## Pseudorandom generators hard for $$k$$-DNF resolution and polynomial calculus resolution.(English)Zbl 1376.03055

Summary: A pseudorandom generator $$G_n:\{0,1\}^n\to \{0,1\}^m$$ is hard for a propositional proof system $$P$$ if (roughly speaking) $$P$$ cannot efficiently prove the statement $$G_n(x_1,\ldots,x_n)\neq b$$ for any string $$b\in\{0,1\}^m$$. We present a function $$(m\geq 2^{n^{\Omega(1)}})$$ generator which is hard for $$\mathrm{Res}(\varepsilon\log n)$$; here $$\mathrm{Res}(k)$$ is the propositional proof system that extends Resolution by allowing $$k$$-DNFs instead of clauses.
As a direct consequence of this result, we show that whenever $$t\geq n^2$$, every $$\mathrm{Res}(\varepsilon\log t)$$ proof of the principle $$\neg \mathrm{Circuit}_t(f_n)$$ (asserting that the circuit size of a Boolean function $$f_n$$ in $$n$$ variables is greater than $$t$$) must have size $$\exp(t^{\Omega(1)})$$. In particular, $$\mathrm{Res}(\log \log N)$$ ($$N\sim 2^n$$ is the overall number of propositional variables) does not possess efficient proofs of $$\mathbf{NP}\not\subseteq \mathbf{P}/\mathrm{poly}$$. Similar results hold also for the system PCR (the natural common extension of Polynomial Calculus and Resolution) when the characteristic of the ground field is different from 2.
As a byproduct, we also improve on the small restriction switching lemma due to N. Segerlind et al. [SIAM J. Comput. 33, No. 5, 1171–1200 (2004; Zbl 1059.03063)] by removing a square root from the final bound. This in particular implies that the (moderately) weak pigeonhole principle $$\mathrm{PHP}^{2n}_n$$ is hard for $$\mathrm{Res}(\varepsilon\log n/\log\log n)$$.

### MSC:

 03F20 Complexity of proofs 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

### Keywords:

proof complexity; pseudorandom generators

Zbl 1059.03063
Full Text:

### References:

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