Ekspans

Ekspans is a formal operation used in string processing and grammar design to enlarge a base sequence by inserting predefined material after each element. It is used to model augmentation, scaffolding in data representations, and certain kinds of derivations in grammar formalisms.

Let S = s1 s2 ... sn be a sequence over an alphabet Σ. Let R be a function that assigns to each symbol a in Σ a (possibly empty) insertion string R(a) in Σ*. The ekspansion of S under R, denoted E_R(S), is the concatenation s1 R(s1) s2 R(s2) ... sn R(sn). In deterministic ekspans, R maps each symbol to a single string; in nondeterministic ekspans, R(a) can be chosen from a set, producing multiple possible results.

Ekspans preserves the left-to-right order of the original symbols, and the length of E_R(S) is |S| plus the total length of the inserted strings. It is monotone with respect to sequence inclusion: extending S can yield extended expansions, subject to the same rules. Variants may differ in whether insertions depend on context.

Example: S = [A, B, C], R(A) = "x", R(B) = "", R(C) = "yz". Then E_R(S) = "AxBCyz".

Applications include data augmentation for natural language generation, scaffolding in code and data encoding, and studies in formal language theory regarding how local insertions affect global structure.

See also: expansion, morphism, string rewriting, grammar-based generation.