First-Order Queries over One Unary Function

This paper investigates the complexity of query problem for first-order formulas on quasi-unary signatures, that is, on vocabularies made of a single unary function and any number of monadic predicates. We first prove a form of quantifier elimination result: any query defined by a quasi-unary first-order formula can be equivalently defined, up to a suitable linear-time reduction, by a quantifier-free formula. We then strengthen this result by showing that first-order queries on quasi-unary signatures can be computed with constant delay i.e. by an algorithm that has a precomputation part whose complexity is linear in the size of the structure followed by an enumeration of all solutions (i.e. the tuples that satisfy the formula) with a constant delay (i.e. depending on the formula size only) between each solution. Among other things, this reproves (see [7]) that such queries can be computed in total time f(|φ|).(|S| + |φ(S)|) where S is the structure, φ is the formula, φ(S) is the result of the query and f is some fixed function. The main method of this paper involves basic combinatorics and can be easily automatized. Also, since a forest of (colored) unranked tree is a quasi-unary structure, all our results apply immediately to queries over that later kind of structures. Finally, we investigate the special case of conjunctive queries over quasiunary structures and show that their combined complexity is not prohibitive, even from a dynamical (enumeration) point of view.