Pumping Lemma For Regular Languages - Formal Statement

Formal Statement

Let L be a regular language. Then there exists an integer p ≥ 1 depending only on L such that every string w in L of length at least p (p is called the "pumping length") can be written as w = xyz (i.e., w can be divided into three substrings), satisfying the following conditions:

  1. |y| ≥ 1;
  2. |xy| ≤ p
  3. for all i ≥ 0, xyiz ∈ L

y is the substring that can be pumped (removed or repeated any number of times, and the resulting string is always in L). (1) means the loop y to be pumped must be of length at least one; (2) means the loop must occur within the first p characters. |x| must be smaller than p (conclusion of (1) and (2)), apart from that there is no restriction on x and z.

In simple words, for any regular language L, any sufficiently long word w (in L) can be split into 3 parts. i.e. w = xyz, such that all the strings xykz for k≥0 are also in L.

Below is a formal expression of the Pumping Lemma.

\begin{array}{l} (\forall L\subseteq \Sigma^*) \\ \quad (\mbox{regular}(L) \Rightarrow \\ \quad ((\exists p\geq 1) ( (\forall w\in L) ((|w|\geq p) \Rightarrow \\ \quad\quad ((\exists x,y,z \in \Sigma^*) (w=xyz \land (|y|\geq 1 \land |xy|\leq p \land (\forall i\geq 0)(xy^iz\in L)))))))) \end{array}

Read more about this topic:  Pumping Lemma For Regular Languages

Famous quotes containing the words formal and/or statement:

    “Good gentlemen, look fresh and merrily.
    Let not our looks put on our purposes,
    But bear it as our Roman actors do,
    With untired spirits and formal constancy.”
    —William Shakespeare (1564–1616)

    “He that writes to himself writes to an eternal public. That statement only is fit to be made public, which you have come at in attempting to satisfy your own curiosity.”
    —Ralph Waldo Emerson (1803–1882)