is a fractional i allowed in pumping lemma??

Question

I checked the pumping lemma in many books(introduction to the theory of computation Michael Sipser) and website(wikipedia). they all give the same explanation:(definition from introduction to the theory of computation, page 78)

Pumping lemma: If $A$ is a regular language, then there is a number $p$ (the pumping length) where if $s$ is any string in $A$ of length at least $p$, then $s$ may be divided into three pieces, $s = xyz$, satisfying the following conditions:

1. for each $i \ge 0$, $xy^iz \in A$,
2. $|y| > 0$, and
3. $|xy| \le p$.

I want to know, is it possible to choose a fractional value such as $\frac{1}{3}$ for $i$?? if it is possible, how to you interpret it??

Answer 1

No. In automata theory unless indicated otherwise all the values being dealt with are natural numbers. The $i$ indicates the length of repeated $y$ substrings and as it is a finite cardinality only natural numbers makes sense.

Answer 2

As a counterexample for one particular interpretation, consider an alphabet with a single symbol $a$, and the language consisting of those words that have even, positive length.

Suppose we have $x,y,z$ with the property that $x y^i z$ is in the language for all integers $i \geq 0$.

We automatically have $y = a^n$, where $n$ is the length of $y$, so we might interpret $y^{1/n} = a$.

But then $xy^{1/n}z$ is not in the language, since it has odd length.