Hierarchies in inclusion logic with lax semantics

We study the expressive power of fragments of inclusion logic under the so-called lax team semantics. The fragments are defined either by restricting the number of universal quantifiers or the arity of inclusion atoms in formulae. In case of universal quantifiers, the corresponding hierarchy collapses at the first level. Arity hierarchy is shown to be strict by relating the question to the study of arity hierarchies in fixed point logics.

💡 Research Summary

This paper investigates the expressive power of two syntactic fragments of inclusion logic (FO(⊆)) when interpreted under the lax team semantics, a variant of team semantics that allows non‑empty sets of assignments to be used for existential quantification and disjunction. The two fragments are defined by (i) limiting the number of universal quantifiers that may appear in a formula, denoted FO(⊆)(k ∀), and (ii) limiting the arity of inclusion atoms (i.e., the length of the tuples involved in x ⊆ y), denoted FO(⊆)(k‑inc).

The authors first recall that under lax semantics FO(⊆) is equivalent to greatest‑fixed‑point logic (GFP), which captures PTIME on finite ordered structures. They also note that, unlike dependence logic, inclusion logic is not downward closed, so the choice between strict and lax semantics matters for its expressive power.

Collapse of the universal‑quantifier hierarchy.
Using a normal‑form transformation (Theorem 3) any FO(⊆) sentence can be written as Q₁x₁ … Qₙxₙ θ where θ is quantifier‑free. The authors then construct, for each such sentence, an equivalent formula that contains at most one universal quantifier. The construction replaces every universal quantifier Qᵢ = ∀ with an existential block followed by a single universal quantifier over a fresh variable y, together with inclusion atoms that enforce the same constraints that the original universal quantifier imposed. The key technical observation is that, because lax semantics permits the team to be expanded arbitrarily for existential quantifiers, the inclusion atoms can simulate the effect of the removed universal quantifiers. Consequently, FO(⊆)(1 ∀) already captures the full logic FO(⊆); the hierarchy collapses at level 1. This result mirrors earlier findings for dependence and independence logics, but the proof here is tailored to the inclusion‑atom setting.

Strictness of the arity hierarchy.
For the arity‑restricted fragments the situation is dramatically different. The authors prove that for every k ≥ 2, FO(⊆)(k‑inc) is strictly more expressive than FO(⊆)(k‑1‑inc). The proof proceeds by encoding a graph property that requires a k‑ary inclusion atom to be expressed. They define a first‑order formula EDGEₖ(x, y) that asserts that two k‑tuples x and y together induce a 2k‑clique in a graph. Using this, they consider the transitive closure TCₖ of EDGEₖ, written as