Lower bounds for DNF-refutations of a relativized weak pigeonhole principle
The relativized weak pigeonhole principle states that if at least 2n out of n(2) 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((log n)3/2-is an element of) for every is an element of > 0 a...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/28085 |
| Acceso en línea: | https://hdl.handle.net/2117/28085 https://dx.doi.org/10.1017/jsl.2014.56 |
| Access Level: | acceso abierto |
| Palabra clave: | Computational complexity Proof complexity Bounded arithmetic Weak pigeonhole principles Approximate counting Bounded-depth frege Propositional proof systems Resolution lower bounds Random formulas Complexity gap Primes Size Complexitat computacional Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica |
| Sumario: | The relativized weak pigeonhole principle states that if at least 2n out of n(2) 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((log n)3/2-is an element of) for every is an element of > 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((log n)O(1)) 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. |
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