## Count(q) does not imply Count(p) (1994)

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Venue: | ANNALS OF PURE AND APPLIED LOGIC |

Citations: | 11 - 4 self |

### BibTeX

@TECHREPORT{Riis94count(q)does,

author = {Søren Riis},

title = {Count(q) does not imply Count(p)},

institution = {ANNALS OF PURE AND APPLIED LOGIC},

year = {1994}

}

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### Abstract

### Citations

326 | The relative efficiency of propositional proof systems
- Cook, Reckhow
- 1977
(Show Context)
Citation Context ...(q) and Count(p). Finally another (related) problem is to examine the efficiency of propositional proof systems. This type of problems has already been studied intensively in the literature [2], [3], =-=[11]-=-, [16], [19], [21], [24]. In S.Cook and Recknow [11] it was shown that the efficiency of propositional proof systems is a natural way of studying the NP versus co-NP problem. Then later [19] these pro... |

281 |
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
- Smolensky
- 1987
(Show Context)
Citation Context ...ependently in [12] and [1]. Later J. Hastad [13] gave a near optimal exponential lower bound. The question becomes particular challenging if we also allow gates which can count modulo q. R. Smolensky =-=[25]-=- settled the case for different prime numbers q and p. The general classification is still open. It has been conjectured (by Barrington, Beigel and Rudich) that the answer is positive exactly when q c... |

122 | Exponential lower bounds for the pigeonhole principle - Pitassi, Beame, et al. - 1993 |

113 |
The complexity of the pigeonhole principle
- Ajtai
- 1994
(Show Context)
Citation Context ...of counting knows that. The underlying logical principle states that a set A has a well-defined cardinality modulo 2. Yet, the Count(2) principle can fail in quite strong systems of Arithmetic (Ajtai =-=[2]-=-,[3]). Similarly for the counting principle modulo p (=Count(p)) where she can be in p states of mind. In 1962 Cohen invented the famous technique of forcing. He used the method to show the independen... |

81 |
Counting problems in bounded arithmetic
- Paris, Wilkie
- 1985
(Show Context)
Citation Context ...t(p). Finally another (related) problem is to examine the efficiency of propositional proof systems. This type of problems has already been studied intensively in the literature [2], [3], [11], [16], =-=[19]-=-, [21], [24]. In S.Cook and Recknow [11] it was shown that the efficiency of propositional proof systems is a natural way of studying the NP versus co-NP problem. Then later [19] these problems was li... |

72 |
Provability of the pigeonhole principle and the existence of infinitely many primes, thisJournal
- Paris, Wilkie, et al.
- 1988
(Show Context)
Citation Context ...General Riemann Hypothesis [17]. A stronger fragment (often denoted S 2 ) are know to show the infinitude of the set of prime numbers. This fact goes hand in hand with Sylvesters prime number theorem =-=[18]-=-. Besides this consider the quantifier elimination phenomenon (the strength of eliminating logic!). Clearly Bounded Arithmetic does not have quantifier elimination. However, one might still be able to... |

68 | Bounded arithmetic
- Buss
- 1986
(Show Context)
Citation Context ...turn out to be intimately linked to deep number theoretical problems/theorems. At present there are only sporadic suggestions of this. One such (which follows as a corollary to S.Buss' main result in =-=[8]-=-) is that if a certain fragment (often denoted by S 1 2 ) proves that the set of prime numbers is in NP (this can be proved in ordinary Arithmetic), then the prime numbers must actually be polynomial ... |

66 | Exponential lower bounds to the size of bounded depth frege proofs of the pigeonhole principle. Random Structure and Algorithms
- Kraj´ı˘cek, ák, et al.
- 1995
(Show Context)
Citation Context ...case where q = 6 and p = 5 has now been open for more than five years (PC. Haastad, Krajicek, Pudlak). Ajtais version of the problem is technically more involved (`presumably more difficult ' to cite =-=[16]-=-). One formulation (given in [10]) concerns the question whether for different primes q and p there exist arithmetical models M , which satisfies the Count(q) principle, but which does not satisfies t... |

61 | Lower bounds on Hilbert’s Nullstellensatz and propositional proofs
- Beame, Krajíček, et al.
- 1996
(Show Context)
Citation Context ...ount(p). So Count(p) is not a consequence (a bounded depth polynomial-size consequence in formulation 2) of Count(q) in this case. Finally I mention the recent and independent developments in [4] and =-=[6]-=-. 2 Constructing the model 2.1 Translating formulas into circuits Let M be a countable non-standard model of Th(N) over a countable first order language L (which extends the language of arithmetic). L... |

53 | Representing Boolean functions as polynomials modulo composite numbers
- Barrington, Beigel, et al.
- 1994
(Show Context)
Citation Context ... the case was settled for different prime numbers q and p. The general classification is still open. It has been conjectured that the answer is positive exactly when q contains all prime factors in p =-=[5]-=-. However even the simple case where q = 6 and p = 5 has now been open for more than five years (PC. Haastad, Krajicek, Pudlak). Ajtais version of the problem is technically more involved (`presumably... |

34 |
On the method of approximations
- Razborov
- 1989
(Show Context)
Citation Context ... to Bounded Arithmetic. And then Ajtai [2] showed that the problems also are tight up with methods and problems from circuit complexity. Recently a fascinating `ultra filter construction' by Razborov =-=[21]-=- even suggest links to higher set-theory. In any case the study of the complexity of elementary counting provides some of the strongest known results in the field of circuit complexity. 1.3 The main r... |

