First-Order Queries over One Unary Function

Abstract

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 quasi-unary structures and show that their combined complexity is not prohibitive, even from a dynamical (enumeration) point of view.