For a language whose strings never exceed length x, what is the smallest…

Concept

The pumping lemma for regular languages guarantees that once a string reaches length p (the pumping length) or more, that string can be decomposed and its middle segment repeated to produce further strings in the language. When the strings under consideration never exceed length x, this guarantee only needs to start being tested from length x+1 onward — any p that clears every string under consideration works, so the smallest such p is the one asked for here.

Application

1. The longest string under consideration has length x, and the next boundary string has length x-1.

2. Test p = x+1: neither the length-x nor the length-(x-1) string reaches length p, so the lemma's clause — "every string of length at least p can be pumped" — ranges over none of them. A statement about an empty set of strings is automatically true, so p = x+1 satisfies the lemma.

Cross-check

Any threshold at or below x pulls at least the length-x string into the "must be pumpable" set, which is not safely guaranteed to hold. Any threshold at or above x+1 (including a much larger one such as the square of x, for x greater than 1) again ranges over no string under consideration, so it is also technically valid — but it is not the smallest one. Since the question asks for the smallest guaranteed pumping length, x+1 is the answer: it is the least threshold at which the lemma's requirement is guaranteed to hold with no unproven assumption needed.

Result: the smallest pumping length is x+1.