Binary relations, matrices and inference developments

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

Factorizing Boolean matrices using formal concepts and iterative usage of essential entries

Very recently, many fixed point results have been introduced in the setting of graphical metric spaces. Due to their intimate links, such works also deal with metric spaces endowed with partial orders. As the reachability relationship in any directed graph (containing all cycles) is a reflexive transitive relation (that is, a preorder), but it is not necessarily a partial order, results on graphical metric spaces are independent from statements on ordered metric spaces. The main aim of this paper is to show that fixed point theorems in the setting of graphical metric spaces can be directly deduced from their corresponding results on measurable spaces endowed with a binary relation. Finally, we also describe the main advantages of involving this last class of spaces.

: Many problems in Artificial Intelligence are intractable due to the exponential growth of the solution space with problem size. Often these problems can benefit from heuristic search or forward-checking techniques which attempt to prune the search space down to a manageable size before or during the actual search procedure. Many interesting search problems can be formulated as consistent labeling problem in which initial problem information is given in the form of a set of binary constraint, for which Boolean matrices are a natural data representation. In this paper optical implementations of Boolean matrix operations are proposed for manipulating the constraint matrices to perform forward-checking and thereby increase the search efficiency. The high degree of parallelism afforded by using optical techniques and the relatively low accuracy requirements of Boolean matrix operations suggest that optical techniques are well matched to this problem.

Boolean matrix factorisation aims to decompose a binary data matrix into an approximate Boolean product of two low rank, binary matrices: one containing meaningful patterns, the other quantifying how the observations can be expressed as a combination of these patterns. We introduce the OrMachine, a probabilistic generative model for Boolean matrix factorisation and derive a Metropolised Gibbs sampler that facilitates efficient parallel posterior inference. On real world and simulated data, our method outperforms all currently existing approaches for Boolean matrix factorisation and completion. This is the first method to provide full posterior inference for Boolean Matrix factorisation which is relevant in applications, e.g. for controlling false positive rates in collaborative filtering and, crucially, improves the interpretability of the inferred patterns. The proposed algorithm scales to large datasets as we demonstrate by analysing single cell gene expression data in 1.3 million mouse brain cells across 11 thousand genes on commodity hardware.

CLIQUE is the generic name for a suite of programs centered around the extraction of maximal complete graphs from a binary symmetric matrix representing a continuous similarity/difference or a binary relation among entities. Its features include the following: derivation of a graph from measures either of similarity or difference, the possibility of employing tests of statistical significance in deriving the input graph or in subsequent clusterings based on the cliques, output allowing investigation of all cliques containing any single element, or permitting a two-stage analysis in which the elements contained in cliques on the second pass correspond to cliques extracted on the first pass. Applicability extends to a considerable range of input data.

Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only finitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is "pointless". Relation algebras were introduced into temporal reasoning by Allen (1983, Communications of the ACM 26(1), 832--843) and into spatial reasoning by Egenhofer and Sharma (1992, Fifth International Symposium on Spatial Data Handling, Charleston, SC). The calculus of relation algebras is also well suited to handle binary constraints as demonstrated e.g. by Ladkin and Maddux (1994, Journal of the ACM 41(3), 435--469). In the present paper I will give an introduction to relation algebras, and an overview of their role in qualitative temporal and spatial reasoning.

This chapter presents the terminology and concepts of graph theory , and describes basic matrix operations that are used in social network analysis. Both graph theory and matrix operations have served as the foundations of many concepts in the analysis of social networks (Hage and Harary 1983; Harary, Norman, and Cartwright 1965). In this chapter, the notation presented in Chapter 3 is used, and more concepts and ideas from graph theory are described and illustrated with examples. The topics covered in this chapter are important for the methods discussed in the remaining chapters of the book, but they are especially important in Chapter 5 (Centrality, Prestige, and Related Actor and Group Measures), Chapter 6 (Structural Balance, Clusterability, and Transitivity), Chapter 7 (Cohesive Subgroups), and Chapter 8 (Affiliations, Co-memberships, and Overlapping Subgroups). We start this chapter with a discussion of some reasons why graph theory and graph theoretic concepts are important for social network analysis. We then define a graph for representing a nondirectional relation. We begin with simple concepts, and progressively build on these to achieve more complicated, and more interesting, graph theoretic concepts. We then define and discuss directed graphs, for representing directional relations. Again, we begin with simple directed graph concepts and build to more complicated ideas. Following this, we discuss signed and valued graphs. We then define and discuss hypergraphs, which are used to represent affiliation networks.

The plurivalent logics considered in Graham Priest's recent paper of that name can be thought of as logics determined by matrices (in the 'logical matrix' sense) whose underlying algebras are power algebras (a.k.a. complex algebras, or 'globals'), where the power algebra of a given algebra has as elements subsets of the universe of the given algebra, and the power matrix of a given matrix has has the power algebra of the latter's algebra as its underlying algebra, with its designated elements being selected in a natural way on the basis of those of the given matrix. The present discussion stresses the continuity of Priest's work on the question of which matrices determine consequence relations (for propositional logics) which remain un-affected on passage to the consequence relation determined by the power matrix of the given matrix with the corresponding (long-settled) question in equational logic as to which identities holding in an algebra continue to hold in its power algebra. Both questions are sensitive to a decision as to whether or not to include the empty set as an element of the power algebra, and our main focus will be on the contrast, when it is included, between the power matrix semantics (derived from the two-element Boolean matrix) and the four-valued Dunn-Belnap semantics for first-degree entailment (à la Anderson and Belnap) in terms of sets of classical values (subsets of {T, F}, that is), in which the empty set figures in a somewhat different way, as Priest had remarked his 1984 study, 'Hyper-contradictions', in which what we are calling the power matrix construction first appeared.