21 | On Frege and Extended Frege Proof Systems - Krajíček - 1995 |

18 | An Exponential separation between the Matching Principle and the pigeonhole principle
- Pitassi
- 1993
(Show Context)
Citation Context ...Finally another (related) problem is to examine the efficiency of propositional proof systems. This type of problems has already been studied intensively in the literature [2], [3], [11], [16], [19], =-=[21]-=-, [24]. In S.Cook and Recknow [11] it was shown that the efficiency of propositional proof systems is a natural way of studying the NP versus co-NP problem. Then later [19] these problems was linked t... |

18 |
Making infinite structures finite in models of second order bounded arithmetic
- Riis
(Show Context)
Citation Context ... clear from the context we let P k := P k (I) and let P := P(I). --- Notice that P 1 ` P 2 ` :::: ` P r ` :::: ` P , for each r 2 !. The idea is to use (P ; `) as the set of forcing conditions. As in =-=[22]-=-: Definition 2.2.3 We say that D ` P is dense if 8g 2 P9h 2 D h ' g: We say that D is quasi-definable if there exists a formula `(x) 2 LM (R ! ) such that D := fm 2 M : M j= `(m)g (the relation R ! is... |

15 | The propositional pigeonhole principle has polynomial size Frege proofs - Buss - 1987 |

6 |
ome open problems in arithmetic, proof theory and computational complexity
- CLOTE, KRAJfEK
- 1993
(Show Context)
Citation Context ...w been open for more than five years (PC. Haastad, Krajicek, Pudlak). Ajtais version of the problem is technically more involved (`presumably more difficult ' to cite [16]). One formulation (given in =-=[10]-=-) concerns the question whether for different primes q and p there exist arithmetical models M , which satisfies the Count(q) principle, but which does not satisfies the Count(p) principle? The method... |

6 |
Independence in Bounded Arithmetic. DPhil dissertation
- Riis
- 1993
(Show Context)
Citation Context ...echnically more involved than the corresponding problem for circuits. Still circuit complexity (especially the method of collapsing circuits by use of random evaluations) is of major importance [15], =-=[23]-=-. In the first part of the paper we reduce Ajtai's question to a purely combinatorial problem. Actually (by elaborating on the ideas in [15]) it is shown that such a reduction (to a nice and purely co... |

2 |
Combinatorial principles in elementary number theory, Annals of Pure and Applied Logic 55
- Berrarducci
- 1991
(Show Context)
Citation Context ...ls of Bounded Arithmetic? This question was first studied intensively by J.Paris, A.Wilkie and many of their students. Many basic number theoretical facts are provable in system of Bounded Arithmetic =-=[7]-=-. Other facts require new proofs. I believe that Bounded Arithmetic raises an important and serious possibility. It seems that the provability (in specific systems of Bounded Arithmetic) of elementary... |

2 |
Almost optimal lower bounds for small depth circuits
- Haastad
- 1986
(Show Context)
Citation Context ... Computer Science, Centre of the Danish National Research Foundation. 1 the number of 1's (in the input string) modulo p. This was answered (negatively) independently in [12] and [1]. Later J. Hastad =-=[13]-=- gave a near optimal exponential lower bound. The question becomes particular challenging if we also allow gates which can count modulo q. R. Smolensky [25] settled the case for different prime number... |

2 |
The strength of weak systems, in: Schriftenreite der Wittgenstein Gessellschaft 13
- Macintyre
- 1987
(Show Context)
Citation Context ...search project is to understand which parts of number theory holds in models of Bounded Arithmetic? This type of question was first studied by J. Paris, A. Wilkie. As pointed out both by A. Macintyre =-=[16]-=-, by J. Paris, A. Wilkie and A. Woods [17] and by A. Berrarducci and B. Intrigila [7] many basic number theoretical facts are provable in system of Bounded Arithmetic. Other facts require new proofs. ... |

1 |
Simpser; Parity circuits and the polynomial time hirarchy
- Furst
- 1981
(Show Context)
Citation Context ...was the mixture of forcing and powerful probabilistic techniques. The Count(q) versus Count(p) problem has various formulations and variants. The most famous variant is from circuit complexity theory =-=[13]-=-. It asks (in the This work was initiated at Oxford University England. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 base case) whether there exist bounde... |

1 |
R.Smolensky; Algebraic methods in the theory of lower bounds for boolean circuit complexity
- Mosses
- 1987
(Show Context)
Citation Context ...red (negatively) independently in [13] and [1]. Later [14] gave a near optimal exponential lower bound. The question becomes particular challenging if we also allow gates which can count modulo q. In =-=[27]-=- the case was settled for different prime numbers q and p. The general classification is still open. It has been conjectured that the answer is positive exactly when q contains all prime factors in p ... |

1 |
Definability in first order structures; Annals of Pure and Applied Logic 24
- Ajtai
- 1983
(Show Context)
Citation Context ...the Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 the number of 1's (in the input string) modulo p. This was answered (negatively) independently in [12] and =-=[1]-=-. Later J. Hastad [13] gave a near optimal exponential lower bound. The question becomes particular challenging if we also allow gates which can count modulo q. R. Smolensky [25] settled the case for ... |

1 |
The independence of the modulo p counting principles, preprint
- Ajtai
(Show Context)
Citation Context ...bounded depth polynomial-size consequence in formulation 2) of Count(q) in this case. This line of research is developed further in [24]. Finally we mention the recent and independent developments in =-=[4]-=- and [6]. 2 Constructing the model 2.1 Translating formulas into circuits Let M be a countable non-standard model of Th(N) over a countable first order language L (which extends the language of arithm... |

1 | Count(p) versus the pigeon-hole principle; Forthcomming in The Arkive of Logic - Riis |