![]() ![]() final section we indicate, by means of an example, how the method of producing lexicodes can be applied optimally to find anncodes. We also give a short proof of the basic properties of the previously known lexicodes, which are defined by means of an exponential algorithm, and are related to game theory. We show that annihilation games provide a potentially polynomial method for computing codes (anncodes). The "losing positions" of certain combinatorial games constitute linear error detecting and correcting codes. We also study whether the fixed-point operator preserves prefix closure, relative closure (also called Lm(G)-closure), and controllability. Fixed-point based characterization of all the above language classes is also given, and their closure under intersection/union is investigated. The computation of super/sublanguages and also computation of their upper/lower bounds has lead to the introduction of other classes of co-observable languages, namely, strongly C&P co-observable languages, strongly D&A co-observable languages, locally observable languages, and strongly locally observable languages. the class of co-observable languages is not closed under intersection/union, we provide upper/lower bound of the super/sublanguage formula we present. We also provide formulas for computing super/sublanguages for each of these classes. We present fixed-point based characterization of several classes of co-observable languages that are of interest in the context of decentralized supervisory control of discrete-event systems, including C&P ∨ D&A co-observable languages, C&P co-observable languages, and D&A co-observable languages. These notes for a series of lectures at the VI-th SERC Numerical Analysis Summer School, Leicester University, apply the principles of validated computatio. In contrast, validated computation ffl checks that the hypotheses of appropriate existence and uniqueness theorems are satisfied, ffl uses interval arithmetic with directed rounding to capture truncation and rounding errors in computation, and ffl organizes the computations to obtain as tight an enclosure of the answer as possible. analysis to help provide this insight, but even the best codes for computing approximate answers can be fooled. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is the answer?" Standard numerical analysis has developed techniques of forward and backward error. Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second.
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