On the Complexity of Existential Positive Queries

TL;DR

This paper analyzes the computational complexity of model checking existential positive first-order logic, providing a classification of tractable and intractable cases, and establishing lower bounds on translation sizes.

Contribution

It introduces a general theorem linking the complexity of existential positive logic to primitive positive logic, and explores fixed-parameter tractability and translation lower bounds.

Findings

Existential positive sentence sets with bounded arity are fixed-parameter tractable iff equivalent to bounded-variable logic.

Some fixed-parameter tractable sets can be NP-complete.

Superpolynomial lower bounds are established for translation lengths to bounded-variable logic.

Abstract

We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted to fall into the set; a natural question is then to classify which sentence sets are tractable and which are intractable. With respect to fixed-parameter tractability, we give a general theorem that reduces this classification question to the corresponding question for primitive positive logic, for a variety of representations of structures. This general theorem allows us to deduce that an existential positive sentence set having bounded arity is fixed-parameter tractable if and only if each sentence is equivalent to one in bounded-variable logic. We then use the lens of classical complexity to study these fixed-parameter tractable sentence sets. We show that such a set can be NP-complete, and consider the length needed by a translation from sentences in such a set to bounded-variable logic; we prove superpolynomial lower bounds on this length using the theory of compilability, obtaining an interesting type of formula size lower bound. Overall, the tools, concepts, and results of this article set the stage for the future consideration of the complexity of model checking on more expressive logics.