Showing papers on "Disjunctive normal form published in 2016"

Approximation Algorithms for Stochastic Submodular Set Cover with Applications to Boolean Function Evaluation and Min-Knapsack

Abstract: We present a new approximation algorithm for the stochastic submodular set cover (SSSC) problem called adaptive dual greedy. We use this algorithm to obtain a 3-approximation algorithm solving the stochastic Boolean function evaluation (SBFE) problem for linear threshold formulas (LTFs). We also obtain a 3-approximation algorithm for the closely related stochastic min-knapsack problem and a 2-approximation for a variant of that problem.We prove a new approximation bound for a previous algorithm for the SSSC problem, the adaptive greedy algorithm of Golovin and Krause.We also consider an approach to approximating SBFE problems using the adaptive greedy algorithm, which we call the Q-value approach. This approach easily yields a new result for evaluation of CDNF (conjunctive / disjunctive normal form) formulas, and we apply variants of it to simultaneous evaluation problems and a ranking problem. However, we show that the Q-value approach provably cannot be used to obtain a sublinear approximation factor for the SBFE problem for LTFs or read-once disjunctive normal form formulas.

Learning sparse two-level boolean rules

Abstract: This paper develops a novel optimization framework for learning accurate and sparse two-level Boolean rules for classification, both in Conjunctive Normal Form (CNF, i.e. AND-of-ORs) and in Disjunctive Normal Form (DNF, i.e. OR-of-ANDs). In contrast to opaque models (e.g. neural networks), sparse two-level Boolean rules gain the crucial benefit of interpretability, which is necessary in a wide range of applications such as law and medicine and is attracting considerable attention in machine learning. This paper introduces two principled objective functions to trade off classification accuracy and sparsity, where 0-1 error and Hamming loss are used to characterize accuracy. We propose efficient procedures to optimize these objectives based on linear programming (LP) relaxation, block coordinate descent, and alternating minimization. We also describe a new approach to rounding any fractional values in the optimal solutions of LP relaxations. Experiments show that our new algorithms based on the Hamming loss objective provide excellent tradeoffs between accuracy and sparsity with improvements over state-of-the-art methods.

Towards a HPC-oriented parallel implementation of a learning algorithm for bioinformatics applications

Abstract: Background: The huge quantity of data produced in Biomedical research needs sophisticated algorithmic methodologies for its storage, analysis, and processing. High Performance Computing (HPC) appears as a magic bullet in this challenge. However, several hard to solve parallelization and load balancing problems arise in this context. Here we discuss the HPC-oriented implementation of a general purpose learning algorithm, originally conceived for DNA analysis and recently extended to treat uncertainty on data (U BRAIN). The U-BRAIN algorithm is a learning algorithm that finds a Boolean formula in disjunctive normal form (DNF), of approximately minimum complexity, that is consistent with a set of data (instances) which may have missing bits. The conjunctive terms of the formula are computed in an iterative way by identifying, from the given data, a family of sets of conditions that must be satisfied by all the positive instances and violated by all the negative ones; such conditions allow the computation of a set of coefficients (relevances) for each attribute (literal), that form a probability distribution, allowing the selection of the term literals. The great versatility that characterizes it, makes U-BRAIN applicable in many of the fields in which there are data to be analyzed. However the memory and the execution time required by the running are of O(n3) and of O(n5) order, respectively, and so, the algorithm is unaffordable for huge data sets.

Sampling frequent and minimal boolean patterns: theory and application in classification

Abstract: We tackle the challenging problem of mining the simplest Boolean patterns from categorical datasets. Instead of complete enumeration, which is typically infeasible for this class of patterns, we develop effective sampling methods to extract a representative subset of the minimal Boolean patterns in disjunctive normal form (DNF). We propose a novel theoretical characterization of the minimal DNF expressions, which allows us to prune the pattern search space effectively. Our approach can provide a near-uniform sample of the minimal DNF patterns. We perform an extensive set of experiments to demonstrate the effectiveness of our sampling method. We also show that minimal DNF patterns make effective features for classification.

New Boolean Equation for Orthogonalizing of Disjunctive Normal Form based on the Method of Orthogonalizing Difference-Building

Abstract: In this paper a new Boolean equation for the orthogonalization of Boolean functions respectively of Ternary-Vector-Lists of disjunctive normal form is presented. It provides the mathematical solution of orthogonalization for the first time. The new equation is based on the new method of orthogonalizing difference-building ?. In contrast to other methods the new method has a faster computation time. Another advantage is the smaller number of product terms respectively of Ternary-Vectors in the orthogonalized result in contrast to other methods. Furthermore, the new equation can be used as a part in the calculation procedure of getting suitable test patterns for combinatorial circuits for verifying feasible logical faults.

