## Pseudorandom Generators Hard for k-DNF Resolution and Polynomial Calculus Resolution (2003)

Citations: | 44 - 4 self |

### BibTeX

@TECHREPORT{Razborov03pseudorandomgenerators,

author = {Alexander A. Razborov},

title = {Pseudorandom Generators Hard for k-DNF Resolution and Polynomial Calculus Resolution},

institution = {},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef- ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing k-DNFs instead of clauses.

### Citations

654 |
How to construct random functions
- Goldreich, Goldwasser, et al.
- 1986
(Show Context)
Citation Context ...= 2. 8 implies that if we compose this generator with itself as many as exp(n 1s) times, the resulting mapping will still be hard for Res(k)/PCR. Applying in particular the classical GGM-construction =-=[GGM-=-86], we get in this way a function generator G n : f0; 1g n ! f0; 1g 2 n that is hard for Res( log n), > 0 a suciently small constant (Theorem 2.12), and for PCR. As we discussed in Section 1.1, thi... |

525 |
Theory and applications of trapdoor functions
- Yao
- 1982
(Show Context)
Citation Context ... the generator approach (to conditional lower bounds for P ). The generator approach certainly does not work for arbitrary mappings G n believed to be pseudorandom generators in the standard sense of =-=[Yao82-=-], and specic counterexamples almost immediately follow from the results in [KP98] on the limitations of EIP. On the positive side, it was observed in [ABRW00] that for proof systems P with EIP there ... |

289 | Hardness vs. randomness
- Nisan, Wigderson
- 1994
(Show Context)
Citation Context ...e generator approach always works for Nisan-Wigderson generators. More specically (assuming that the constructions are based on combinatorial designs with the same parameters as in the seminal paper [=-=NW94]-=-), Conjecture 1. Any NW-generator based on any poly-time function that is hard on average for NC 1 =poly, is hard for the Frege proof system. Conjecture 2. Any NW-generator based on any function in NP... |

202 | The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, A Wiley-Interscience Publication - Alon, Spencer - 1992 |

180 | Natural proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ... of the original principle :Circuit t (f n ) for general circuits even more intriguing. The connection of this question to the generator approach above is the same as in the context of Natural Proofs =-=[RR97]-=-. Namely, if we have a function pseudorandom generator G n : f0; 1g t 0 ! f0; 1g 2 n that is hard for a proof system P , and such that the associated predicate G(x) y (x 2 f0; 1g t 0 ; y 2 f0; 1g n ) ... |

137 | Lower bounds for resolution and cutting plane proofs and monotone computations - Pudlák - 1997 |

106 | The complexity of propositional proofs - Urquhart - 1995 |

103 | Simplified and improved resolution lower bounds
- Beame, Pitassi
- 1996
(Show Context)
Citation Context ...long. Let ussx for the rest of the section an (mn) (r; d)-lossless expander A, its ordering and an integer k 1 such that (2) holds. The overall strategy of our proof for thesrst time appeared in [BP=-=9-=-6] and since that has become a standard tool in proof complexity. Namely, we want to design a random partial assignment of the variables V ars (A) that has the following two properties: Height-reduc... |

101 | Using the Groebner basis algorithm to find proofs of unsatisfiability - Clegg, Edmonds, et al. - 1996 |

84 | CREW PRAMs and decision trees - Nisan - 1991 |

77 | Lower bounds for cutting planes proofs with small coefficients - Bonet, Pitassi, et al. - 1997 |

75 | Some consequences of cryptographical conjectures for S 2
- Kraj́ıček, Pudlák
- 1998
(Show Context)
Citation Context ...h certainly does not work for arbitrary mappings G n believed to be pseudorandom generators in the standard sense of [Yao82], and specic counterexamples almost immediately follow from the results in [=-=KP98]-=- on the limitations of EIP. On the positive side, it was observed in [ABRW00] that for proof systems P with EIP there is an easy and general way of converting any pseudorandom generator that is comput... |

73 | Propositional proof complexity: past, present, and future - Beame, Pitassi - 1998 |

60 | On the complexity of a concentrator
- Pinsker
- 1973
(Show Context)
Citation Context ...orem 2.12 immediately follows from Theorem 2.10. 7. Random matrices have good expansion properties The statements of this sort have been re-appearing in the literature at a steady rate beginning from =-=[Pin73]-=-. We, however, have not been able tosnd any particular source handling the matter in the generality needed for our purposes. Thus, we prove Theorem 2.5 from the scratch (this is not hard anyway). 46 L... |

