Redundanttiset

Redundanttiset is a theoretical construct used in discussions of redundancy within collections of items. In this sense, a Redundanttiset is a finite multiset of elements drawn from a universe, augmented with a redundancy relation that marks some elements as redundant because their information is implied by others in the collection.

Formally, let U be a universe and S be a finite multiset of elements from U. A redundancy predicate R assigns to each element e in S a boolean value indicating whether e is redundant, with respect to a chosen irredundant core C subset of S. An element is redundant if its information is contained in the information carried by the nonredundant elements; the irredundant core is the smallest subcollection whose information equals that of S under a specified information operation (such as a closure operator). Under suitable conditions, for example when the information operation defines a matroid structure, this core is unique.

Properties of a Redundanttiset include a redundancy degree, defined as the number of redundant elements or a weighted sum, and recoverability, where S is recoverable from its irredundant core via the redundancy relations. Construction methods often involve duplicating elements or encoding schemes that preserve total information while increasing resilience to loss.

Example: a Redundanttiset S = {x, y, x', y'} where x' encodes x and y' encodes y. The irredundant core could be {x, y}, with {x', y'} providing redundancy.

Applications appear in data storage design, error-correcting codes, fault-tolerant systems, and theoretical studies of information redundancy and matroid theory. The concept is primarily used in thought experiments and hypothetical models rather than standard mathematical practice.

See also: redundancy, multiset, matroid, information theory, data integrity.