The running time and memory requirement of a parallel program with dynamic, lightweight threads depends heavily on the underlying thread scheduler. In this paper we present a new asynchronous, space-efficient scheduling algorithm for shared memory machines that combines the low scheduling overheads and good locality of work stealing with the low space requirements of depth-first schedulers. For a nested-parallel program with depth D and serial space requirement S 1 , we show that the expected space requirement is S 1 + O(K⋅ p⋅ D) on p processors. Here, K is a user-adjustable runtime parameter, which provides a tradeoff between running time and space requirement. Our algorithm achieves good locality and low scheduling overheads by automatically increasing the granularity of the work scheduled on each processor. We have implemented the new scheduling algorithm in the context of a native, user-level implementation of Posix standard threads or Pthreads, and evaluated its performance using a set of C-based benchmarks that have dynamic or irregular parallelism. We compare the performance of our scheduler with that of two previous schedulers: the thread library's original scheduler (which uses a FIFO queue), and a provably space-efficient depth-first scheduler. At a fine thread granularity, our scheduler outperforms both these previous schedulers, but requires marginally more memory than the depth-first scheduler. We also present simulation results on synthetic benchmarks to compare our scheduler with space-efficient versions of both a work-stealing scheduler and a depth-first scheduler. The results indicate that unlike these previous approaches, the new algorithm covers a range of scheduling granularities and space requirements, and allows the user to trade the space requirement of a program with the scheduling granularity.

The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted m-edge and n-node graphs require Ω(min{nω, mn}) time (for 2 ≤ ω > 2.373). In this paper, we drastically improve these runtimes as follows: • Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in O(m) time computes an O(n)-multiplicative approximation of the girth as well as an O(n)-multiplicative roundtrip spanner with O(n) edges with high probability (w.h.p). • Nearly Tight Additive Approximations: For unweighted graphs and any a ∈ (0, 1) we give an algorithm that in O(mn1−a) time computes an 0(na)-additive approximation of the girth, w.h.p. We show that the run-time of our algorithm cannot be significantly improved without a breakthrough in combinatorial boolean matrix multiplication. We also show that if the girth is 0(na), then the same guarantee can be achieved via a deterministic algorithm. Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than Ω(mn) time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.

Buffer insertion and wire sizing are critical in deep submicron VLSI design. This paper studies the problem of constructing routing trees with simultaneous buffer insertion and wire sizing in the presence of routing and buffer obstacles. No previous algorithms consider all these factors simultaneously. Previous dynamic programming based algorithm is first extended to solve the problem. However, with the size of routing graph increasing and with wire sizing taken into account, the time and space requirement increases enormously. Then a new approach is proposed to formulate the problem as a series of graph problems. The routing tree solution is obtained by finding shortest paths in a series of graphs. In the new approach, wire sizing can be handled almost without any additional time and space requirement, Moreover, the time and space requirement is only polynomial in terms of the size of routing graph. Our algorithm differs from traditional dynamic programming, and is capable of addressing the problem of inverter insertion and sink polarity. Both theoretical and experimental results show that the graph-based algorithm outperforms the DP-based algorithm by a large margin. We also propose a hierarchical approach to construct routing tree for a large number of sinks.

Binary matrixes (matrixes having only 0 and 1 elements) occur in many theoretical and practical areas (e.g. in the theory of graphs and relations, grammars, coding, etc.) therefore the binary matrix operations have high importance in these topics. It is an obvious idea to use cellular processor consisting of micro-cells for the execution of these operations. The present paper gives a powerful cellular algorithm for the multiplication of two binary matrixes, taking into account the cellprocessor architecture developed by T. Legendi. Moreover, on the basis of this solution cellular algorithms are given for coding and for the computation of the m-th power of a binary matrix.

Boundary value problems (BVP) arise in many mathematical model of frontier science. In short, a BVP is a second order ordinary differential equation with boundary conditions prescribed at two distinct points. Usually, we prescribe three types of boundary conditions: Dirichlet, Neumann and Mixed. Among them mixed boundary conditions is little difficult to approach. One of the method to solve the BVP with mixed boundary conditions is the shooting method. In this method, we first convert the BVP into an initial value problems (IVP) and then solve it. We then check whether the solution of IVP is a solution of the BVP. There are standard ways of solving IVPs. Recently, some alternate methods are also used to solve them. For example, several authors use Haar wavelet. Frames are considered as an extension to wavelet. In this paper, we construct operational matrices from a frame and then apply them to solve the IVP which is obtained from the given BVP by shooting method. This method is simple and easy to use because we construct the operational matrix once, store it and then apply it any times we want. Moreover, we also show that the given operational matrix method converges quadratically.