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Given a finite collection of classifiers one might wish to combine, or fuse, the classifiers in hopes that the multiple classifier system (MCS) will perform better than the individuals. One method of fusing classifiers is to combine their final decision using Boolean rules (e.g., a logical OR, AND, or a majority vote of the classifiers in the system). An established method for evaluating a classifier is measuring some aspect of its Receiver Operating Characteristic (ROC) curve, which graphs the trade-off between the conditional probabilities of detection and false alarm. This work presents a unique method of estimating the performance of an MCS in which Boolean rules are used to combine individual decisions. The method requires performance data similar to the data available in the ROC curves for each of the individual declassifiers, and the method can be used to estimate the ROC curve for the entire system. A consequence of this result is that one can save time and money by effectively evaluating the performance of an MCS without performing experiments.

Given a finite collection of classifiers one might wish to combine, or fuse, the classifiers in hopes that the multiple classifier system (MCS) will perform better than the individuals. One method of fusing classifiers is to combine their final decision using Boolean rules (e.g., a logical OR, AND, or a majority vote of the classifiers in the system). An established method for evaluating a classifier is measuring some aspect of its Receiver Operating Characteristic (ROC) curve, which graphs the trade-off between the conditional probabilities of detection and false alarm. This work presents a unique method of estimating the performance of an MCS in which Boolean rules are used to combine individual decisions. The method requires performance data similar to the data available in the ROC curves for each of the individual declassifiers, and the method can be used to estimate the ROC curve for the entire system. A consequence of this result is that one can save time and money by effectively evaluating the performance of an MCS without performing experiments.

We study the class of cooperative Boolean networks whose only regulatory functions are COPY, binary AND, and binary OR. We prove that for all sufficiently large N and c < 2 there exist Boolean networks in this class that have an attractor of length > c^N whose basin of attraction comprises an arbitrarily large fraction of the state space. The existence of such networks contrasts with results on various other types of dynamical systems that show nongenericity or absence of non-steady state attractors under the assumption of cooperativity.

We consider the extensions of modal transition systems (MTS), namely Boolean MTS and parametric MTS and we investigate the refinement problems over both classes. Firstly, we reduce the problem of modal refinement over both classes to a problem solvable by a QBF solver and provide experimental results showing our technique scales well. Secondly, we extend the algorithm for thorough refinement of MTS providing better complexity then via reductions to previously studied problems. Finally, we investigate the relationship between modal and thorough refinement on the two classes and show how the thorough refinement can be approximated by the modal refinement.

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day?night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of complex clocks.

Parameterised Boolean Equation Systems (PBESs) are sequences of Boolean fixed point equations with data variables, used for, e.g., verification of modal mu-calculus formulae for process algebraic specifications with data. Solving a PBES is usually done by instantiation to a Parity Game and then solving the game. Practical game solvers exist, but the instantiation step is the bottleneck. We enhance the instantiation in two steps. First, we transform the PBES to a Parameterised Parity Game (PPG), a PBES with each equation either conjunctive or disjunctive. Then we use LTSmin, that offers transition caching, efficient storage of states and both distributed and symbolic state space generation, for generating the game graph. To that end we define a language module for LTSmin, consisting of an encoding of variables with parameters into state vectors, a grouped transition relation and a dependency matrix to indicate the dependencies between parts of the state vector and transition groups. Benchmarks on some large case studies, show that the method speeds up the instantiation significantly and decreases memory usage drastically.

