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Parallel web automata (PWAÂ¿a generalization of cellular arrays to arbitrary graph structuresÂ¿are defined. It is shown that such automata make poor acceptors (i.e., they are not good recognizers of their own graph structures). However, they can be used to perform parallel local pattern matching (with respect to patterns of bounded size) on themselves, so that they have potential usefulness as models for ``semantic memory.'' They can also be used to find greatest compatible sets of graph labelings subject to given local constraints; this problem arises, for example, in the analysis of scenes containing three-dimensional objects.