The term "information algebra" refers to mathematical techniques of information processing. Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources and that it is therefore usually combined. It has furthermore been neglected in classical information theory that one wants to extract those parts out of a piece of information that are relevant to specific questions. A mathematical phrasing of these operations leads to an algebra of information, describing basic modes of information processing. Such an algebra involves several formalisms of computer science, which seem to be different on the surface: relational databases, multiple systems of formal logic or numerical problems of linear algebra. It allows the development of generic procedures of information processing and thus a unification of basic methods of computer science, in particular of distributed information processing. Information relates to precise questions, comes from different sources, must be aggregated, and can be focused on questions of interest. Starting from these considerations, information algebras are two-sorted algebras (\Phi,D): Where \Phi is a semigroup, representing combination or aggregation of information, and D is a lattice of domains (related to questions) whose partial order reflects the granularity of the domain or the question, and a mixed operation representing focusing or extraction of information.

Information and its operations

More precisely, in the two-sorted algebra (\Phi,D), the following operations are defined Additionally, in D the usual lattice operations (meet and join) are defined.

Axioms and definition

The axioms of the two-sorted algebra (\Phi,D), in addition to the axioms of the lattice D: A two-sorted algebra (\Phi,D) satisfying these axioms is called an Information Algebra.

Order of information

A partial order of information can be introduced by defining if. This means that \phi is less informative than \psi if it adds no new information to \psi. The semigroup \Phi is a semilattice relative to this order, i.e. . Relative to any domain (question) x \in D a partial order can be introduced by defining if. It represents the order of information content of \phi and \psi relative to the domain (question) x.

Labeled information algebra

The pairs (\phi,x) , where and x \in D such that form a labeled Information Algebra. More precisely, in the two-sorted algebra (\Phi,D) , the following operations are defined

Models of information algebras

Here follows an incomplete list of instances of information algebras:

Worked-out example: relational algebra

Let be a set of symbols, called attributes (or column names). For each let U_\alpha be a non-empty set, the set of all possible values of the attribute \alpha. For example, if, then could be the set of strings, whereas and are both the set of non-negative integers. Let. An x-tuple is a function f so that and for each \alpha\in x The set of all x-tuples is denoted by E_x. For an x-tuple f and a subset the restriction f[y] is defined to be the y-tuple g so that for all \alpha\in y. A relation R over x is a set of x-tuples, i.e. a subset of E_x. The set of attributes x is called the domain of R and denoted by d(R). For the projection of R onto y is defined as follows: The join of a relation R over x and a relation S over y is defined as follows: As an example, let R and S be the following relations: Then the join of R and S is: A relational database with natural join \bowtie as combination and the usual projection \pi is an information algebra. The operations are well defined since It is easy to see that relational databases satisfy the axioms of a labeled information algebra:

Connections

Historical Roots

The axioms for information algebras are derived from the axiom system proposed in (Shenoy and Shafer, 1990), see also (Shafer, 1991).