This article is from Gut Pathogens , volume 5 . Abstract Background: Three pathogenicity islands, viz. SPI-1 (Salmonella pathogenicity island 1), SPI-2 (Salmonella pathogenicity island 2) and T6SS (Type VI Secretion System), present in the genome of Salmonella typhimurium have been implicated in the virulence of the pathogen. While the regulation of SPI-1 and SPI-2 (both encoding components of the Type III Secretion System - T3SS) are well understood, T6SS regulation is comparatively less studied. Interestingly, inter-connections among the regulatory elements of these three virulence determinants have also been suggested to be essential for successful infection. However, till date, an integrated view of gene regulation involving the regulators of these three secretion systems and their cross-talk is not available. Results: In the current study, relevant regulatory information available from literature have been integrated into a single Boolean network, which portrays the dynamics of T3SS (SPI-1 and SPI-2) and T6SS mediated virulence. Some additional regulatory interactions involving a two-component system response regulator YfhA have also been predicted and included in the Boolean network. These predictions are aimed at deciphering the effects of osmolarity on T6SS regulation, an aspect that has been suggested in earlier studies, but the mechanism of which was hitherto unknown. Simulation of the regulatory network was able to recreate in silico the experimentally observed sequential activation of SPI-1, SPI-2 and T6SS. Conclusions: The present study integrates relevant gene regulatory data (from literature and our prediction) into a single network, representing the cross-communication between T3SS (SPI-1 and SPI-2) and T6SS. This holistic view of regulatory interactions is expected to improve the current understanding of pathogenesis of S. typhimurium.

This article is from Gut Pathogens , volume 5 . Abstract Background: Three pathogenicity islands, viz. SPI-1 (Salmonella pathogenicity island 1), SPI-2 (Salmonella pathogenicity island 2) and T6SS (Type VI Secretion System), present in the genome of Salmonella typhimurium have been implicated in the virulence of the pathogen. While the regulation of SPI-1 and SPI-2 (both encoding components of the Type III Secretion System - T3SS) are well understood, T6SS regulation is comparatively less studied. Interestingly, inter-connections among the regulatory elements of these three virulence determinants have also been suggested to be essential for successful infection. However, till date, an integrated view of gene regulation involving the regulators of these three secretion systems and their cross-talk is not available. Results: In the current study, relevant regulatory information available from literature have been integrated into a single Boolean network, which portrays the dynamics of T3SS (SPI-1 and SPI-2) and T6SS mediated virulence. Some additional regulatory interactions involving a two-component system response regulator YfhA have also been predicted and included in the Boolean network. These predictions are aimed at deciphering the effects of osmolarity on T6SS regulation, an aspect that has been suggested in earlier studies, but the mechanism of which was hitherto unknown. Simulation of the regulatory network was able to recreate in silico the experimentally observed sequential activation of SPI-1, SPI-2 and T6SS. Conclusions: The present study integrates relevant gene regulatory data (from literature and our prediction) into a single network, representing the cross-communication between T3SS (SPI-1 and SPI-2) and T6SS. This holistic view of regulatory interactions is expected to improve the current understanding of pathogenesis of S. typhimurium.

We prove that cooperativity in Boolean networks precludes a strong notion of sensitive dependence on initial conditions. Weaker notions of sensitive dependence are shown to be consistent with cooperativity, but if each regulatory functions is binary AND or binary OR, in N-dimensional networks they impose an upper bound of approximately sqrt(3)^N on the lengths of attractors that can be reached from a fraction p approaching 1 of initial conditions. The upper bound is shown to be sharp.

A parameterised Boolean equation system (PBES) is a set of equations that defines sets as the least and/or greatest fixed-points that satisfy the equations. This system is regarded as a declarative program defining functions that take a datum and returns a Boolean value. The membership problem of PBESs is a problem to decide whether a given element is in the defined set or not, which corresponds to an execution of the program. This paper introduces reduced proof graphs, and studies a technique to solve the membership problem of PBESs, which is undecidable in general, by transforming it into a reduced proof graph. A vertex X(v) in a proof graph represents that the data v is in the set X, if the graph satisfies conditions induced from a given PBES. Proof graphs are, however, infinite in general. Thus we introduce vertices each of which stands for a set of vertices of the original ones, which possibly results in a finite graph. For a subclass of disjunctive PBESs, we clarify some conditions which reduced proof graphs should satisfy. We also show some examples having no finite proof graph except for reduced one. We further propose a reduced dependency space, which contains reduced proof graphs as sub-graphs if a proof graph exists. We provide a procedure to construct finite reduced dependency spaces, and show the soundness and completeness of the procedure.

