It is well known that random $ k $-CNF formulas over $ n $ variables with $ cn $ clauses are unsatisfiable (i.e. they are contradictions) with high probability, for sufficiently large constant $ c $. Thus, random $ k $-CNF formulas (for $ c $ large enough) constitute a natural distribution over unsatisfiable Boolean formulas (or dually, over tautologies, i.e. negations of contradictions). This distribution has been studied extensively.

My question is the following: are there any other established distributions over propositional tautologies or contradictions, that can be considered as capturing the "average-case" of tautologies or unsatisfiable formulas? Have these distributions been intensively studied?

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    $\begingroup$ @Iddo Tautologies do not exists in a "true" CNF model because otherwise you would need to have a literal and its complement in the same clause.... Tautologies are not interesting to study in CNF. $\endgroup$
    – Tayfun Pay
    Mar 2, 2011 at 19:00
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    $\begingroup$ @Pay, the negation of an unsatisfiable formula is obviously a tautology. Hence, we can consider random k-CNF's as a distribution over tautologies (when the clause-to-variable ratio is large enough, and where there is an o(1) probability for a k-CNF to be satisfiable). $\endgroup$ Mar 3, 2011 at 6:41
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    $\begingroup$ I think that Tayfun is right. You should speak of either CNF formulas being unsatisfiable or DNF formulas being tautologies. In the current question, you are mixing the two. $\endgroup$ Mar 3, 2011 at 23:18
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    $\begingroup$ This is my last comment on this matter: I do not know why you insist to keep the word “tautologies,” which is clearly wrong as Tayfun explained. But I am fine if you do not want to incorporate other people’s comments to improve the wording of your question. $\endgroup$ Mar 4, 2011 at 2:49
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    $\begingroup$ I prefer to keep the term 'tautologies' in the title because I'm asking about distributions on either tautologies or contradictions, and the question is phrased accordingly. $\endgroup$ Mar 4, 2011 at 3:52

1 Answer 1


Paul Beame has two papers (with various coauthors) where the resolution complexity of certain distributions of random formulas are studied. These formulas arise by expressing properties, such as k-colorability or having independent sets of size k, of random graphs from the usual distribution $G(n,p)$. Here are the links:

Paul Beame, Russell Impagliazzo, and Ashish Sabharwal. The resolution complexity of independent sets and vertex covers in random graphs. Computational Complexity, 16(3):245-297, 2007.

Paul Beame, Joe Culberson, David Mitchell, and Cristopher Moore. The resolution complexity of random graph k-colorability. Discrete Applied Mathematics, 153:25-47, 2005.


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