Safe Dependency Atoms and Possibility Operators in Team Semantics

Abstract

I consider the question of which dependencies are safe for a Team Semantics-based logic FO(D), in the sense that they do not increase its expressive power over sentences when added to it. I show that some dependencies, like totality, non-constancy and non-emptiness, are safe for all logics FO(D), and that other dependencies, like constancy, are not safe for FO(D) for some choices of D despite being strongly first order (that is, safe for FO( / 0)). I furthermore show that the possibility operator ⋄φ , which holds in a team if and only if φ holds in some nonempty subteam, can be added to any logic FO(D) without increasing its expressive power over sentences.

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

“…2. As shown in [11] and recalled above, the constancy atom itself is not safe for all families of dependencies.…”

“…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.…”

“…Clearly, a dependency is strongly first order if and only if it is safe for the empty set of dependencies. However, as shown in [11], a strongly first order dependency is not necessarily safe for all families of dependencies: in particular, the constancy atom is not safe for the unary 11 inclusion atom v 1 ⊆ v 2 , in which v 1 and v 2 must be single variables (rather than tuples of variables). 12 On the other hand, in [12] it was shown that strongly first order dependencies are safe for any family of downwards closed dependencies: It is also worth mentioning here that the same notions of safety and strong first orderness can be easily generalized to operators.…”

“…12 On the other hand, in [12] it was shown that strongly first order dependencies are safe for any family of downwards closed dependencies: It is also worth mentioning here that the same notions of safety and strong first orderness can be easily generalized to operators. For example, in [11] it was shown that the possibility operator This is different from the disjunction ∨ of Definition 5: for example, a team X of the form {(v : 0, w :…”

Characterizing Downwards Closed, Strongly First-Order, Relativizable Dependencies

“…A property related to strong first orderness and introduced in the recent article [17] is safety. In brief, a dependency (or a set of dependencies) is safe for some logic if it can be added to it without increasing the expressive power (wrt sentences) of the resulting formalism.…”