven rotational degrees of mobility has been considered. Technological processes in robotic industries can be associated with the positioning of the grip, at the required points, in the predefined coordinates, or with the execution of the movement of a working body along the predefined trajectories, which can also be determined using the required points in the predefined coordinates. A necessary condition for a manipulation robot to execute a specified process is that all the required positioning points should be within a working space. To solve this task, a method is proposed that involves the analysis of the kinematic scheme of a manipulation robot in order to acquire a graphic image of the working space to identify boundary surfaces, as well as identify additional surfaces. The working space is limited by a set of boundary surfaces where additional surfaces are needed to highlight parts of the working space. Specifying each surface as a logic function, the working space is described piece by piece. Next, the resulting parts are combined with a logical expression, which is a disjunctive normal form of logic functions, which is an analytical description of the working space. The correspondence of the obtained analytical description to the original graphic image of working space is verified by simulating the disjunctive normal form of logic functions using MATLAB (USA).

An Empirical Investigation Into Deep and Shallow Rule Learning

Inductive rule learning is arguably among the most traditional paradigms in machine learning. Although we have seen considerable progress over the years in learning rule-based theories, all state-of-the-art learners still learn descriptions that directly relate the input features to the target concept. In the simplest case, concept learning, this is a disjunctive normal form (DNF) description of the positive class. While it is clear that this is sufficient from a logical point of view because every logical expression can be reduced to an equivalent DNF expression, it could nevertheless be the case that more structured representations, which form deep theories by forming intermediate concepts, could be easier to learn, in very much the same way as deep neural networks are able to outperform shallow networks, even though the latter are also universal function approximators. However, there are several non-trivial obstacles that need to be overcome before a sufficiently powerful deep rule learning algorithm could be developed and be compared to the state-of-the-art in inductive rule learning. In this paper, we therefore take a different approach: we empirically compare deep and shallow rule sets that have been optimized with a uniform general mini-batch based optimization algorithm. In our experiments on both artificial and real-world benchmark data, deep rule networks outperformed their shallow counterparts, which we take as an indication that it is worth-while to devote more efforts to learning deep rule structures from data.

Logical minimization for combinatorial structure in FPGA

The paper describes the research results of application efficiency of minimization programs of functional descriptions of combinatorial logic blocks, which are included in digital devices projects that are implemented in FPGA. Programs are designed for shared and separated function minimization in a disjunctive normal form (DNF) class and minimization of multilevel representations of fully defined Boolean functions based on Shannon expansion with finding equal and inverse cofactors. The graphical form of such representations is widely known as binary decision diagrams (BDD). For technological mapping the program of "enlargement" of obtained Shannon expansion formulas was applied in a way that each of them depends on a limited number of k input variables and can be implemented on one LUT-k – a programmable unit of FPGA with k input variables. It is shown that a preliminary logic minimization, which is performed on the domestic programs, allows improving design results of foreign CAD systems such as Leonardo Spectrum (Mentor Graphics), ISE (Integrated System Environment) Design Suite and Vivado (Xilinx). The experiments were performed for FPGA families’ Virtex-II PRO, Virtex-5 and Artix-7 (Xilinx) on standard threads of industrial examples, which define both DNF systems of Boolean functions and systems represented as interconnected logical equations.

Unsupervised DNF Blocking for Efficient Linking of Knowledge Graphs and Tables

Entity Resolution (ER) is the problem of identifying co-referent entity pairs across datasets, including knowledge graphs (KGs). ER is an important prerequisite in many applied KG search and analytics pipelines, with a typical workflow comprising two steps. In the first ’blocking’ step, entities are mapped to blocks. Blocking is necessary for preempting comparing all possible pairs of entities, as (in the second ‘similarity’ step) only entities within blocks are paired and compared, allowing for significant computational savings with a minimal loss of performance. Unfortunately, learning a blocking scheme in an unsupervised fashion is a non-trivial problem, and it has not been properly explored for heterogeneous, semi-structured datasets, such as are prevalent in industrial and Web applications. This article presents an unsupervised algorithmic pipeline for learning Disjunctive Normal Form (DNF) blocking schemes on KGs, as well as structurally heterogeneous tables that may not share a common schema. We evaluate the approach on six real-world dataset pairs, and show that it is competitive with supervised and semi-supervised baselines.

AIGEN: Random Generation of Symbolic Transition Systems

AIGEN is an open source tool for the generation of transition systems in a symbolic representation. To ensure diversity, it employs a uniform random sampling over the space of all Boolean functions with a given number of variables. AIGEN relies on reduced ordered binary decision diagrams (ROBDDs) and canonical disjunctive normal form (CDNF) as canonical representations that allow us to enumerate Boolean functions, in the former case with an encoding that is inspired by data structures used to implement ROBDDs. Several parameters allow the user to restrict generation to Boolean functions or transition systems with certain properties, which are then output in AIGER format. We report on the use of AIGEN to generate random benchmark problems for the reactive synthesis competition SYNTCOMP 2019, and present a comparison of the two encodings with respect to time and memory efficiency in practice.

