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Elucidating the architecture and dynamics of large scale genetic regulatory networks of cells is an important goal in systems biology. We study the system level dynamical properties of the genetic network of Escherichia coli that regulates its metabolism, and show how its design leads to biologically useful cellular properties. Our study uses the database (Covert et al., Nature 2004) containing 583 genes and 96 external metabolites which describes not only the network connections but also the boolean rule at each gene node that controls the switching on or off of the gene as a function of its inputs. We have studied how the attractors of the boolean dynamical system constructed from this database depend on the initial condition of the genes and on various environmental conditions corresponding to buffered minimal media. We find that the system exhibits homeostasis in that its attractors, that turn out to be fixed points or low period cycles, are highly insensitive to initial conditions or perturbations of gene configurations for any given fixed environment. At the same time the attractors show a wide variation when external media are varied implying that the system mounts a highly flexible response to changed environmental conditions. The regulatory dynamics acts to enhance the cellular growth rate under changed media. Our study shows that the reconstructed genetic network regulating metabolism in {\it E. coli} is hierarchical, modular, and largely acyclic, with environmental variables controlling the root of the hierarchy. This architecture makes the cell highly robust to perturbations of gene configurations as well as highly responsive to environmental changes. The twin properties of homeostasis and response flexibility are achieved by this dynamical system even though it is not close to the edge of chaos.

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group. of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.

A boolean term order is a total order on subsets of [n]={1,...,n} such that \emptyset < alpha for all nonempty alpha contained in [n], and alpha < beta implies alpha \cup gamma < beta \cup gamma for all gamma which do not intersect alpha or beta. Boolean term orders arise in several different areas of mathematics, including Gr\"obner basis theory for the exterior algebra, and comparative probability. The main result of this paper is that boolean term orders correspond to one element extensions of the oriented matroid M(B_n), where B_n is the root system {e_i:1 \leq i \leq n \} \cup {e_i \pm e_j :1 \leq i < j \leq n}. This establishes boolean term orders in the frame work of the Baues problem. We also define a notion of coherence for a boolean term order, and a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of boolean term orders for small values of n.

An evaluation of an information retrieval system using a Boolean-based retrieval engine and inverted file architecture and WAIS, which uses a vector-based engine, was conducted. Four research questions in aeronautical engineering were used to retrieve sets of citations from the NASA Aerospace Database which was mounted on a WAIS server and available through Dialog File 108 which served as the Boolean-based system (BBS). High recall and high precision searches were done in the BBS and terse and verbose queries were used in the WAIS condition. Precision values for the WAIS searches were consistently above the precision values for high recall BBS searches and consistently below the precision values for high precision BBS searches. Terse WAIS queries gave somewhat better precision performance than verbose WAIS queries. In every case, a small number of relevant documents retrieved by one system were not retrieved by the other, indicating the incomplete nature of the results from either retrieval system. Relevant documents in the WAIS searches were found to be randomly distributed in the retrieved sets rather than distributed by ranks. Advantages and limitations of both types of systems are discussed.

Interaction graphs provide an important qualitative modeling approach for System Biology. This paper presents a novel approach for construction of interaction graph with the help of Boolean function decomposition. Each decomposition part (Consisting of 2-bits) of the Boolean functions has some important significance. In the dynamics of a biological system, each variable or node is nothing but gene or protein. Their regulation has been explored in terms of interaction graphs which are generated by Boolean functions. In this paper, different classes of Boolean functions with regards to Interaction Graph with biologically significant properties have been adumbrated.

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are R-related to all B. Such primitives formalize sentences from natural language like `All students read some textbooks'. Here A and B denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.

We consider a random system of equations $x_i+x_j=b_{(i,j)} (\text{mod }2)$, $(x_u\in \{0,1\},\, b_{(u,v)}=b_{(v,u)}\in\{0,1\})$, with the pairs $(i,j)$ from $E$, a symmetric subset of $[n]\times [n]$. $E$ is chosen uniformly at random among all such subsets of a given cardinality $m$; alternatively $(i,j)\in E$ with a given probability $p$, independently of all other pairs. Also, given $E$, $\pr\{b_{e}=0\}=\pr\{b_e=1\}$ for each $e\in E$, independently of all other $b_{e^\prime}$. It is well known that, as $m$ passes through $n/2$ ($p$ passes through $1/n$, resp.), the underlying random graph $G(n,\#\text{edges}=m)$, ($G(n,\pr(\text{edge})=p)$, resp.) undergoes a rapid transition, from essentially a forest of many small trees to a graph with one large, multicyclic, component in a sea of small tree components. We should expect then that the solvability probability decreases precipitously in the vicinity of $m\sim n/2$ ($p\sim 1/n$), and indeed this probability is of order $(1-2m/n)^{1/4}$, for $m0$. Mike Molloy noticed that the Boolean system with $b_e\equiv 1$ is solvable iff the underlying graph is $2$-colorable, and asked whether this connection might be used to determine an order of probability of $2$-colorability in the near-critical case. We answer Mike's question affirmatively and show that probability of $2$-colorability is $\lesssim 2^{-1/4}e^{1/8}c(\lambda)n^{-1/12}$, and asymptotic to $2^{-1/4}e^{1/8}c(\la)n^{-1/12}$ at a critical phase $\la=O(1)$, and for $\la\to -\infty$. (Submitted to Electronic Journal of Combinatorics on September 7, 2009.)

