DEPENDENCE LOGIC IN PREGEOMETRIES ANDω-STABLE THEORIES

Abstract

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

Independence logic and abstract independence relations

“…In [15], the cases of (in)dependence occurring in pregeometries and ω-stable theories were also shown to be instances of this theory. We now generalize these results to the other cases of independence listed in the above list.…”

“…In the following three important examples of federated independent relations. Example 3.7 (vector spaces; [15]) Let VS inf K denote the theory of infinite vector spaces over a fixed field K. Let : M → M be such that A → A , i.e., the linear span of A, then is a pregeometric operator. Notice that the theory VS inf K is superstable (if K is countable it actually is ω-stable) and strongly minimal.…”

“…Example 3.8 (algebraically closed fields; [15]) Let ACF p denote the theory of algebraically closed fields of characteristic p, where p is either 0 or a prime. Let acl : M → M be such that A → acl(A), i.e., the algebraic closure of A, then acl is a pregeometric operator.…”

A finite axiomatization of conditional independence and inclusion dependencies

“…Then we derive p u r w ⊆ pr (26) from p u ≡ p w and (25) by Identity Rule. By Chase Rule we then derive p u q u r w ⊆ pqr (27) from the assumption q ⊥ p r, (24) and (26). Now it can be the case that x i ∈ pq and x i ∈ r, but pr i (x u ) = pr i (x w ).…”

“…For example, Arrow's theorem from social choice theory has recently been axiomatized and formally proved in this framework [25]. Also, the pure independence atom y⊥z and its axioms has various concrete interpretations such as independence X ⊥ ⊥ Y between two sets of random variables [13], and independence in vector spaces and algebraically closed fields [26].…”