An improved characteristic set algorithm for solving Boolean polynomial systems is pro- posed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain pseudo-remainders. Three important techniques are applied in the algorithm. The first one is eliminating variables by new gener- ated linear polynomials. The second one is optimizing the strategy of choosing polynomial for zero decomposition. The third one is to compute add-remainders to eliminate the leading variable of new generated monic polynomials. By analyzing the depth of the zero decompo- sition tree, we present some complexity bounds of this algorithm, which are lower than the complexity bounds of previous characteristic set algorithms. Extensive experimental results show that this new algorithm is more efficient than previous characteristic set algorithms for solving Boolean polynomial systems.

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.

This paper introduces an approach decentralized to fault detection and isolation (FDI) in manufacturing systems using a Boolean discrete event model. The method incorporates diverse information sources to create distinct models for plant systems and control. The objective is to enhance the understanding of process operations by employing various representation tools tailored to each information source. It is to reduce the number of explosion problems combinatorial and detect faults in the shortest possible time. This comprehensive representation facilitates the fulfillment of three crucial diagnosis functions: detection, localization, and identification. The approach involves Boolean modeling of each process actuator along with its corresponding sensors, a temporal model based on fuzzy expectations of event occurrences, and a set of if...then rules. The goal of this decentralized approach minimize both the complexity and the manual construction effort required for the model. The paper demonstrates the effectiveness of thi

We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph.

With this paper, we contribute to the understanding of ant colony optimization (ACO) algorithms by formally analyzing their runtime behavior. We study simple MAX-MIN ant systems on the class of linear pseudo-Boolean functions defined on binary strings of length 'n'. Our investigations point out how the progress according to function values is stored in pheromone. We provide a general upper bound of O((n^3 \log n)/ \rho) for two ACO variants on all linear functions, where (\rho) determines the pheromone update strength. Furthermore, we show improved bounds for two well-known linear pseudo-Boolean functions called OneMax and BinVal and give additional insights using an experimental study.

128 pages :

We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.

A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in $4\log_2 n\,2^n$ operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. Under precise algebraic assumptions on the input system, we show that the deterministic variant of our algorithm has complexity bounded by $O(2^{0.841n})$ when $m=n$, while a probabilistic variant of the Las Vegas type has expected complexity $O(2^{0.792n})$. Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to~1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as~200, and thus very relevant for cryptographic applications.

This paper presents an algorithm for approximating certain types of dynamical systems given by a system of ordinary delay differential equations by a Boolean network model. Often Boolean models are much simpler to understand than complex differential equations models. The motivation for this work comes from mathematical systems biology. While Boolean mechanisms do not provide information about exact concentration rates or time scales, they are often sufficient to capture steady states and other key dynamics. Due to their intuitive nature, such models are very appealing to researchers in the life sciences. This paper is focused on dynamical systems that exhibit bistability and are desc ribedby delay equations. It is shown that if a certain motif including a feedback loop is present in the wiring diagram of the system, the Boolean model captures the bistability of molecular switches. The method is appl ied to two examples from biology, the lac operon and the phage lambda lysis/lysogeny switch.

Chaotic functions are characterized by sensitivity to initial conditions, transitivity, and regularity. Providing new functions with such properties is a real challenge. This work shows that one can associate with any Boolean network a continuous function, whose discrete-time iterations are chaotic if and only if the iteration graph of the Boolean network is strongly connected. Then, sufficient conditions for this strong connectivity are expressed on the interaction graph of this network, leading to a constructive method of chaotic function computation. The whole approach is evaluated in the chaos-based pseudo-random number generation context.

Signed systems were introduced as a general, syntax-independent framework for paraconsistent reasoning, that is, non-trivialised reasoning from inconsistent information. In this paper, we show how the family of corresponding paraconsistent consequence relations can be axiomatised by means of quantified Boolean formulas. This approach has several benefits. First, it furnishes an axiomatic specification of paraconsistent reasoning within the framework of signed systems. Second, this axiomatisation allows us to identify upper bounds for the complexity of the different signed consequence relations. We strengthen these upper bounds by providing strict complexity results for the considered reasoning tasks. Finally, we obtain an implementation of different forms of paraconsistent reasoning by appeal to the existing system QUIP.

