# Information algebra

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 Template:Harv are two-sorted algebras , where is a semigroup, representing combination or aggregation of information, 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 , the following operations are defined

Additionally, in the usual lattice operations (meet and join) are defined.

## Axioms and definition

The axioms of the two-sorted algebra , in addition to the axioms of the lattice :

A two-sorted algebra 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 is less informative than if it adds no new information to . The semigroup is a semilattice relative to this order, i.e. . Relative to any domain (question) a partial order can be introduced by defining if . It represents the order of information content of and relative to the domain (question) .

## Labeled information algebra

The pairs , where and such that form a **labeled Information Algebra**. More precisely, in the two-sorted algebra , the following operations are defined

## Models of information algebras

Here follows an incomplete list of instances of information algebras:

- Relational algebra: The reduct of a relational algebra with natural join as combination and the usual projection is a labeled information algebra, see Example.
- Constraint systems: Constraints form an information algebra Template:Harv.
- Semiring valued algebras: C-Semirings induce information algebras Template:Harv;Template:Harv;Template:Harv.
- Logic: Many logic systems induce information algebras Template:Harv. Reducts of cylindric algebras Template:Harv or polyadic algebras are information algebras related to predicate logic Template:Harv.
- Module algebras: Template:Harv;Template:Harv.
- Linear systems: Systems of linear equations or linear inequalities induce information algebras Template:Harv.

### Worked-out example: relational algebra

Template:Cleanup section
Let be a set of symbols, called *attributes* (or *column
names*). For each let be a non-empty set, the
set of all possible values of the attribute . For example, if
, then could
be the set of strings, whereas and are both
the set of non-negative integers.

Let . An *-tuple* is a function so that
and for each The set
of all -tuples is denoted by . For an -tuple and a subset
the restriction is defined to be the
-tuple so that for all .

A *relation over * is a set of -tuples, i.e. a subset of .
The set of attributes is called the *domain* of and denoted by
. For the *projection* of onto is defined
as follows:

The *join* of a relation over and a relation over is
defined as follows:

As an example, let and be the following relations:

A relational database with natural join as combination and the usual projection 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:

- semigroup
- and
- transitivity
- If , then .
- combination
- If and , then .
- idempotency
- If , then .
- support
- If , then .

## Connections

- Valuation algebras
- Dropping the idempotency axiom leads to valuation algebras. These axioms have been introduced by Template:Harv to generalize
*local computation schemes*Template:Harv from Bayesian networks to more general formalisms, including belief function, possibility potentials, etc. Template:Harv. For a book-length exposition on the topic see Template:Harvtxt. - Domains and information systems
*Compact Information Algebras*Template:Harv are related to Scott domains and Scott information systems Template:Harv;Template:Harv;Template:Harv.- Uncertain information
- Random variables with values in information algebras represent
*probabilistic argumentation systems*Template:Harv. - Semantic information
- Information algebras introduce semantics by relating information to questions through focusing and combination Template:Harv;Template:Harv.
- Information flow
- Information algebras are related to information flow, in particular classifications Template:Harv.
- Tree decomposition
- ...
- Semigroup theory
- ...

## Historical Roots

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

## References

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