Disjunctive Normal Form of Weak Classifiers for Online Learning based Object Tracking

Abstract: The use of a strong classifier that is combined by an ensemble of weak classifiers has been prevalent in tracking, classification etc. In the conventional ensemble tracking, one weak classifier selects a 1D feature, and the strong classifier is combined by a number of 1D weak classifiers. In this paper, we present a novel tracking algorithm where weak classifiers are 2D disjunctive normal form (DNF) of these 1D weak classifiers. The final strong classifier is then a linear combination of weak classifiers and 2D DNF cell classifiers. We treat tracking as a binary classification problem, and one full DNF can express any particular Boolean function; therefore 2D DNF classifiers have the capacity to represent more complex distributions than original weak classifiers. This can strengthen any original weak classifier. We implement the algorithm and run the experiments on several video sequences.

A Duality-Aware Calculus for Quantified Boolean Formulas

Abstract: Learning and backjumping are essential features in search-based decision procedures for Quantified Boolean Formulas (QBF). To obtain a better understanding of such procedures, we present a formal framework, which allows to simultaneously reason on prenex conjunctive and disjunctive normal form. It captures both satisfying and falsifying search statesin a symmetric way. This symmetry simplifies the framework and offers potential for further variants.

Disjunctive Normal Form Schemes for Heterogeneous Attributed Graphs.

Abstract: Several 'edge-discovery' applications over graph-based data models are known to have worst-case quadratic complexity, even if the discovered edges are sparse. One example is the generic link discovery problem between two graphs, which has invited research interest in several communities. Specific versions of this problem include link prediction in social networks, ontology alignment between metadata-rich RDF data, approximate joins, and entity resolution between instance-rich data. As large datasets continue to proliferate, reducing quadratic complexity to make the task practical is an important research problem. Within the entity resolution community, the problem is commonly referred to as blocking. A particular class of learnable blocking schemes is known as Disjunctive Normal Form (DNF) blocking schemes, and has emerged as state-of-the art for homogeneous (i.e. same-schema) tabular data. Despite the promise of these schemes, a formalism or learning framework has not been developed for them when input data instances are generic, attributed graphs possessing both node and edge heterogeneity. With such a development, the complexity-reducing scope of DNF schemes becomes applicable to a variety of problems, including entity resolution and type alignment between heterogeneous RDF graphs, and link prediction in networks represented as attributed graphs. This paper presents a graph-theoretic formalism for DNF schemes, and investigates their learnability in an optimization framework. Experimentally, the DNF schemes learned on pairs of heterogeneous RDF graphs are demonstrated to achieve high complexity-reductions (98.25% across ten RDF test cases) at little cost to coverage, and with high reliability (<2.5% standard deviation). Finally, one extant class of RDF blocking schemes is shown to be a special case of DNF schemes.

Data search method and device

Abstract: The embodiment of the invention discloses a data search method and device The method comprises the following steps: converting a search rule into a disjunctive normal form rule from a Boolean rule; decomposing the disjunctive normal form rule into a plurality of simple conjunctive form rules; and comparing data in a database with each simple conjunctive form rule in the disjunctive normal form rule, and obtaining data which hits all objects contained in any one simple conjunctive form rule, and taking the data as search data corresponding to the search rule The technical scheme of the invention solves the problems of overhigh expenditure of time and space required by indexing and low search efficiency brought by the Boolean operation of a regular expression in a traditional data search algorithm

Adaptive Candidate Generation for Scalable Edge-discovery Tasks on Data Graphs

Abstract: Several `edge-discovery' applications over graph-based data models are known to have worst-case quadratic time complexity in the nodes, even if the discovered edges are sparse. One example is the generic link discovery problem between two graphs, which has invited research interest in several communities. Specific versions of this problem include link prediction in social networks, ontology alignment between metadata-rich RDF data, approximate joins, and entity resolution between instance-rich data. As large datasets continue to proliferate, reducing quadratic complexity to make the task practical is an important research problem. Within the entity resolution community, the problem is commonly referred to as blocking. A particular class of learnable blocking schemes is known as Disjunctive Normal Form (DNF) blocking schemes, and has emerged as state-of-the art for homogeneous (i.e. same-schema) tabular data. Despite the promise of these schemes, a formalism or learning framework has not been developed for them when input data instances are generic, attributed graphs possessing both node and edge heterogeneity. With such a development, the complexity-reducing scope of DNF schemes becomes applicable to a variety of problems, including entity resolution and type alignment between heterogeneous graphs, and link prediction in networks represented as attributed graphs. This paper presents a graph-theoretic formalism for DNF schemes, and investigates their learnability in an optimization framework. We also briefly describe an empirical case study encapsulating some of the principles in this paper.

