Datalog-Expressibility for Monadic and Guarded Second-Order Logic

Abstract

We characterise the sentences in Monadic Second-order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class C of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all , k ∈ N, there exists a canonical Datalog program Π of width ( , k), that is, a Datalog program of width ( , k) which is sound for C (i.e., Π only derives the goal predicate on a finite structure A if A ∈ C) and with the property that Π derives the goal predicate whenever some Datalog program of width ( , k) which is sound for C derives the goal predicate. The same characterisations also hold for Guarded Second-order Logic (GSO), which properly extends MSO. To prove our results, we show that every class C in GSO whose complement is closed under homomorphisms is a finite union of constraint satisfaction problems (CSPs) of ω-categorical structures.