 
Summary: http://www.elsevier.com/locate/jcss
Journal of Computer and System Sciences 68 (2004) 238268
A sharp threshold in proof complexity yields lower bounds for
satisfiability search
Dimitris Achlioptas,a
Paul Beame,b,Ã,1
and Michael Molloyc,2
a
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
b
Computer Science and Engineering, University of Washington, Seattle, WA 981952350, USA
c
Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S 1A4
Received 31 December 2001; revised 18 April 2003
Abstract
Let Fðrn; DnÞ denote a random CNF formula consisting of rn randomly chosen 2clauses and Dn
randomly chosen 3clauses, for some arbitrary constants r; DX0: It is wellknown that, with probability
1 À oð1Þ; if r41 then Fðrn; DnÞ has a linearsize resolution refutation. We prove that, with probability
1 À oð1Þ; if ro1 then Fðrn; DnÞ has no subexponentialsize resolution refutation.
Our result also yields the first proof that random 3CNF formulas with densities well below the generally
