nietequivalent

Nietequivalent is a term encountered in some theoretical and applied discourses to describe the degree to which two objects fail to be equivalent under a specified equivalence relation. It is not a standard or universally defined concept in mainstream mathematics or computer science, and its exact meaning varies with context and author.

Formal idea. Let X be a set and ~ an equivalence relation on X. Suppose there is a allowed set of transformations or operations that can be applied to objects in X, each with a nonnegative cost. The nietequivalent between a and b, denoted N(a,b), is defined as the minimum total cost of a sequence of allowed transformations that makes a equivalent to b (or to some object equivalent to b). If a and b are already equivalent, N(a,b) = 0; if no transforming sequence exists within the allowed operations, N(a,b) is taken to be infinite. In many treatments, N is linked to a similarity measure S(a,b) in [0,1] by N(a,b) = 1 − S(a,b), so higher similarity corresponds to lower nietequivalence. The exact construction therefore depends on the chosen operations, invariants, and cost function.

Applications and scope. The concept is used informally to compare objects in fields such as graph theory, data deduplication, and formal languages, where one wants a quantitative sense of non-equivalence or closeness to equivalence under a given theory. It can serve as a backdrop for algorithm design, for example in graph edit distance calculations or model comparison, where the goal is to minimize transformation cost while respecting structural invariants.

See also. Non-equivalence, equivalence relation, edit distance, graph edit distance, similarity measure.