## Bounded Arithmetic and Lower Bounds in Boolean Complexity (1993)

Venue: | Feasible Mathematics II |

Citations: | 44 - 5 self |

### BibTeX

@INPROCEEDINGS{Razborov93boundedarithmetic,

author = {Alexander A. Razborov},

title = {Bounded Arithmetic and Lower Bounds in Boolean Complexity},

booktitle = {Feasible Mathematics II},

year = {1993},

pages = {344--386},

publisher = {Birkhauser}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 (which, in view of RSUV -isomorphism, is equivalent to S 2 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 . As a by-product of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.

### Citations

290 | Algebraic methods in the theory of lower bounds for Boolean circuit complexity
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Citation Context ...unctions is one of the most challenging tasks in computational complexity. This theory met with remarkable success at least twice: in the 60's (see e.g. [36, 31, 32, 37, 38]) and in more recent time (=-=[9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]-=-). A nice survey of many major results known in Boolean complexity at present can be found in [5]. Both times, however, the period of enthusiasm was followed by understanding that it is not quite clea... |

256 | Small-bias probability spaces: Efficient constructions and applications
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Citation Context ... 1 (ffi) is exactly the theory capturingsNC-computations. Examples of fis needed in Facts 1, 2 are known to be NC-computable from ff. For Fact 1 one could apply the standard derandomization procedure =-=[16]-=-; the NC-algorithm for Fact 2 is based upon computing the matrix rank [15]. Hence U 0 1 (ffi) can define relationals ffi witnessing Facts 1, 2 in the real world. It is not clear, however, to which ext... |

187 |
Separating the polynomial-time hierarchy by oracles
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Citation Context ...unctions is one of the most challenging tasks in computational complexity. This theory met with remarkable success at least twice: in the 60's (see e.g. [36, 31, 32, 37, 38]) and in more recent time (=-=[9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]-=-). A nice survey of many major results known in Boolean complexity at present can be found in [5]. Both times, however, the period of enthusiasm was followed by understanding that it is not quite clea... |

186 |
Computational Limitations of Small Depth Circuits
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Citation Context ...unctions is one of the most challenging tasks in computational complexity. This theory met with remarkable success at least twice: in the 60's (see e.g. [36, 31, 32, 37, 38]) and in more recent time (=-=[9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]-=-). A nice survey of many major results known in Boolean complexity at present can be found in [5]. Both times, however, the period of enthusiasm was followed by understanding that it is not quite clea... |

175 |
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Citation Context ...g. [36, 31, 32, 37, 38]) and in more recent time ([9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]). A nice survey of many major results known in Boolean complexity at present can be found in =-=[5]-=-. Both times, however, the period of enthusiasm was followed by understanding that it is not quite clear to which extent the methods developed so far can be useful for attacking central open problems ... |

174 | Natural proofs
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Citation Context ...S 1 2 and PV [6, Chapter 6]. 1.2. Recent developments A purely complexity framework for analyzing the methods developed so far in non-uniform Boolean complexity was proposed by Razborov and Rudich in =-=[22]-=-. Namely, in that paper we introduced the notion of natural proof and argued that the known proofs of lower bounds on the complexity of explicit Boolean functions in non-monotone models fall within th... |

147 |
Monotone Circuits for Connectivity Require Super-logarithmic Depth
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130 | The monotone circuit complexity of boolean functions
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Bounded Arithmetic, Bibliopolis
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Citation Context ...nstraint becomes unessential (see Theorem 3.9 below), and the corresponding families of circuits can compute exactly functions in P . Thus, our result for this case is analogous to the main result of =-=[6]-=- concerning \Sigma b 1 -definable in S 1 2 functions. The first order theory characterizing NC computable functions was in various forms introduced by Allen [2] and Clote [7]. Takeuti [26] later showe... |

120 |
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Citation Context ...ircuits are exactly NC-circuits) provides a new proof of the result by Allen and Clote. Finally, if we appropriately adjust the languages, the theory V 0 1 (ffi) resembles Cook's equational system PV =-=[8]-=-, only instead of introducing function symbols for polynomially computable functions, we introduce relationals for evaluating polynomial size circuits. Respectively, the proof of our main result corre... |

