Question

Suppose is a regular language and let be a finite automaton that recognizes . which of the following statements justifies the existence of a finite pumping length with respect to ?

a. length need not be finite.
b. all strings in are of finite length.
c. length exists since must include a directed cycle.
d. the pigeon-hole principle demonstrates that is a candidate for

Answer 1

The existence of a finite pumping length for a regular language recognized by a finite automaton is justified by the fact that the automaton must have a directed cycle, as stated by the Pumping Lemma for regular languages. option c. length exists since must include a directed cycle is the correct answer.

Step-by-step explanation:

The question is asking to justify the existence of a finite pumping length for a regular language recognized by a finite automaton. The correct statement that justifies the existence of a finite pumping length is c. length exists since must include a directed cycle. The Pumping Lemma for regular languages states that for any regular language L there exists some integer p (the pumping length) such that any string s in L of length at least p can be divided into three parts, s = xyz, satisfying the conditions:

• |y| > 0
• |xy| ≤ p
• For all i ≥ 0, the string xy^iz belongs in L

This lemma takes advantage of the fact that a finite automaton has a limited number of states, and any sufficiently long string must revisit some state due to the pigeon-hole principle, forming a cycle that can be pumped. Hence, the pumping length corresponds to the number of states in the automaton or less.