57 | In search of an easy witness: Exponential time vs. probabilistic polynomial time
- IMPAGLIAZZO, KABANETS, et al.
- 2002
(Show Context)
Citation Context ...h is the uniform counterpart of propositional proof complexity. At the suggestion of Jan Krajcek, in later papers it was re-casted in more convenient framework of propositional proof complexity. 6 [IK=-=W02-=-] proved that under the assumption NEXP P=poly, it is the speci c tautologies :Circuit t (f n ) (for any f n whatsoever) that are hard for any proof system. R. Impagliazzo (see a footnote in [Kra02, ... |

57 | Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic - Razborov - 1995 |

53 | The lengths of proofs - Pudlák |

52 | Resolution lower bounds for the weak pigeonhole principle - Raz - 2001 |

51 | Lower bounds for the polynomial calculus
- Razborov
- 1998
(Show Context)
Citation Context ...l. And we (lower bounds provers) try to fool it by feeding into the potential proof something which looks like a (non-existing) manifold to such an extent that P can not discern the dif4 ference. See =-=[Raz98]-=- for (apparently) the cleanest implementation of this intuitive scheme. Anyway, the moving forces that make the Nisan-Wigderson generator work in the computational world are of so general nature that ... |

49 | On interpolation and automatization for Frege systems - Bonet, Pitassi, et al. |

46 | Bounded arithmetic and lower bounds in Boolean complexity
- Razborov
- 1995
(Show Context)
Citation Context ...s study even more interesting is to look at the proof complexity of statements whose validity is also not known, and whose importance stretches well beyond any proof-theoretical studies. To that end, =-=[Raz95a]-=- proposed 1 to study the proof complexity of the principle :Circuit t (f n ) expressing that the circuit size of the Boolean functionsf n in n variables, given as its truth-table, is lower bounded by ... |

46 | A switching lemma for small restrictions and lower bounds for 푘-DNF resolution
- Segerlind, Buss, et al.
(Show Context)
Citation Context ...al common extension of Polynomial Calculus and Resolution) when the characteristic of the groundseld is dierent from 2. As a by-product, we also improve on the small restriction switching lemma from [=-=SBI0-=-2] by removing a square root from thesnal bound. This in particular implies that the (moderately) weak pigeonhole principlesPHP 2n n is hard for Res( log n= log log n). Institute for Advanced Study, ... |

45 |
Poisson approximation for large deviations, Random Structures Algorithms 1
- Janson
- 1990
(Show Context)
Citation Context ...d let be a partial assignment so that for every line F of P , h(F j ) h. Then j has a resolution refutation of width kh. Finally, we recall a combinatorial inequality originally proved in [Jan90=-=-=-] and further generalized in [AS00, Section 8.1]. Denition 3.7 For propositional variables x 1 ; : : : ; x n and probabilities p 1 ; : : : ; p n 2 [0; 1], denote by a ~ p a random assignment that inde... |

43 | Resolution lower bounds for perfect matching principles, in
- Razborov
- 2007
(Show Context)
Citation Context ... after we are done with this general overview). Some arguments advocating this choice (as opposed to other classical cryptographic constructions) were presented in [ABRW00] and in the introduction to =-=[Raz02b]-=-: the principle expressing hardness of NW-generators is tightly related to such familiar personages in proof complexity as the pigeonhole principle and Tseitin tautologies. Elaborating on these argume... |

40 | Space complexity in propositional calculus
- Alekhnovich, Ben-Sasson, et al.
- 2002
(Show Context)
Citation Context ...mial equation f = 0). It has polynomials x 2 i x i (i 2 [n]) as default axioms and has two inference rules: f g f +sg ; ;s2 F (Scalar Addition); f x i f (Variable Multiplication): Denition 2.18 ([ABR=-=-=-W02]) Let again F be asxedseld. Polynomial Calculus with Resolution (PCR) is the proof system whose lines are polynomials from F [x 1 ; : : : ; x n ; x 1 ; : : : ; x n ], where x 1 ; : : : ; x n a... |

38 | Lower bounds for polynomial calculus: non-binomial case - Alekhnovich, Razborov - 2003 |

36 | Candidate one-way functions based on expander graphs
- Goldreich
(Show Context)
Citation Context ...gestion to use Nisan-Wigderson generators for lower bounds in proof complexity has been recently re-iterated in [Kra02]. That paper also proposes a paradigm similar in spirit to the construction from =-=[Gol00]-=- in the context of computational complexity: hardness of the resulting mapping should depend on the randomness of the base functions rather than their complexity. So far all known results on the hardn... |

