Boolean function

About: Boolean function is a research topic. Over the lifetime, 10089 publications have been published within this topic receiving 201604 citations. The topic is also known as: Boolean operation.

Analysis and Control of Boolean Networks: A Semi-tensor Product Approach

Abstract: A Boolean network is a logical dynamic system, which has been used to describe cellular networks. Using a new matrix product, called semi-tensor product of matrices, a logical function can be expressed as an algebraic function. This expression can covert the Boolean networks into discrete-time linear dynamic systems. Similarly, the Boolean control networks can also be converted into discrete time bilinear dynamic systems. Under these forms the standard matrix analysis can be used to consider the structure and the control problems of Boolean (control) networks. After the detailed description of this new approach, the controllability of Boolean control networks is considered in the paper as an application.

Constant depth circuits, Fourier transform, and learnability

Abstract: In this paper, Boolean functions in ,4C0 are studied using harmonic analysis on the cube. The main result is that an ACO Boolean function has almost all of its "power spectrum" on the low-order coefficients. An important ingredient of the proof is Hastad's switching lemma (8). This result implies several new properties of functions in -4C(': Functions in AC() have low "average sensitivity;" they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators. Perhaps the most interesting application is an O(n POIYIOg(n ')-time algorithm for learning func- tions in ACO. The algorithm observes the behavior of an AC'" function on O(nPO'Y'Og(n)) randomly chosen inputs, and derives a good approximation for the Fourier transform of the function. This approximation allows the algorithm to predict, with high probability, the value of the function on other randomly chosen inputs.

Learning in Embedded Systems

Abstract: This dissertation addresses the problem of designing algorithms for learning in embedded systems. This problem differs from the traditional supervised learning problem. An agent, finding itself in a particular input situation must generate an action. It then receives a reinforcement value from the environment, indicating how valuable the current state of the environment is for the agent. The agent cannot, however, deduce the reinforcement value that would have resulted from executing any of its other actions. A number of algorithms for learning action strategies from reinforcement values are presented and compared empirically with existing reinforcement-learning algorithms. The interval-estimation algorithm uses the statistical notion of confidence intervals to guide its generation of actions in the world, trading off acting to gain information against acting to gain reinforcement. It performs well in simple domains but does not exhibit any generalization and is computationally complex. The cascade algorithm is a structural credit-assignment method that allows an action strategy with many output bits to be learned by a collection of reinforcement-learning modules that learn Boolean functions. This method represents an improvement in computational complexity and often in learning rate. Two algorithms for learning Boolean functions in k-DNF are described. Both are based on Valiant's algorithm for learning such functions from input-output instances. The first uses Sutton's techniques for linear association and reinforcement comparison, while the second uses techniques from the interval estimation algorithm. They both perform well and have tractable complexity. A generate-and-test reinforcement-learning algorithm is presented. It allows symbolic representations of Boolean functions to be constructed incrementally and tested in the environment. It is highly parametrized and can be tuned to learn a broad range of function classes. Low-complexity functions can be learned very efficiently even in the presence of large numbers of irrelevant input bits. This algorithm is extended to construct simple sequential networks using a set-reset operator, which allows the agent to learn action strategies with state. These algorithms, in addition to being studied in simulation, were implemented and tested on a physical mobile robot.

Random networks of automata: a simple annealed approximation

Abstract: Kauffman's model is a random complex automata where nodes are randomly assembled. Each node σi receives K inputs from K randomly chosen nodes and the values of σi at time t + 1 is a random Boolean function of the K inputs at time t. Numerical simulations have shown that the behaviour of this model is very different for K > 2 and K ≤ 2. It is the purpose of this work to give a simple annealed approximation which predicts K = 2 as the critical value of K. This approximation gives also quantitative predictions for distances between iterated configurations. These predictions agree rather well with the numerical simulations. A possible way of improving this annealed approximation is proposed.

Algebraic decision diagrams and their applications

Abstract: In this paper we present theory and experiments on the algebraic decision diagrams (ADDs). These diagrams extend BDD's by allowing values from an arbitrary finite domain to be associated with the terminal nodes. We present a treatment founded in Boolean algebras and discuss algorithms and results in applications like matrix multiplication and shortest path algorithms. Furthermore, we outline possible applications of ADD's to logic synthesis, formal verification, and testing of digital systems.