Independence and Functional Dependence Relations on Secrets

We study logical principles connecting two relations: independence, which is known as nondeducibility in the study of information flow, and functional dependence. Two different epistemic interpretations for these relations are discussed: semantics of secrets and probabilistic semantics. A logical system sound and complete with respect to both of these semantics is introduced and is shown to be decidable. Introduction In this paper, we study the properties of interdependencies between pieces of information. We call these pieces secrets to emphasize the fact that they might be unknown to some parties. One of the simplest relations between two secrets is functional dependency. We denote it as a b. It means that the value of secret a reveals the value of secret b. This relation is reflexive and transitive. A more general and less trivial form of functional dependency is functional dependency between sets of secrets. If A and B are two sets of secrets, then A B means that, together, the values of all secrets in A reveal the values of all secrets in B. Armstrong (1974) presented the following sound and complete axiomatization of this relation: 1. Reflexivity: A B, if A ⊇ B, 2. Augmentation: A B → A,C B,C, 3. Transitivity: A B → (B C → A C), where here and everywhere below A,B denotes the union of sets A and B. The above axioms are known in database literature as Armstrong’s axioms (Garcia-Molina, Ullman, and Widom 2009, p. 81). Beeri, Fagin, and Howard (1977) suggested a variation of Armstrong’s axioms that describe properties of multi-valued dependency. Not all dependencies between two secrets are functional. For example, if secret a is a pair 〈x, y〉 and secret b is a pair 〈y, z〉, then there is an interdependency between these secrets in the sense that not every value of secret a is compatible with every value of secret b. However, neither a b nor b a is necessarily true. If there is no interdependency between two secrets, then we will say that the two secrets Copyright c © 2010, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. are independent. In other words, secrets a and b are independent if any possible value of secret a is compatible with any possible value of secret b. We denote this relation between two secrets by a ‖ b. This relation was introduced by Sutherland (1986) and is also known as nondeducibility in the study of information flow. Halpern and O’Neill (2008) proposed a closely related notion called f -secrecy. More and Naumov (2009b) gave a complete axiomatization of the independence relation if secrets are generated over a collaboration network with a fixed topology. Like functional dependence, independence also can be generalized to relate two sets of secrets. If A and B are two such sets, thenA ‖ B means that any consistent combination of values of secrets in A is compatible with any consistent combination of values of secrets in B. Note that “consistent combination” is an important condition here since some interdependencymay exist between secrets in setA even while the entire set of secrets A is independent from the secrets in set B. A sound and complete axiomatization of this independence relation between sets was given by More and Naumov (2009a): 1. Empty Set: ∅ ‖ A, 2. Monotonicity: A,B ‖ C → A ‖ C, 3. Symmetry: A ‖ B → B ‖ A, 4. Public Knowledge: A ‖ A → (B ‖ C → A,B ‖ C), 5. Exchange: A,B ‖ C → (A ‖ B → A ‖ B,C). Essentially the same axiomswere shown by Geiger, Paz, and Pearl (1991) to provide a complete axiomatization of the independence relation between random variables in probability theory. In this paper, we study properties that connect the independence ‖ and functional dependence relations. An example of such a property, which we call the Substitution Axiom, is a ‖ b → (b c → a ‖ c). Its soundness with respect to a formally-defined semantics of secrets is shown in Theorem 1. Themain focus of this work is the independence and functional dependence relations between single secrets, not sets of secrets, as the single-secret setting already provides a non-trivial system of properties. We describe a sound and complete axiomatization of these properties, prove the decidability of our logical system, and establish the indepen528 Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning (KR 2010)