30 |
Randomness conductors and constant-degree expansion beyond the degree/2 barrier
- Capalbo, Reingold, et al.
- 2002
(Show Context)
Citation Context ...act the dierence between s and c that matters, and for most applications we need not know s (and in fact even do not need regularity). Also, the \ordinary" lossless expanders recently constructed=-= in [CRVW0-=-2] correspond to the case d = s for a small constant > 0, whence our choice of the name. Despite recent progress, no explicit constructions of (r; d)-lossless expanders with the parameters sucient fo... |

29 | Linear gaps between degrees for the Polynomial Calculus modulo distinct primes
- Buss, Grigoriev, et al.
- 2001
(Show Context)
Citation Context ...uch that jJ i (A)j 2d for all i 2 [m]. Then for every ordering and every b 2 f0; 1g m , every PCR refutation of (A; b) must have degree > r=8. Proof. Although this follows by the technique of [BGI=-=P99-=-] (cf. remark in [ABRW00] before Theorem 3.10), it is easier to apply more general result from [BI99] that for our purposes can be stated as follows. If a CNF results from expanding a set of F 2 -lin... |

28 | bounds for propositional proofs and independence results in bounded arithmetic, Automata, languages, and programming: 23rd international colloquium, icalp ’96 - Lower - 1996 |

23 | Bounded Arithmetic, Propositional Logic and Complexity Theory - Krajcek - 1996 |

15 |
R.: Using the Groebner basis algorithm to proofs of unsatis
- Clegg, Edmonds, et al.
- 1996
(Show Context)
Citation Context ...n) 1 ), for every b 2 f0; 1g m every Res(k) refutation of (A; b) must have size exp(n 41 ). Let us now recall another extension of Resolution that is of more algebraicsavour. 16 Denition 2.17 ([CEI=-=96]-=-) Let F be asxedseld. Polynomial Calculus (PC for short) is the proof system whose lines are polynomials f 2 F[x 1 ; : : : ; x n ] (a polynomial f is interpreted as the polynomial equation f = 0). It ... |

12 |
Pseudorandom generators in propositional complexity
- Alekhnovich, Ben-Sasson, et al.
- 2000
(Show Context)
Citation Context ...andom generators in the standard sense of [Yao82], and specic counterexamples almost immediately follow from the results in [KP98] on the limitations of EIP. On the positive side, it was observed in [=-=ABRW00]-=- that for proof systems P with EIP there is an easy and general way of converting any pseudorandom generator that is computationally hard (in the standard sense) into a pseudorandom generator that is ... |

12 | Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic - Krajcek - 1997 |

10 | Proof complexity of pigeonhole principles - Razborov - 2001 |

8 | On the weak pigeonhole principle - Krajcek - 2001 |

5 |
Random CNF"s are Hard for the Polynomial Calculus
- Ben-Sasson, Impagliazzo
- 1998
(Show Context)
Citation Context ...refutation of (A; b) must have degree > r=8. Proof. Although this follows by the technique of [BGIP99] (cf. remark in [ABRW00] before Theorem 3.10), it is easier to apply more general result from [B=-=I9-=-9] that for our purposes can be stated as follows. If a CNF results from expanding a set of F 2 -linear equations, then every PCR refutation of over F (remember that char(F) 6= 2) gives rise to a re... |

3 | Lower bounds to the size of constant-depth propositional proofs - Krajcek - 1994 |

2 |
Lower bounds for a proof system with an exponential speed-up over constant-depth Frege systems and over polynomial calculus
- Krajcek
- 1997
(Show Context)
Citation Context ... diculty occurred in the proof of Theorem 2.21 leads 62 to the system PCRes(2) which is a natural hybrid of PC and Res(2). Now, lower bounds for this system are known (see much more general result in =-=[Kra97b]-=-), but what we really need is a pseudo-random generator hard for it. Last, but not the least, construct explicit lossless expanders (ideally, expanders with parameters close to those in Theorem 2.5). ... |

2 |
Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds
- Krajcek
- 2002
(Show Context)
Citation Context ...98, Raz02a]. For strong proof systems the current situation is by far more miserable. Although there are no rigorous results along these lines (and, moreover, this feeling is not universal { see e.g. =-=[Kra02]-=-), the empirical evidence strongly suggests that lower bounds for a proof system P are even harder to attain than computational lower bounds for the companion class C P . Therefore, with our current u... |

1 |
Tautologies from pseudo-random generators
- Krajcek
- 2001
(Show Context)
Citation Context ...diculty by showing that it is indeed the case if hardness is replaced by a stronger notion of s-iterability (the latter, in turn, being a variant of a similar notion of freeness earlier introduced in =-=[Kra01b]-=-). As asrst application of this approach, [Kra02] showed that one particular construction of the Nisan generator from [ABRW00] can be iterated with itself once, thus giving a pseudorandom generator wi... |