Propositional logics of dependence

Abstract

In this paper, we study logics of dependence on the propositional level. We prove that several interesting propositional logics of dependence, including propositional dependence logic, propositional intuitionistic dependence logic as well as propositional inquisitive logic, are expressively complete and have disjunctive or conjunctive normal forms. We provide deduction systems and prove the completeness theorems for these logics.

A finite axiomatization of conditional independence and inclusion dependencies

“…(30) Let s list the variables in x \ pqr. From the inclusion atom in (30) we derive by Inclusion Introduction p u q u r * s * ⊆ pqrs (31) where s * is a sequence of pairwise distinct new variables. Then p u q u r * s * has repetitions at least where pqrs has, and hence we can define x v new as the sequence of length |x| where …”

“…For instance, join dependencies and functional dependencies both belong to the class of typed dependencies (TDs) that enjoys chase-based complete axiomatizations [3,4]. In particular, TDs have a representation in algebraic terms that bears a striking similarity to the approach of this article [32,1]. A project-join (PJ) expression is an algebraic expression using projection, natural join and inclusion as its building blocks.…”

“…Furthermore, the set of valid formulas of dependence logic has the same complexity as that of full second-order logic, hence it is not possible to give a complete axiomatization of dependence logic [29]. However, by restricting attention to syntactic fragments [31,15,20] or by modifying the semantics [9] complete axiomatizations have recently been obtained. The axiomatization presented in this article is based on the classical characterization of logical implication between dependencies in terms of the Chase procedure [22].…”

Logics for First-Order Team Properties

“…The invertible rule ∀ 1 Ext is an adaption of a similar rule in the system of D in [27], and it is inspired also by a similar equivalence given in [14]. The rules =(·)wI and =(·)wE for dependence atoms were introduced in [34] in the propositional context.…”

Complexity Thresholds in Inclusion Logic

“…One motivation behind dependence logic is to find a unifying logical framework for analyzing dependency notions from different contexts. Since its introduction, versions of dependence logic have been formulated and investigated in a variety of logical environments, including propositional logic [15,28,30], modal logic [7,26], probabilistic logics [5], and two-variable logics [21]. Recent research has also pursued connections and applications of dependence logic to fields such as database theory [13,14], Bayesian networks [4], and social choice theory [23].…”