Matching and Mediation of Heterogeneous Context Information

Abstract

With the further emergence of mobile and ubiquitous computing, also the number of context-aware applications grows. Often these applications require similar types of context information (e.g. a position or the current activity). An aggravating factor is that the number of sources for context information (e.g. sensors, reasoners, or databases) grows and the requirements on these sources and purposes for using the information are different. As a result, the level of heterogeneity of these sources and the provided information significantly increases. In this article, we present a system for matching and mediating context offers of loosely coupled context sources while taking into account the requirements of context-aware applications expressed in form of context requests. The matching and mediation process allows an autonomous establishment of mediation processes in order to transfer information from an offered representation into a requested representation and thus to overcome heterogeneous context information. [The dissertation of the first author contains a more comprehensive discussion of the subject, see (Wagner, Context as a service, Ph.D. thesis, University of Kassel, FB 16: Elektrotechnik/Informatik, Distributed Systems Group, 2013.)]

Appendix: Analytic Tableaux Method for Propositional Logic

The analytic tableaux (or semantic tableaux) method is a decision procedure (a method for solving a decision problem) for propositional logic and a proof procedure (a systematic method for producing proofs) for formulae of first-order logic. The tableaux method can also determine the satisfiability of finite sets of formulae of various logics. In this section we present the analytic tableaux method as introduced by Smullyan [16].

Definition 1 (Analytic Tableau).

An analytic tableau for X is an ordered dyadic tree, whose points are (occurrences of) formulae, which is constructed as follows: we start by placing X in the origin. Now suppose $\mathcal{T}$ is a tableau for X, which has already been constructed; let Y be an end point. Then we may extend $\mathcal{T}$ by either of the following two operations:

• A) If some α, as depicted in the first column of the following table, occurs on the path P _Y , then we may adjoin α ₁ or α ₂ as the sole successor of Y.

  Table 1

• B) If some β, as depicted in the first column of the following table, occurs on the path P _Y , then we may simultaneously adjoin β ₁ as the left successor of Y and β ₂ as the right successor of Y.

  Table 2

The above inductive definition of tableaux for X can be made explicitly as follows: given two ordered dyadic trees $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, whose points are occurrences of formulae, we call $\mathcal{T}_{2}$ a direct extension of $\mathcal{T}_{1}$ if $\mathcal{T}_{2}$ can be obtained from $\mathcal{T}_{1}$ by one application of the operation (A) or (B) above. Then $\mathcal{T}$ is a tableau for X iff there exists a finite sequence $(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{n} = \mathcal{T} )$ such that $\mathcal{T}_{1}$ is a 1-point tree whose origin is X and such that for each i < n, $\mathcal{T}_{i+1}$ is a direct extension of $\mathcal{T}_{i}$. A branch θ of a tableau is closed if it contains some formula and its negation. And $\mathcal{T}$ is called closed if every branch of $\mathcal{T}$ is closed. A proof of a formula X means that the tableau for ¬X is closed.

Definition 2 (Complete Branch, Complete Tableau).

A branch θ of a tableau is complete if for every α which occurs in θ, both α ₁ and α ₂ occur in θ, and for every β which occurs in θ at least one of β ₁ and β ₂ occurs in θ. A tableau $\mathcal{T}$ is called completed if every branch of $\mathcal{T}$ is either closed or complete.

Theorem 3.

If $\mathcal{T}$ is any completed open tableau (open in the sense that at least one branch is not closed), the origin of $\mathcal{T}$ is satisfiable. (For proof see [16] .)