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear models with application in estimation and control of Boolean processes based on noisy and incomplete measurements. The optimal minimum mean square error (MMSE) algorithms for POBDS state estimation, namely, the Boolean Kalman filter (BKF) and Boolean Kalman smoother (BKS), are intractable in the case of large systems, due to computational and memory requirements. To address this, we propose approximate MMSE filtering and smoothing algorithms based on the auxiliary particle filter (APF) method from sequential Monte-Carlo theory. These algorithms are used jointly with maximum-likelihood (ML) methods for simultaneous state and parameter estimation in POBDS models. In the presence of continuous parameters, ML estimation is performed using the expectation-maximization (EM) algorithm; we develop for this purpose a special smoother which reduces the computational complexity of the EM algorithm. The resulting particle-based adaptive filter is applied to a POBDS model of Boolean gene regulatory networks observed through noisy RNA-Seq time series data, and performance is assessed through a series of numerical experiments using the well-known cell cycle gene regulatory model.

An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over the field with two elements. For systems that can be described by monomials (including Boolean AND systems), one can obtain information about the limit cycle structure from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. This condition depends on the cycle structure of the dependency graph of the system and can be verified in polynomial time.

The well-known problem of state space explosion in model checking is even more critical when applying this technique to programming languages, mainly due to the presence of complex data structures. One recent and promising approach to deal with this problem is the construction of an abstract and correct representation of the global program state allowing to match visited states during program model exploration. In particular, one powerful method to implement abstract matching is to fill the state vector with a minimal amount of relevant variables for each program point. In this paper, we combine the on-the-fly model-checking approach (incremental construction of the program state space) and the static analysis method called influence analysis (extraction of significant variables for each program point) in order to automatically construct an abstract matching function. Firstly, we describe the problem as an alternation-free value-based mu-calculus formula, whose validity can be checked on the program model expressed as a labeled transition system (LTS). Secondly, we translate the analysis into the local resolution of a parameterised boolean equation system (PBES), whose representation enables a more efficient construction of the resulting abstract matching function. Finally, we show how our proposal may be elegantly integrated into CADP, a generic framework for both the design and analysis of distributed systems and the development of verification tools.

The perfection of the transistor and the subsequent birth of a vast digital technology has focused special attention on the methods and mathematics of Boolean algebra. The result of this attention has been the development of powerful methods of synthesis and minimization of logic circuits. This area continues to be the subject of extensive research. The purpose of this paper is to explore the applicability of Boolean minimization methods to the design of large relay interlock and control systems. The investigation of an existing and representative system from the Boolean standpoint was judged to be the best way to accomplish this purpose. Accordingly, the system selected was the Livermore variable-energy 90-inch cyclotron located at the Lawrence Radiation Laboratory, Livermore, California. The conclusion is that for this class of system a working knowledge of elementary Boolean algebra and an engineer's normal intuition would be sufficient to achieve minimal design. As a bonus result, the symbolic notation known as \"gate notation\" was found to be a valuable aid in the representation and understanding of the interlock and control logic of the cyclotron.

Linear rules have played an increasing role in structural proof theory in recent years. It has been observed that the set of all sound linear inference rules in Boolean logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left- and right-)linear rewrite rule. In this paper we study properties of systems consisting only of linear inferences. Our main result is that the length of any 'nontrivial' derivation in such a system is bound by a polynomial. As a consequence there is no polynomial-time decidable sound and complete system of linear inferences, unless coNP=NP. We draw tools and concepts from term rewriting, Boolean function theory and graph theory in order to access some required intermediate results. At the same time we make several connections between these areas that, to our knowledge, have not yet been presented and constitute a rich theoretical framework for reasoning about linear TRSs for Boolean logic.

Includes index

133 pages : 28 cm

We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph.

Thesis (MS)--U.S. Naval Postgraduate School, 1963

The development of Boolean calculus for its application to developing digital system design methodologies that would reduce system complexity, size, cost, speed, power requirements, etc., is discussed. Synthesis procedures for logic circuits are examined particularly asynchronous circuits using clock triggered flip flops.

