Generating probabilistic Boolean networks from a prescribed stationary distribution

Modeling gene regulation is an important problem in genomic research. Boolean networks (BN) and its generalization probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. BN is a deterministic model while PBN is a stochastic model. In a PBN, on one hand, its stationary distribution gives important information about the long-run behavior of the network. On the other hand, one may be interested in system synthesis which requires the construction of networks from the observed stationary distribution. This results in an inverse problem which is ill-posed and challenging. Because there may be many networks or no network having the given properties and the size of the inverse problem is huge. In this paper, we consider the problem of constructing PBNs from a given stationary distribution and a set of given Boolean Networks (BNs). We first formulate the inverse problem as a constrained least squares problem. We then propose a heuristic method based on Conjugate Gradient (CG) algorithm, an iterative method, to solve the resulting least squares problem. We also introduce an estimation method for the parameters of the PBNs. Numerical examples are then given to demonstrate the effectiveness of the proposed methods.     

Generating minimal fault detecting test suites for general Boolean specifications

Context

Boolean expressions are a central aspect of specifications and programs, but they also offer dangerously many ways to introduce faults. To counter this effect, various criteria to generate and evaluate tests have been proposed. These are traditionally based on the structure of the expressions, but are not directly related to the possible faults. Often, they also require expressions to be in particular formats such as disjunctive normal form (DNF), where a strict hierarchy of faults is available to prove fault detection capability.

Objective

This paper describes a method that generates test cases directly from an expression’s possible faults, guaranteeing that faults of any chosen class will be detected. In contrast to many previous criteria, this approach does not require the Boolean expressions to be in DNF, but allows expressions in any format, using any deliberate fault classes.

Method

The presented approach is based on creating test objectives for individual faults, such that efficient, modern satisfiability solvers can be used to derive test cases that directly address the faults. Although the number of such test objectives can be high depending on the considered fault classes, a number of optimizations can be applied to reduce the test generation effort.

Results

Evaluation on a set of commonly used benchmarks shows that despite guaranteeing fault coverage, the number of test cases can be reduced even further than that produced by other state of the art strategies. At the same time, the fault detection capability is not affected negatively, and clearly improves over state of the art criteria for general form Boolean expressions.

Conclusion

The approach presented in this paper is shown to improve over the state of the art with respect to the types of expressions that can be handled, the fault classes that are guaranteed to be covered, and the sizes of test suites generated automatically. This has implications for several fields of software testing: A main application is specification based testing, but Boolean expressions also exist in normal source code and need to be tested there as well.     

Boolean Functions as Models for Quantified Boolean Formulas

In this paper, we introduce the notion of models for quantified Boolean formulas. For various classes of quantified Boolean formulas and various classes of Boolean functions, we investigate the problem of determining whether a model exists. Furthermore, we show for these classes the complexity of the model checking problem, which is to check whether a given set of Boolean functions is a model for a formula. For classes of Boolean functions, we establish some characterizations in terms of classes of quantified Boolean formulas that have such a model. This research has been supported in part by the Air Force Office of Scientific Research under grant FA9550-06-1-0050. This research has been supported in part by the NSFC under grants 60573011 and 10410638.     

Boolean sets and most general solutions of Boolean equations

The aim of this paper is twofold. First we determine the most general form of the subsumptive general solution of a Boolean equation (Theorem 1 and Theorem 2). Then we discuss several characterizations of Boolean sets, meaning sets of zeros of Boolean functions, and prove that every Boolean transformation X=Φ(T) is the parametric general solution of a certain Boolean equation.     

多值Boole过程

采用文献[1]中定义的扩展Allen-Givone代数概念将Boo1e过程沦扩充,提出了多值Boo1e过程的概念及其运算,为精确统一描述多值逻辑电路的逻辑功能和定时行为提供了一种解析途径。提出基于Allen-Givone代数的带状波形概念,用实值的加、减、乘、除运算为电路的异步特性提出了解析化的理论基础。这种数学分析与离散数学相结合的途径能相对精确地描述电路的时滞模型。在多值逻辑电路设计自动化技术的测试、模拟、综合等领域中,这种方法有它的应用前景。     

Boolean dominated MV-algebras

In this paper we study an obvious generalization of the hyperarchimedean MV-algebras: boolean dominated MV-algebras. Particularly we point out the wide difference between the class of the hyperarchimedean MV-algebras and the class of the boolean dominated MV-algebras.     

