# Induction on all polynomial runtimes?

Has there ever been a proof technique to show that a language isn't in $$\mathrm{P}$$, by showing inductively there isn't any $$k$$ for which the language is in $$\mathrm{TIME}(n^k)$$?

e.g.: $$L\notin \mathrm{TIME}(n^0)$$,

$$L\notin \mathrm{TIME}(n^k)\rightarrow L\notin \mathrm{TIME}(n^{k+1})$$,

$$\therefore L\notin \mathrm{P}$$.

Is something of the sort feasible? Has it been used?

## 1 Answer

Something like this is used in the definition of creative sets by Joseph and Young (1985). See the Wikipedia article on polynomial creativity. But their use is to concoct NP-hard sets with unusual properties (potential counterexamples to the Berman–Hartmanis conjecture rather than to say anything nontrivial about the complexity of any natural algorithmic problem.