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On the complexity of proof in propositional calculus. (Russian) Zbl 0197.00102
the rule of “expansion”: If $$\alpha$$, $$\beta$$, $$\gamma$$, are variables, neither $$\alpha$$ nor $$\overline\alpha$$ appearing in $$\Sigma$$, $$\alpha\vee\beta$$, $$\alpha\vee\gamma$$, $$\overline\alpha\vee\overline\beta\vee\overline\gamma$$ may be added to $$\Sigma$$. The following estimates are proved by a series of graph-theoretical theorems (which might be of independent interest):
$(N(\Sigma)+1)/2 \leq L(\Sigma) \leq (3/2)^{N(\Sigma)-1}.$
2) For every $$c<1/4$$ and every $$n$$ there is a contradictory system $$\Sigma$$, such that
$n < c (\log_2\overline N(\Sigma))^2 \leq \log_2L(\Sigma)\quad\text{and}\quad n < (\log_2 L^*(\Sigma))^2 \leq \log_2 L(\Sigma).$
3) There exists $$c>0$$, such that for every $$n$$ there is a contradictory system $$\Sigma$$, such that
$n < c\cdot \root 3\of{N^*(\Sigma)} \leq \log_2 \overline N(\Sigma),\quad n < c\cdot \sqrt{\lambda(\Sigma)} \leq \log_2 \overline N(\Sigma).$
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 03F20 Complexity of proofs
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