Journal of Statistical Mechanics: Theory and Experiment

Information Processing Letters, 2011

We consider the relationship between size and depth for layered Boolean circuits, synchronous circuits and planar circuits as well as classes of circuits with small separators. In particular, we show that every layered Boolean circuit of size s can be simulated by a layered Boolean circuit of depth O(√ s log s). For planar circuits and synchronous circuits of size s, we obtain simulations of depth O(√ s). The best known result so far was by Paterson and Valiant [16], and Dymond and Tompa [6], which holds for general Boolean circuits and states that D(f) = O(C(f)/ log C(f)), where C(f) and D(f) are the minimum size and depth, respectively, of Boolean circuits computing f. The proof of our main result uses an adaptive strategy based on the two-person pebble game introduced by Dymond and Tompa [6]. Improving any of our results by polylog factors would immediately improve the bounds for general circuits.

2011

The multi-scale strategy in studying biological regulatory networks analysis is based on two level of analysis. The first level is structural and consists in examining the architecture of the interaction graph underlying the network and the second level is functional and analyse the regulatory properties of the network. We apply this dual approach to the "immunetworks" involved in the control of the immune system. As a result, we show that the small number of attractors of these networks is due to the presence of intersecting circuits in their interaction graphs. We obtain an upper bound of the number of attractors of the whole network by multiplying the number of attractors of each of its strongly connected components. We detect first the strongly connected components in the architecture of the interaction digraph of the network. Secondly, we study the dynamical function of the attractors by looking further inside these components, notably when they form circuits (intersecting or not).

Arxiv preprint nlin/0408006, 2004

Bulletin of Mathematical Biology, 2013

It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states J.-P. Comet () • M. Noual • A. Richard Lab.

IEEE International Conference on Neural Networks, 1988

The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled "Dynamics of Random Boolean Networks Governed by a Generalization of Rule 22 of Elementary Cellular Automata" by Gary L. Beck in partial fulfillment of the requirements for the degree

Arxiv preprint adap-org/9909004, 1999

During the last few years an area of active research in the field of complex systems is that of their information storing and processing abilities. Common opinion has it that the most interesting beaviour of these systems is found "at the edge of chaos", which would seem to suggest that complex systems may have inherently non-trivial information proccesing abilities in the vicinity of sharp phase transitions. A comprenhensive, quantitative understanding of why this is the case is however still lacking. Indeed, even "experimental" (i.e., often numerical) evidence that this is so has been questioned for a number of systems. In this paper we will investigate, both numerically and analitically, the behavior of Random Boolean Networks (RBN's) as they undergo their order-disorder phase transition. We will use a simple mean field approximation to treat the problem, and without lack of generality we will concentrate on a particular value for the connectivity of the system. In spite of the simplicity of our arguments, we will be able to reproduce analitically the amount of mutual information contained in the system as measured from numerical simulations.

2012

Reservoir Computing (RC) is a computational model in which a trained readout layer interprets the dynamics of a component called a reservoir that is excited by external input stimuli. The reservoir is often constructed using homogeneous neural networks in which a neuron's in-degree distributions as well as its functions are uniform. RC lends itself to computing with physical and biological systems. However, most such systems are not homogeneous. In this paper, we use Random Boolean Networks (RBN) to build the reservoir. We explore the computational capabilities of such a RC device using the temporal parity task and the temporal density classification. We study the sufficient dynamics of RBNs using kernel quality and generalization rank measures. We verify findings by that the critical connectivity of RBNs optimizes the balance between the high memory capacity of RBNs with K < 2 and the higher information processing of RBNs with K > 2. We show that in a RBN-based RC system, the optimal connectivity for the parity task, a processing intensive task, and the density classification task, a memory intensive task, agree with Lizier et al.'s theoretical results. Our findings may contribute to the development of optimal selfassembled nanoelectronic computer architectures and biologically-inspired computing paradigms.

Physical Review Letters, 2009

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. This framework paves the way for obtaining new results on error-rates, function-depth and sensitivity, and their dependence on the gate-type and noise model used.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010

Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors and are closer to criticality when chaotic dynamics would be expected, compared to classical RBNs.

Journal of Physics A: …, 2005

Several order parameters have been considered to predict and characterize the transition between ordered and disordered phases in random Boolean networks, such as the Hamming distance between replicas or the stable core, which have been successfully used. In this work, we propose a natural and clear new order parameter: the temporal variance. We compute its value analytically and compare it with the results of numerical experiments. Finally, we propose a complexity measure based on the compromise between temporal and spatial variances. This new order parameter and its related complexity measure can be easily applied to other complex systems.

Information Processing Letters, 2004

From a theorem of Markov, the minimum number of negation gates in a circuit sufficient to compute any collection of Boolean functions on n variable is ℓ = ⌈log(n + 1)⌉. Santha and Wilson [SIAM Journal of Computing 22 : 294-302 (1993)] showed that in some classes of bounded-depth circuits ℓ negation gates are no longer sufficient for some explicitly defined Boolean function. In this paper, we consider a general class of bounded-depth circuits in which each gate computes an arbitrary monotone Boolean function or its negation. Our purpose is to extend the theorem of Markov for such a general class of circuits. We first show that a lower bound shown by Santha and Wilson becomes an extension of Markov's lower bound by a small refinement. Then, we present tight upper bounds on the number of negations for computing an arbitrary collection of Boolean functions.

2021

Lucas Kluge, 2 Joshua E. S. Socolar, and Eckehard Schöll 1 Potsdam Insitute for Climate Impact Research, Telegrafenberg, 14473 Potsdam, Germany Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany∗ Department of Physics, Duke University, Durham, NC, 27708, USA Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

Theory of Computing Systems / Mathematical Systems Theory, 2007

Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, what Boolean functions can be defined depends on these gate functions: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes have been known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover, we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f, but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f, but still know that it is from A? For nearly all possible combinations, we show that this is not the case, and that the problem is either in P or coNP-complete.

Discrete Applied Mathematics, 2012

In line with fields of theoretical computer science and biology that study Boolean automata networks to model regulation networks, we present some results concerning the dynamics of networks whose underlying structures are oriented cycles, that is, Boolean automata circuits. In the context of biological regulation, former studies have highlighted the importance of circuits on the asymptotic dynamical behaviour of the biological networks that contain them. Our work focuses on the number of attractors of Boolean automata circuits whose elements are updated in parallel. In particular, we give the exact value of the total number of attractors of a circuit of arbitrary size n as well as, for every positive integer p, the number of its attractors of period p depending on whether the circuit has an even or an odd number of inhibitions. As a consequence, we obtain that both numbers depend only on the parity of the number of inhibitions and not on their distribution along the circuit. We also relate the counting of attractors of Boolean automata circuits to other known combinatorial problems and give intuition about how circuits interact by studying their dynamics when they intersect one another in one point.