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We generalize the notion of self-similar groups of infinite treeautomorphisms to allow for groups which are defined on a tree but do not actfaithfully on it. The elements of such a group correspond to labeled treeswhich may be recognized by a tree automaton (e.g. Rabin, B\"{u}chi, etc.), orconsidered as elements of a tree shift (e.g. of finite type, sofic) as insymbolic dynamics. We give examples to show that the various classes ofself-similar groups defined in this way do not coincide. As the main result,extending the classical result of Kitchens on one-dimensional group shifts, weprovide a sufficient condition for a self-similar group whose elements form asofic tree shift to be a tree shift of finite type. As an application, we showthat the closure of certain self-similar groups of tree automorphisms are notRabin-recognizable. \end{abstract}

A Language Hierarchy and Kitchens-Type Theorem for Self-Similar Groups