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Technical Report

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Technical Report Number



In this paper, we formalize the concept of tracking in a sensor network and develop a rigorous theory of {\em trackability} that investigates the rate of growth of the number of consistent tracks given a sequence of observations made by the sensor network. The phenomenon being tracked is modelled by a nondeterministic finite automaton and the sensor network is modelled by an observer capable of detecting events related, typically ambiguously, to the states of the underlying automaton. More formally, an input string, $Z^t$, of $t+1$ symbols (the sensor network observations) that is presented to a nondeterministic finite automaton, $M$, (the model) determines a set, ${\cal H}_M(Z^t)$, of state sequences (the tracks or hypotheses) that are capable of generating the input string $Z^t$. We study the growth of the size of this set, $|{\cal H}_M(Z^t)|$, as a function of the length of the input string, $t+1$. Our main result is that for a given automaton and sensor coverage, the worst-case rate of growth is either polynomial or exponential in $t$, indicating a kind of phase transition in tracking accuracy. The techniques we use include the Joint Spectral Radius, $\rho(\Sigma)$, of a finite set, $\Sigma$, of $(0,1)$-matrices derived from $M$. Specifically, we construct a set of matrices, $\Sigma$, corresponding to $M$ with the property that $\rho(\Sigma) \leq 1$ if and only if $|{\cal H}_M(Z^t)|$ grows polynomially in $t$. We also prove that for $(0,1)$-matrices, the decision problem $\rho(\Sigma)\leq 1$ is Turing decidable and, therefore, so is the problem of deciding whether worst case state sequence growth for a given automaton is polynomial or exponential. These results have applications in sensor networks, computer network security and autonomic computing as well as various tracking problems of recent interest involving detecting phenomena using noisy observations of hidden states.