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**ABSTRACT:**The traditional problem in binary decision diagrams (BDDs) has been to minimize the number of nodes since this reduces the memory needed to store the BDD. Recently, a new problem has emerged: minimizing the average path length (APL). APL is a measure of the time needed to evaluate the function by applying a sequence of variable values. It is of special significance when BDDs are used in simulation and design verification. A main result of this paper is that the APL for benchmark functions is typically much smaller than for random functions. That is, for the set of all functions, we show that the average APL is close to the maximum path length, whereas benchmark functions show a remarkably small APL. Surprisingly, however, typical functions do not achieve the absolute maximum APL. We show that the parity functions are unique in having that distinction. We show that the APL of a BDD can vary considerably with variable ordering. We derive the APL for various functions, including the AND, OR, threshold, Achilles' heel, and certain arithmetic functions. We show that the unate cascade functions uniquely achieve the absolute minimum APL.IEEE Transactions on Computers 10/2005; 54(9):1041- 1053. · 1.47 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Agent-based modeling and simulation is a useful method to study biological phenomena in a wide range of fields, from molecular biology to ecology. Since there is currently no agreed-upon standard way to specify such models, it is not always easy to use published models. Also, since model descriptions are not usually given in mathematical terms, it is difficult to bring mathematical analysis tools to bear, so that models are typically studied through simulation. In order to address this issue, Grimm et al. proposed a protocol for model specification, the so-called ODD protocol, which provides a standard way to describe models. This paper proposes an addition to the ODD protocol which allows the description of an agent-based model as a dynamical system, which provides access to computational and theoretical tools for its analysis. The mathematical framework is that of algebraic models, that is, time-discrete dynamical systems with algebraic structure. It is shown by way of several examples how this mathematical specification can help with model analysis. This mathematical framework can also accommodate other model types such as Boolean networks and the more general logical models, as well as Petri nets.Bulletin of Mathematical Biology 09/2010; 73(7):1583-602. · 2.02 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Analysis and synthesis techniques for a class of sequential discrete-state networks are discussed. These networks, made up of arbitrary interconnections of unit-delay elements (or of trigger flip-flops), modulo-p adders, and scalar multipliers (modulo alpha , prime p ), are of importance in unconventional radar and communication systems, in automatic error-correction circuits, and in the control circuits of digital computers. In addition, these networks are of theoretical significance to the study of more general sequential networks. The basic problem with which this paper is concerned is that of finding economical realizations of such networks for prescribed autonomous (excitation-free) behavior. To this end, an analytical-algebraic model is described which permits the investigation of the relation between network logical structure and state-sequential behavior. This relation is studied in detail for nonsingular networks (those with purely cyclic behavior). Among the results of this investigation is the establishment of relations between the state diagram of the network and a characteristic polynomial derived from its logical structure, An operation of multiplication of state diagrams is shown to correspond to multiplication of the corresponding polynomials. A criterion is established for the realizability of prescribed cyclic behavior by means of linear autonomous sequential networks. An effective procedure for the economical realization of such networks is described, and it is shown that linear feedback shift registers constitute a canonical class of realizations. Examples are given of the realization procedure. The problem of synthesis with only one-cycle length specified is also discussed. A partial solution is obtained to this "don't care" problem. Some special families of feedback shift registers are investigated in detail, and the state-diagram structures are obtained for an arbitrary number of stages and an arbitrary (prime) modulus. Mathematical appendixes are included which summarize the pertinent results in Galois field theory and in the factorization of cyclotomic polynomials into irreducible factors over a modular field. The relation of the theory developed in this paper to Huffman's description of linear sequence tr- ansducers in terms of the D operator is discussed, as well as unsolved problems and directions for further generalization.IRE Transactions on Circuit Theory 04/1959;

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