Dependence Logic in Algebra and Model Theory

Systems In this chapter we define the abstract versions of the systems that we are going to deal with for the rest of the thesis. They qualify as abstract because their semantics intend to give an abstract account of the concepts of dependence and independence, and do not refer to any particular version of these notions. The variants of dependence and independence that are conceivable are several. In our analysis we will focus on the following: y is dependent on x, x is independent, x is independent from y, x is independent over z, and x is independent from y over z. To these variants of the notions we associate respectively the following atoms: =(x, y), ⊥(x), x ⊥ y, ⊥z(x) and x ⊥z y. For completeness of the study that we develop in Chapter 3 we also consider the atom x = y. All the atoms except for ⊥(x) and ⊥z(x) are present in the literature; these two have been introduced by the author. Each kind of atom gives rise to an atomic language, and for each atomic language we define a team semantics and a deductive system. The resulting systems are: Atomic Equational Logic, Atomic Dependence Logic, Atomic Absolute Independence Logic, Atomic Independence Logic, Atomic Absolute Conditional Independence Logic and Atomic Conditional Independence Logic. The first four admit finite complete axiomatizations while the last one does not. Indeed, Parker and Parsaye-Ghomi [32] proved that it is not possible to find a finite complete axiomatization for the conditional independence atoms. Furthermore, in [23] and [22] Hermann proved that the consequence relation between these atoms is undecidable. However, in [31] Naumov and Nicholls developed a recursively enumerable axiomatization of them. 2.1 Atomic Equational Logic In this section we define the system Atomic Equational Logic (AEL).