V-МІНІМІЗАЦІЯ БУЛЕВИХ ФУНКЦІЙ ЗА МАТРИЦЕЮ ВІДСТАНЕЙ ТА ЗВЕДЕННЯМ ДО ЗАДАЧІ МАТЕМАТИЧНОГО ПРОГРАМУВАННЯ

The further development of the analytical method for minimizing Boolean functions (BF) in the class of disjunctive normal forms by the numbers of sets of argument values, called -minimization, is proposed. Such an approach allows to reduce the process of minimizing the BF to the use of an exclusively analytical description of all its steps using functions of the number of the set of the BF argument values. In general, the idea is to develop such a toolkit that allows at all stages of the minimization process to operate only with numerical objects – Boolean vectors, not involve a visual analysis of the results of the intermediate steps, and not resort to attracting letter objects. In the future, the implementation of this idea at the software level for computers is assumed. The article consists of two parts. The first part is about calculating the Hamming distance between two Boolean vectors. Its equality to unity is a necessary and sufficient condition for gluing together the constituents of unity (zero) or elementary conjunctions (disjunctions). The Hamming distance between two sets of variable values was calculated using the Zhegalkin operation (inversion of equivalence), the distance matrix was compiled, and the abbreviated disjunctive normal form (DNF) was determined from it. From the abbreviated DNF deadlock form, and among them the minimal ones, were obtained by sorting out subsets of the set of units of the distance matrix. Deadlock forms are found by the units to which the implicants, covering all units of the set of BF values, correspond. The final result is presented, of course, in letter form. The second part of the article is devoted to the formulation and solution of the problem of minimizing BF as a problem of linear integer mathematical programming. The goal function is the arithmetic sum of all implicants of the abbreviated DNF. The system of restrictions is based on the fact that among the set of simple implicants, it is necessary to choose the smallest number of them so that all constituents of the unit of the original BF are covered. Restrictions are based on how much the constituent of a unit is covered by one or another implicant, and they are written in the form of linear inequalities through the symbol "greater than or equal to" with the right-hand sides equal to unity. The solution of the mathematical programming problem requires the use of a distance matrix. Examples of -minimization of BF are given.

Switching function based on hypergraphs with algorithm and python programming

According to Boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions (it can also be described as an OR of AND’s). For each table an arbitrary T.B.T is given (total binary truth table) Boolean expression can be written as a disjunctive normal form. This paper considers a notation of a T.B.T, introduces a new concept of the hypergraphable Boolean functions and the Boolean functionable hypergraphs with respect to any given T.B.T. This study defines a notation of unitors set on switching functions and proves that every T.B.T corresponds to a minimum Boolean expression via unitors set and presents some conditions on a T.B.T to obtain a minimum irreducible Boolean expression from switching functions. Indeed, we generate a switching function in different way via the concept of hypergraphs in terms of Boolean expression in such a way that it has a minimum irreducible Boolean expression, for every given T.B.T. Finally, an algorithm is presented. Therefore, a Python programming(with complete and original codes) such that for any given T.B.T, introduces a minimum irreducible switching expression.

A heuristic method for bi-decomposition of partial Boolean functions

The problem of decomposition of a Boolean function is to represent a given Boolean function in the form of a superposition of some Boolean functions whose number of arguments are less than the number of given function. The bi-decomposition represents a given function as a logic algebra operation, which is also given, over two Boolean functions. The task is reduced to specification of those two functions. A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0. The complete graph of orthogonality of Boolean vectors that constitute the definitional domain of the given function is considered. In the graph, the edges are picked out, any of which has its ends corresponding the elements of Boolean space where the given function has different values. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs (bicliques). Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is a pair of certain parameters of assigned DNF. According to each biclique of obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF. A technique for constructing the mentioned cover for two kinds of output function is described.

Canalyzing minors of Boolean functions

Canalyzing functions are a special type of Boolean functions. For a canalyzing function, there is at least one argument, in which taking a certain value can determine the value of the function. Identification of variables can also shrink the resulting function into constant or function depending on one variable. In this paper, we discuss a particular disjunctive normal form for representation of Boolean function with its identification minors. Then an upper bound of the number of canalyzing minors is obtained. Finally, the number of canalyzing minors for Boolean functions with five essential variables is discussed.

De novo design of protein logic gates

The design of modular protein logic for regulating protein function at the posttranscriptional level is a challenge for synthetic biology. Here, we describe the design of two-input AND, OR, NAND, NOR, XNOR, and NOT gates built from de novo–designed proteins. These gates regulate the association of arbitrary protein units ranging from split enzymes to transcriptional machinery in vitro, in yeast and in primary human T cells, where they control the expression of the TIM3 gene related to T cell exhaustion. Designed binding interaction cooperativity, confirmed by native mass spectrometry, makes the gates largely insensitive to stoichiometric imbalances in the inputs, and the modularity of the approach enables ready extension to three-input OR, AND, and disjunctive normal form gates. The modularity and cooperativity of the control elements, coupled with the ability to de novo design an essentially unlimited number of protein components, should enable the design of sophisticated posttranslational control logic over a wide range of biological functions.