Lcontaining

Lcontaining is a term used in formal language theory to describe strings that contain a substring belonging to a given language L.

Definition: Let L be a language over an alphabet Σ. A word w ∈ Σ* is called L-containing if there exists a substring s of w such that s ∈ L. Equivalently, there exist indices i and j with 1 ≤ i ≤ j ≤ |w| such that w[i..j] ∈ L.

Examples: If L = { "ab", "ba" } over Σ = {a, b}, then w = "cabd" is L-containing because it contains the substring "ab". A word like "cdef" is not L-containing with this L.

Checking and representations: For finite L, checking whether a word is L-containing reduces to searching for any member of L as a substring of w. For infinite L, the method depends on how L is represented (automaton, regular expression, or grammar). In practice, automata-based approaches (such as building an automaton that accepts all strings containing a member of L) are used.

Variants and related concepts: A related notion is L-free, where a word contains no substring that belongs to L. L-containing words are closed upwards under string extension: any superstring of an L-containing word remains L-containing, since the original substring still appears. Variants may impose constraints on the position or length of the containing substring or consider prefixes, suffixes, or specific subword relations.

Applications: L-containing concepts appear in pattern matching, text processing, pattern avoidance studies, and analyses of subword complexity within formal languages.