The logic of pumping lemma is an example of __________ .

Explanation:

The pumping lemma for regular languages uses a counting argument about a finite number of states, which is an application of the pigeonhole principle. A concise proof sketch:

1. Assume a language is regular, so there exists a deterministic finite automaton (DFA) with p states recognizing it.

2. Take any string in the language of length at least p + 1. As the DFA reads the first p + 1 symbols, it visits p + 1 positions but only p states.

3. By the pigeonhole principle, two of those positions must correspond to the same DFA state. This gives a decomposition of the string into xyz where the middle part y corresponds to the loop between the repeated states and y is nonempty.

4. Because the DFA can traverse that loop any number of times and still end in the same state, the string segments x y^i z are also accepted for all i ≥ 0. This is the pumping property.

Thus, the key logical tool is the pigeonhole principle (finite states forcing a repeated state), which leads to the pumping behavior.