scholarly journals Exponential lower bounds for the pigeonhole principle

1993 ◽  
Vol 3 (2) ◽  
pp. 97-140 ◽  
Author(s):  
Toniann Pitassi ◽  
Paul Beame ◽  
Russell Impagliazzo
Keyword(s):  
Lower Bounds ◽  
Pigeonhole Principle

scholarly journals LOWER BOUNDS FOR DNF-REFUTATIONS OF A RELATIVIZED WEAK PIGEONHOLE PRINCIPLE

2015 ◽  
Vol 80 (2) ◽  
pp. 450-476 ◽  
Author(s):  
ALBERT ATSERIAS ◽  
MORITZ MÜLLER ◽  
SERGI OLIVA
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Pigeonhole Principle ◽  
Low Degree ◽  
Robustness Condition ◽  
Image Position

AbstractThe relativized weak pigeonhole principle states that if at least 2n out of n2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size $2^{\left( {{\rm{log\ }}n} \right)^{3/2 - \varepsilon } } $ for every ε﹥0 and every sufficiently large n. By reducing it to the standard weak pigeonhole principle with 2n pigeons and n holes, we also show that this lower bound is essentially tight in that there exist DNF-refutations of size $2^{\left( {{\rm{log\ }}n} \right)^{O\left( 1 \right)} } $ even in R(log). For the lower bound proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.

Lower bounds to the size of constant-depth propositional proofs

1994 ◽  
Vol 59 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bounds ◽  
Propositional Logic ◽  
Sequent Calculus ◽  
Pigeonhole Principle ◽  
Constant Depth ◽  
Total Size ◽  
Natural Modification ◽  
Gentzen Sequent Calculus

AbstractLK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and ∧,∨ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set of depth d sequents of total size O(n3+d) which are refutable in LK by depth d + 1 proof of size exp(O(log2n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets express a weaker form of the pigeonhole principle.

scholarly journals Exponential lower bounds for the pigeonhole principle

1992 ◽  
Author(s):  
Paul Beame ◽  
Russell Impagliazzo ◽  
Jan Krajíček ◽  
Toniann Pitassi ◽  
Pavel Pudlák ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Pigeonhole Principle

scholarly journals Lower Bounds for the Weak Pigeonhole Principle and Random Formulas beyond Resolution

2002 ◽  
Vol 176 (2) ◽  
pp. 136-152 ◽  
Author(s):  
Albert Atserias ◽  
Maria Luisa Bonet ◽  
Juan Luis Esteban
Keyword(s):  
Lower Bounds ◽  
Pigeonhole Principle

Regular Resolution Lower Bounds For The Weak Pigeonhole Principle

COMBINATORICA ◽  
2004 ◽  
Vol 24 (3) ◽  
pp. 503-524 ◽  
Author(s):  
Toniann Pitassi* ◽  
Ran Raz†
Keyword(s):  
Lower Bounds ◽  
Pigeonhole Principle
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