Universal Boolean Systems

Boolean interaction systems and hard interaction systems define nets of interacting cells. They are based on the same local interaction principle between two cells as interaction nets but do not allow that the structure of nets may evolve. With boolean nets, it is not possible to create or destroy cells or links between existing cells. They are very similar to hardware circuits but based on an implicit rendez-vous information exchange mechanism.If we want to implement such systems using hardware circuits, it is important to define a set of universal combinators that reduces this task to the implementation of a fixed number of known agents. Here, we show how we can simulate every hard interaction system by a universal boolean interaction system composed of three combinators: a duplicator, a NAND gate and a three-state input/output channel.     

Negative Boolean constraints

Systems of Boolean constraints which allow negative constraints such as f g are investigated. The results form a basis for algorithms to determine satisfiability, validity, implication, equivalence and variable elimination for such systems. These algorithms have applications in spatial query decomposition, machine reasoning and constraint logic programming. Proofs of the results rely on independence of inequations, which enables results for systems with a single inequation to be lifted to systems with many inequations.     

Stochastic Boolean Satisfiability

Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSAT, which can function for probabilistic domains as SAT does for deterministic domains. It shows the connection between SSAT and well-studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSAT instances. These algorithms are validated on random SSAT formulae generated under the fixed-clause model. In spite of the large complexity gap between SSAT (PSPACE) and SAT (NP), the paper suggests that much of what we have learned about SAT transfers to the probabilistic domain.     

Boolean Expression Diagrams

This paper presents a new data structure called boolean expression diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of binary decision diagrams (BDDs) which can represent any Boolean circuit in linear space. Two algorithms are described for transforming a BED into a reduced ordered BDD. One is a generalized version of the BDD apply-operator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two different circuit implementations of a 16-bit multiplier implement the same Boolean function. Using BEDs, this verification problem is solved efficiently, while using standard BDD techniques this problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the end-result as a reduced ordered BDD is small. Moreover, using operators for substitution and existential quantification they allow for the verification of large hierarchical circuits.     

Scannerless Boolean Parsing

Scannerless generalized parsing techniques allow parsers to be derived directly from unified, declarative specifications. Unfortunately, in order to uniquely parse existing programming languages at the character level, disambiguation extensions beyond the usual context-free formalism are required.This paper explains how scannerless parsers for boolean grammars (context-free grammars extended with intersection and negation) can specify such languages unambiguously, and can also describe other interesting constructs such as indentation-based block structure.The sbp package implements this parsing technique and is publicly available as Java source code.     

Boolean Algebra as a Fragment of the Theory of Boolean Toposes

Basic identities of Boolean algebra are considered, and their categorical analogs are proved.     

Boolean invariants of databases

The article considers some consequences of the application of Boolean algebra to relational database models.Translated from Kibernetika, No. 5, pp. 32–35, September–October, 1989.     

Impact of Boolean factorization as preprocessing methods for classification of Boolean data

We explore a utilization of Boolean matrix factorization for data preprocessing in classification of Boolean data. In our previous work, we demonstrated that preprocessing that consists in replacing the original Boolean attributes by factors, i.e. new Boolean attributes obtained from the original ones by Boolean matrix factorization, can improve classification quality. The aim of this paper is to explore the question of how the various Boolean factorization methods that were proposed in the literature impact the quality of classification. In particular, we compare five factorization methods, present experimental results, and outline issues for future research.     

Complexity of Boolean algebras

We show that the elementary theory of Boolean algebras is _log-complete for the Berman complexity class _c<ω STA(*, 2^cn, n), the class of sets accepted by alternating Turing machines running in time 2^cn for some constant c and making at most n alternations on inputs of length n; thus the theory is computationally equivalent to the theory of real addition with order. We extend the completeness results to various subclasses of Boolean algebras, including the finite, free, atomic, atomless, and complete Boolean algebras. Finally we show that the theory of any finite collection of finite Boolean algebras is complete for PSPACE, while the theory of any other collection is _log-hard for _c<ω STA(*, 2^cn, n).     

Activity in Boolean networks

In this paper we extend the notion of activity for Boolean networks introduced by Shmulevich and Kauffman (Phys Rev Lett 93(4):48701:1–4, 2004). In contrast to existing theory, we take into account the actual graph structure of the Boolean network. The notion of activity measures the probability that a perturbation in an initial state produces a different successor state than that of the original unperturbed state. It captures the notion of sensitive dependence on initial conditions, and provides a way to rank vertices in terms of how they may impact predictions. We give basic results that aid in the computation of activity and apply this to Boolean networks with threshold functions and nor functions for elementary cellular automata, d-regular trees, square lattices, triangular lattices, and the Erd?s–Renyi random graph model. We conclude with some open questions and thoughts on directions for future research related to activity, including long-term activity.