On Strongly First-Order Dependencies

Abstract

We prove that the expressive power of first-order logic with team semantics plus contradictory negation does not rise beyond that of first-order logic (with respect to sentences), and that the totality atoms of arity k + 1 are not definable in terms of the totality atoms of arity k. We furthermore prove that all first-order nullary and unary dependencies are strongly first order, in the sense that they do not increase the expressive power of first order logic if added to it.

Characterizing Strongly First Order Dependencies: The Non-Jumping Relativizable Case

“…are strongly first order, as they can be defined in terms of upwards closed first order dependencies and constancy atoms; and as mentioned in [10], the same type of argument can be used to show that all first order dependencies D(R) where R has arity one are also strongly first order. This led to the following Conjecture 1 ( [11]) Every strongly first order dependency D(R) is definable in terms of upwards closed dependencies and constancy atoms.…”

“…FO(D)) is equivalent to some first order sentence. 10 Additionally, it is clear that any dependency E that is definable in FO(D ↑ , =(·)), in the sense that there exists some formula φ (v) ∈ FO(D ↑ , =(·)) over the empty signature such that M |= X Ev ⇔ M |= X φ (v), is itself strongly first order. This can be used, as discussed in [9], to show that for instance the negated inclusion atoms…”

“…The choice of the variable v is of course irrelevant here, and we could have defined NE as a 0-ary dependency instead; but treating it as a 1-ary dependency is formally simpler 9. That is, M |= X =(v) iff for all s, s ′ ∈ X, s(v) = s ′ (v) 10. It is worth pointing out here that if D is strongly first order then it is first order in the sense of Definition 8, because (M, R) ∈ D ⇔ (M, R) |= ∀x(¬Rx ∨ (Rx ∧ Dx)).…”

Logics for First-Order Team Properties

“…In related previous work, it was shown in [10,13,28] that first-order logic extended with constancy atoms =(x) and FO extended with classical negation ∼ are both equivalent to FO over sentences, whereas on the level of formulas they are both strictly less expressive than FO, and thus fail to capture all first-order team properties. It was also illustrated in [24] that a certain simple disjunction of dependence atoms already defines an NP-complete team property.…”

Approximation and Dependence via Multiteam Semantics

“…In this article we establish that, in the multiteam semantics setting, independence atoms can be naturally interpreted exactly as statistical conditional independence. Probabilistic versions of dependence logic have been previously studied by Galliani and Mann [8,11].…”