Improving Connectionist Energy Minimization

Symmetric networks designed for energy minimization such as Boltzman machines and Hopfield nets are frequently investigated for use in optimization, constraint satisfaction and approximation of NP-hard problems. Nevertheless, finding a global solution (i.e., a global minimum for the energy function) is not guaranteed and even a local solution may take an exponential number of steps. We propose an improvement to the standard local activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm, called activate, is uniform and does not assume that the network is tree-like. It can identify tree-like subnetworks even in cyclic topologies (arbitrary networks) and avoid local minima along these trees. For acyclic networks, the algorithm is guaranteed to converge to a global minimum from any initial state of the system (self-stabilization) and remains correct under various types of schedulers. On the negative side, we show that in the presence of cycles, no uniform algorithm exists that guarantees optimality even under a sequential asynchronous scheduler. An asynchronous scheduler can activate only one unit at a time while a synchronous scheduler can activate any number of units in a single time step. In addition, no uniform algorithm exists to optimize even acyclic networks when the scheduler is synchronous. Finally, we show how the algorithm can be improved using the cycle-cutset scheme. The general algorithm, called activate-with-cutset, improves over activate and has some performance guarantees that are related to the size of the network's cycle-cutset.Comment: See http://www.jair.org/ for any accompanying file

Self-Stabilizing Wavelets and r-Hops Coordination

We introduce a simple tool called the wavelet (or, r-wavelet) scheme. Wavelets deals with coordination among processes which are at most r hops away of each other. We present a selfstabilizing solution for this scheme. Our solution requires no underlying structure and works in arbritrary anonymous networks, i.e., no process identifier is required. Moreover, our solution works under any (even unfair) daemon. Next, we use the wavelet scheme to design self-stabilizing layer clocks. We show that they provide an efficient device in the design of local coordination problems at distance r, i.e., r-barrier synchronization and r-local resource allocation (LRA) such as r-local mutual exclusion (LME), r-group mutual exclusion (GME), and r-Reader/Writers. Some solutions to the r-LRA problem (e.g., r-LME) also provide transformers to transform algorithms written assuming any r-central daemon into algorithms working with any distributed daemon

Synchronous Counting and Computational Algorithm Design

Consider a complete communication network on $n$ nodes, each of which is a state machine. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are "odd" and which are "even". We require that the solution is self-stabilising (reaching the correct operation from any initial state) and it tolerates $f$ Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms are expensive to implement in hardware: they require a source of random bits or a large number of states. This work consists of two parts. In the first part, we use computational techniques (often known as synthesis) to construct very compact deterministic algorithms for the first non-trivial case of $f = 1$. While no algorithm exists for $n < 4$, we show that as few as 3 states per node are sufficient for all values $n \ge 4$. Moreover, the problem cannot be solved with only 2 states per node for $n = 4$, but there is a 2-state solution for all values $n \ge 6$. In the second part, we develop and compare two different approaches for synthesising synchronous counting algorithms. Both approaches are based on casting the synthesis problem as a propositional satisfiability (SAT) problem and employing modern SAT-solvers. The difference lies in how to solve the SAT problem: either in a direct fashion, or incrementally within a counter-example guided abstraction refinement loop. Empirical results suggest that the former technique is more efficient if we want to synthesise time-optimal algorithms, while the latter technique discovers non-optimal algorithms more quickly.Comment: 35 pages, extended and revised versio

Reliability of Transcriptional Cycles and the Yeast Cell-Cycle Oscillator

A recently published transcriptional oscillator associated with the yeast cell cycle provides clues and raises questions about the mechanisms underlying autonomous cyclic processes in cells. Unlike other biological and synthetic oscillatory networks in the literature, this one does not seem to rely on a constitutive signal or positive auto-regulation, but rather to operate through stable transmission of a pulse on a slow positive feedback loop that determines its period. We construct a continuous-time Boolean model of this network, which permits the modeling of noise through small fluctuations in the timing of events, and show that it can sustain stable oscillations. Analysis of simpler network models shows how a few building blocks can be arranged to provide stability against fluctuations. Our findings suggest that the transcriptional oscillator in yeast belongs to a new class of biological oscillators

Self-stablizing cuts in synchronous networks

Consider a synchronized distributed system where each node can only observe the state of its neighbors. Such a system is called self-stabilizing if it reaches a stable global state in a finite number of rounds. Allowing two different states for each node induces a cut in the network graph. In each round, every node decides whether it is (locally) satisfied with the current cut. Afterwards all unsatisfied nodes change sides independently with a fixed probability p. Using different notions of satisfaction enables the computation of maximal and minimal cuts, respectively. We analyze the expected time until such cuts are reached on several graph classes and consider the impact of the parameter p and the initial cut