Reconstructing a Logic from Tractatus: Wittgenstein’s Variables and Formulae

Abstract: It is and has been widely assumed, e.g., in Hintikka and Hintikka (1986), that the logical theory available from Wittgenstein’s Tractatus Logico-Philosophicus (Wittgenstein 1984) affords a foundation for (or is at least consistent with) the conventional logic represented in standard formulations of classical propositional, first-order predicate, and perhaps higher-order formal systems. The present article is a detailed attempt at a mathematical demonstration, or as much demonstration as the sources will allow, that this assumption is false by contemporary lights and according to a preferred account of argument validity. When Wittgenstein’s description of the forms of propositions or Satze in the 5-numbered remarks and Remark 6 is given a close reconstruction, one sees that no Tractarian proposition is logically equivalent to a simple universally or existentially quantified formula of first-order predicate logic. Therefore, although Wittgenstein employs the sign ∀—or (x), in his notation—occasionally in explanations and illustrations, e.g., 4.0411 and 5.1311, when it comes to logic, the sign ∀ should receive in Tractatus a semantical treatment that is nonstandard. En route to that result, we show that the hierarchy of variables–and, hence, of propositions–defined at 5.501 incorporates the expressive power of (at least) finitary classical propositional logic. Also, when constructed over a first-order language for arithmetic, formulae corresponding to the specification of 5.501 (but salted with parameters) pick out all and only arithmetic sets of numbers. Consequently, Wittgenstein’s hierarchy of iterated N-propositions–as described in vide Remark 6–does not collapse: at any level k, one finds propositions at k + 1 or above that are not logically equivalent to any proposition formed at k or below.

A New Method to Calculate the Appropriateness Measures of Label Expressions in Uncertainty Model

Abstract: The appropriateness measure of label expression is a basal concept in uncertainty modelling based on label semantics theory for dealing with vague concepts. In the paper, the concept of disjunctive normal forms is presented. It is proved that each label expression is semantic equivalent to a disjunctive normal form. Further, a new method of calculating the appropriateness measures of label expressions is provided.

Transmission exchange method and device of logic expression

Abstract: The embodiment of the invention discloses a transmission exchange method and device of a logic expression. The transmission exchange method comprises the following steps: a client side obtains a search condition and generates a service request according to the search condition; the client side carries out disjunctive normal form processing by aiming at the logic expression contained in the service request; the client side sends the service request subjected to the disjunctive normal form processing to a service side; the service side receives the service request sent from the client side; the service side analyzes the logic expression in the received service request according to the disjunctive normal form and carries out searching according to the analyzed logic expression to obtain a search result; and the service side returns the search result to the client side. Since the embodiment of the invention carries out disjunctive form processing on the logic expression before data is subjected to transmission exchange, the service side can more friendly identify the logic expressions contained in the service request transmitted from the client side or accurately compare whether the logic expressions are consistent or not so as to obtain the more accurate search result.

Applying Boolean discrete methods in the production of a real-valued probabilistic programming model.

Abstract: In this paper we explore the application of some notable Boolean methods, namely the Disjunctive Normal Form representation of logic table expansions, and apply them to a real-valued logic model which utilizes quantities on the range [0,1] to produce a probabilistic programming of a game character's logic in mathematical form.

A Logical Characterization of Small 2NFAs

Abstract: Let 2N be the class of families of problems solvable by families of two-way nondeterministic finite automata of polynomial size. We characterize 2N in terms of families of formulas of transitive-closure logic. These formulas apply the transitive-closure operator on a quantifier-free disjunctive normal form of first-order logic with successor and constants, where (i) apart from two special variables, all others are equated to constants in every clause, and (ii) no clause simultaneously relates these two special variables and refers to fixed input cells. We prove that automata with polynomially many states are as powerful as formulas with polynomially many clauses and polynomially large constants. This can be seen as a refinement of Immerman’s theorem that nondeterministic logarithmic space matches positive transitive-closure logic (NL = FO+pos TC).

Experimental comparison of decomposition methods for systems of Boolean function

Abstract: In this paper, we describe the results of the experimental comparison of programs that implement various decomposition methods for disjunctive normal forms of systems of completely defined Boolean functions. The complexity of a system of disjunctive normal forms is expressed in two ways: by the area of a programmable logic array that implements a system of disjunctive normal forms, or by the number of vertices of a binary decision diagram, which represents a system of Boolean functions. The complexity of the functional expansion of a system's functions is determined as the sum of the complexities of the subsystem of the functions included in this expansion. The estimates of the complexity are oriented on the synthesis of combinational circuits based on the programmed logical arrays and the library's logical elements.

A theory of contemplation

Abstract: In this paper you can explore the application of some notable Boolean-derived methods, namely the Disjunctive Normal Form representation of logic table expansions, and extend them to a real-valued logic model which is able to utilize quantities on the range [0,1], [-1,1], [a,b], (x,y), (x,y,z), and etc. so as to produce a logical programming of arbitrary range, precision, and dimensionality, thereby enabling contemplation at a logical level in notions of arbitrary data, colors, and spatial constructs, with an example of the production of a game character's logic in mathematical form.