Abstract

this paper we define a sub-language of SnS that characterizes the local sets. The key insight behind this result is the fact that the local sets are characterized by "subtree substitution closure": A recognizable set is local iff it is closed under substitution of sub-trees at similarly labeled nodes. As a corollary, we get that the local sets are closed under intersection but not under union. This, of course, is in contrast to the class of context-free string languages which is closed under union but not intersection, and is in contrast to the recognizable sets as well which are closed under all Boolean operations (since the language of SnS is closed wrt the Boolean connectives). The corollary implies that any language characterizing the local sets will not be closed wrt disjunction. On the other hand, the language must support some disjunction---the labels on nodes must be contingent on the labels of the nodes that occur in the same local trees. Similarly, the language will need some form of universal quantification since the productions of a CFG specify conditions that hold for every local tree occurring in the set. But unrestricted quantification, even just first-order universal quantification, allows definition of non-local sets. For instance (8x; y)[(A(x)B(x)) (B(x)B(y) ! x ß y)] picks out our example non-local set. We need, then, a restricted form of quantification. We are left with a language, Loc, employing restricted quantification in the form of what are, in essence, modal operators for parent, child and left- and rightsibling. Nesting of these operators and of disjunction is restricted in such a way that labels on nodes can only depend on the labels of its parent, its children, or its siblings. We show that a set of finite trees with bounded branching is def...

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