Wikipedia:Reference desk/Archives/Mathematics/2017 October 28

= October 28 =

Resolution Lower Bound Example
I am looking for an example of a 3-CNF formula, $$\varphi$$, and a clause $$C$$ with at most 3 variables, that can be derived from $$\varphi$$ using resolution, and any resolution derivation of $$C$$ from $$\varphi$$ must derive also some 5-variable clause (in some derivation step).

For example, any derivation of $$C=(b+c+d)$$ from $$\varphi=(a+b+c)(\bar a+d+e)(b+c+\bar e)$$ must derive also some 4-variable clause (i.e, must derive either $$(b+c+d+e)$$ or $$(\bar a+d+b+c)$$). עברית (talk) 14:46, 28 October 2017 (UTC)
 * Try $$C=(c+d+e)$$ and $$\varphi=(b+h+i)(c+f+\bar h)(d+e+\bar f)(c+g+\bar i)(d+e+\bar g)(\bar b+c+j)(d+e+\bar j).$$ Not my area of expertise and I only checked by hand so there could have been errors, but allowing only clauses with 4 or fewer variables seems to only lead to dead ends no matter which order you eliminate the variables. No real methodology here, just took an expression with a 5 term clause, where the expression resolves to a 3 term clause, then added more variables to get an expression with only 3 term clauses, then checked for alternate derivations with no 5 term clauses. Seems like with a bit more systematic approach you could generate expressions which require clauses of arbitrary length. --RDBury (talk) 06:55, 29 October 2017 (UTC)