Feature Trees over Arbitrary Structures

This paper presents a family of rst order feature tree theories, indexed by the theory of the feature labels used to build the trees. A given feature label theory, which is required to carry an appropriate notion of sets, is conservatively extended to a theory of feature trees with the predicates xt]y (feature t leads from the root of tree x to the tree y), where we have to require t to be a ground term, and xt# (feature t is deened at the root of tree x). In the latter case, t might be a variable. Together with the notion of sets provided by the feature label theory, this yields a rst-class status of arities. We present a quantiier elimination procedure to reduce any sentence of the feature tree theory to an equivalent sentence of the feature label theory. Hence, if the feature label theory is decidable, the feature tree theory is too. If the feature label theory is the theory of innnitely many constants and nite sets over innnitely many constants, we obtain an extension of the feature theory CFT, giving rst-class status to arities. As an another application, we obtain decidability of the theory of feature trees, where the feature labels are words, and where the language includes the successor function on words, lexical comparison of words and rst-class status of arities.