Under the auspices of the Computational Complexity Foundation (CCF)

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### Paper:

TR22-013 | 5th February 2022 18:36

#### On properties that are non-trivial to test

TR22-013
In this note we show that all sets that are neither finite nor too dense are non-trivial to test in the sense that, for every $\epsilon>0$, distinguishing between strings in the set and strings that are $\epsilon$-far from the set requires $\Omega(1/\epsilon)$ queries.
Specifically, we show that if, for infinitely many $n$'s, the set contains at least one $n$-bit long string and at most $2^{n-\Omega(n)}$ many $n$-bit strings, then it is non-trivial to test.