We present a stochastic, time-discrete boolean model which mimics the mesoscopic dynamics of the desorption reactions $A+A\to A+S$ and $A+A\to S+S$ in a 1D lattice. In the continuous-time limit, we derive a hierarchy of dynamical equations for the subset of moments involving contiguous lattice sites. The solution of the hierarchy allows to compute the exact dynamics of the mean coverage for both microscopic and coarse-grained initial conditions, which turn out to be different from the mean field predictions. The evolution equations for the mean coverage and the second order moments are shown to be equivalent to those provided by a time-continuous Master equation. The important role of higher order fluctuations is brought out by the failure of a truncation scheme retaining only two-particle fluctuation correlations.

We present a method for fighting the state space explosion of parameterised Boolean equation systems (PBESs); these essentially are systems of mutually recursive Boolean fixed point equations, parameterised with data. PBESs can encode equivalence checking problems and model checking problems for symbolic, process algebraic specifications. Our method essentially consists of three phases: (1) the control flow in the PBES is reconstructed, detecting control flow parameters that were encoded in the description of the process, as well as control flow parameters that were introduced during the construction of the PBES; (2) we use a data flow analysis based on the control flow in the PBES to detect irrelevant data parameters and (3) we reset those data parameters of the equations in the PBES that were found to be irrelevant. Our reduction preserves the solution to the PBES, and never increases the size of the underlying Boolean equation system. The reduction is evaluated using a number of case studies.

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.

In this paper we introduce the notion of {\it core} for two specific classes of boolean maps on finite involution posets (which are a generalization of the boolean lattices) and we prove some extension results for such families of boolean maps. Through the properties of the core, we provide a complete characterization of such maps. The main purpose of such abstract results is their application to the study of the compatibility of a particular class of systems of linear inequalities related to a conjecture of Manickam, Mikl\"os and Singhi (\cite{ManSin88}, \cite{ManMik87}), still unsolved and that can be considered dual to the theorem of Erd\"os-Ko-Rado \cite{erd-ko-rad}.

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.

We associate to each discrete partial dynamical system a universal C*-algebra generated by partial isometries satisfying relations given by a Boolean algebra connected to the discrete partial dynamical system in question. We show that for symbolic dynamical systems like one-sided and two-sided shift spaces and topological Markov chains with an arbitrary state space the C*-algebras usually associated to them, can be obtained in this way. As a consequence of this, we will be able to show that for two-sided shift spaces having a certain property, the crossed product of the two-sided shift space is a quotient of the C*-algebra associated to the corresponding one-sided shift space.

Discrete dynamical systems defined on the state space {0,1,...,p-1}^n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n=m+1, as in the Smale theorem for the continuous case, but we show that this is possible for n=m+2 as long as p is sufficiently large. We also prove that a natural discrete analogue of strong cooperativity implies nontrivial bounds on the lengths of periodic orbits and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strong cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strong cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.

The perfection of the transistor and the subsequent birth of a vast digital technology has focused special attention on the methods and mathematics of Boolean algebra. The result of this attention has been the development of powerful methods of synthesis and minimization of logic circuits. The purpose of this paper is to explore the applicability of Boolean minimization methods to the design of large relay interlock and control systems. The investigation of an existing and representative system from the Boolean standpoint was judged the best way to accomplish this purpose. Accordingly, the system selected was the Livermore variable-energy 90-inch cyclotron located at the Lawrence Radiation Laboratory, Livermore, California. The conclusion is that for this class of system a working knowledge of elementary Boolean algebra and an engineer's normal intuition would be sufficient to achieve minimal design. As a bonus result, the symbolic notation known as gate notation was found to be a valuable aid in the representation and understanding of the interlock and control logic of the cyclotron.

This textbook is one of a series of publications designed to provide information needed by Navy personnel whose duties require an elementary and general knowledge of the fundamental concepts of number systems, logic circuits, and Boolean algebra. Topic 1, Number Systems, describes the radix; the positional notation; the decimal, binary, octal, and hexadecimal number systems; and the conversion techniques needed to convert from one system to another. Topic 2, Boolean Algebra, includes rules, laws, mechanization, and simplification techniques. Topic 3, Logic Circuits, includes logic computation; logic polarity; and the six basic logic circuits of AND, OR, NOT, NAND, NOR, and Exclusive OR. A glossary of terms is included. (MNS)

US Navy Training Course Introduction to Statistics, Number Systems, and Boolean Algebra