boolean_icons_system_tray_by_inaseparatesky

The goal of this thesis is to design and implement a computer aided diagnostic system for combinational circuits, a type of circuit used in the design of all computers. The diagnostic system is to accept a description of the circuit, supervise an adaptive input-output experiment on a potentially faulty implementation of the circuit, and return the locations of all faults in the circuit. The description of the circuit which will be input to this system can be in one of two forms: Boolean equations, or statements in the VHSIC Hardware Description Language (VHDL). Based on the circuit structure, a fault model is developed which can be used to mathematically model the state of faults in the circuit. The circuit description is processed to derive a single Boolean characteristic equation; information gained from testing is used to update this equation. The characteristic equation is manipulated to generate test vectors which are used as inputs to the actual circuit being diagnosed. After a given test vector has been input to the circuit, the output is observed. The state of the circuit output is then input to the diagnostic system which uses it to derive new knowledge about the actual circuit. Such tests are conducted repetively until the diagnostic system determines that further information cannot be derived from testing. At this point, the diagnostic system determines the nature and location of faults in the actual circuit as well as the function actually performed by the circuit.

Most reliability analysis techniques and tools assume that a system is used for a mission consisting of a single phase. However, multiple phases are natural in many missions. The failure rates of components, system configuration, and success (failure) criteria may vary from phase to phase. In addition, the duration of a phase may be deterministic or random. We describe a new technique for phased-mission system reliability analysis based on Boolean algebraic methods. Our technique is computationally efficient and is applicable to a large class of systems for which the failure criterion in each phase can be expressed as a fault tree (or an equivalent representation). Our technique avoids state space explosion that commonly plague Markov chain-based analysis. We develop a phase algebra to account for the effects of variable configurations and failure criteria from phase to phase. Our technique yields exact (as opposed to approximate) results. We demonstrate the use our technique by means of an example and present numerical results to show the effects of mission phases on the system reliability.

Most reliability analysis techniques and tools assume that a system is used for a mission consisting of a single phase. However, multiple phases are natural in many missions. The failure rates of components, system configuration, and success criteria may vary from phase to phase. In addition, the duration of a phase may be deterministic or random. Recently, several researchers have addressed the problem of reliability analysis of such systems using a variety of methods. We describe a new technique for phased-mission system reliability analysis based on Boolean algebraic methods. Our technique is computationally efficient and is applicable to a large class of systems for which the failure criterion in each phase can be expressed as a fault tree (or an equivalent representation). Our technique avoids state space explosion that commonly plague Markov chain-based analysis. We develop a phase algebra to account for the effects of variable configurations and success criteria from phase to phase. Our technique yields exact (as opposed to approximate) results. We demonstrate the use our technique by means of an example and present numerical results to show the effects of mission phases on the system reliability.

In warfare, historical records indicate that a larger force usually defeats a smaller force. The U.S. military will most likely fight outnumbered in any large scale conflict in the future. To redress this imbalance, Force Multipliers are used to increase the combat effectiveness of systems. The need to design systems to increase combat effectiveness in a limited resource environment is growing every day. Decision processes for design occur in five areas (material, force structure, doctrine/tactics, training, and the principles of war). This research focuses on a design of a weapon system within the material development area using Force Multipliers as the measure of effectiveness. This research develops and implements a design methodology for weapon systems that is based on pseudo-Boolean models. The model and methodology developed allows the user to participate in a design process through the specifications and situational/tactical inputs. The Indicators of Force Multipliers (INFORM) Model is developed to implement this methodology to design a main battle tank. The main battle tank, designed by this model, was potentially 30-50% more combat effective than a previous main battle tank.

Most reliability analysis techniques and tools assume that a system is used for a mission consisting of a single phase. However, multiple phases are natural in many missions. The failure rates of components, system configuration, and success criteria may vary from phase to phase. In addition, the duration of a phase may be deterministic or random. Recently, several researchers have addressed the problem of reliability analysis of such systems using a variety of methods. A new technique for phased-mission system reliability analysis based on Boolean algebraic methods is described. Our technique is computationally efficient and is applicable to a large class of systems for which the failure criterion in each phase can be expressed as a fault tree (or an equivalent representation). Our technique avoids state space explosion that commonly plague Markov chain-based analysis. A phase algebra to account for the effects of variable configurations and success criteria from phase to phase was developed. Our technique yields exact (as opposed to approximate) results. The use of our technique was demonstrated by means of an example and present numerical results to show the effects of mission phases on the system reliability.