115 | Metamathematics of First-Order Arithmetics - Hájek, Pudlák - 1993 |

85 |
1 -formulae on finite structures
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73 | Monotone circuits for matching require linear depth
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69 |
Provability of the pigeonhole principle and the existence of infinitely many primes
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Citation Context ...ize issn 10 . V 3 1 can prove the existence of Boolean functions with exponential circuit size (since S 3 2 proves WPHP , the weak pigeon hole principle stating that 2a pigeons can not sit in a holes =-=[17]-=-). D. On the mathematics and metamathematics of V 1 1 Many people believe that the theory S 1 2 is the most important and natural theory among various fragments of Bounded Arithmetic. The same applies... |

64 |
A fast parallel algorithm to compute the rank of a matrix over an arbitrary field
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(Show Context)
Citation Context ...needed in Facts 1, 2 are known to be NC-computable from ff. For Fact 1 one could apply the standard derandomization procedure [16]; the NC-algorithm for Fact 2 is based upon computing the matrix rank =-=[15]-=-. Hence U 0 1 (ffi) can define relationals ffi witnessing Facts 1, 2 in the real world. It is not clear, however, to which extent U 0 1 (ffi) can prove the desired properties of these relationals. The... |

58 |
The gap between monotone and non-monotone circuit complexity is exponential
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54 | Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic
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Citation Context ... link between mathematics and metamathematics already has turned out important for research on the possibility of solving major open problems in Boolean complexity by means of Bounded Arithmetic (see =-=[21]-=-). E. Proofs in subsystems of V 1 1 In Section B we formulated the thesis that V 1 1 supports proofs of lower bounds for explicit functions existing in Boolean complexity at the moment. It is conceiva... |

33 |
Applications of matrix methods to the theory of lower bounds in computational complexity, Combinatorica 10
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Citation Context ...c and Lower Bounds in Boolean Complexity 375 ffl lower bounds for constant-depth circuits with MOD \Gamma q gates [35, 24, 4], ffl lower bounds for monotone formulae based on communication complexitys=-=[12, 19, 18]-=-. The reader wishing to learn more about these and other results is referred to the excellent survey [5]. A. The formalization Apparently, L 1 is the most natural and elegant formal language for forma... |

31 |
RSUV isomorphisms
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- 1993
(Show Context)
Citation Context ...ese methods in a natural, "straightforward" way. We carefully present both formal and informal arguments supporting the claim that the desired fragment is V 1 1 . Note that, due to RSUV - is=-=omorphism [25, 26, 20]-=-, this system is equivalent to S 1 2 , the latter being considered as the most important among various fragments of Bounded Arithmetic. For several reasons, however, it is more natural and elegant to ... |

29 | An equivalence between second order bounded domain bounded arithmetic and first order bounded arithmetic
- Razborov
- 1993
(Show Context)
Citation Context ...understanding that it is not quite clear to which extent the methods developed so far can be useful for attacking central open problems in Boolean complexity. This paper (as well as the earlier paper =-=[20]-=-) mainly stemmed from the author's intention to look at this situation from the logical point of Supported by the grant # 93-011-16015 of the Russian Foundation for Fundamental Research 344 Bounded Ar... |

18 |
Arithmetizing uniform NC
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Citation Context ...case is analogous to the main result of [6] concerning \Sigma b 1 -definable in S 1 2 functions. The first order theory characterizing NC computable functions was in various forms introduced by Allen =-=[2]-=- and Clote [7]. Takeuti [26] later showed that this theory is equivalent to R 1 2 and, via the RSUV - isomorphism, to U 1 1 . With these equivalencies in mind, our characterization of \Sigma 1;b 1 -de... |

13 | cek. Bounded arithmetic, propositional logic and complexity theory - Kraj - 1994 |

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Citation Context ...on the complexity of explicitly given Boolean functions is one of the most challenging tasks in computational complexity. This theory met with remarkable success at least twice: in the 60's (see e.g. =-=[36, 31, 32, 37, 38]-=-) and in more recent time ([9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]). A nice survey of many major results known in Boolean complexity at present can be found in [5]. Both times, howeve... |

5 |
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Citation Context ...ous to the main result of [6] concerning \Sigma b 1 -definable in S 1 2 functions. The first order theory characterizing NC computable functions was in various forms introduced by Allen [2] and Clote =-=[7]-=-. Takeuti [26] later showed that this theory is equivalent to R 1 2 and, via the RSUV - isomorphism, to U 1 1 . With these equivalencies in mind, our characterization of \Sigma 1;b 1 -definable in U 1... |

4 |
The number of two terminal series parallel networks
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Citation Context ...th the phenomenon that the inherently hard problems making the core of the field become trivial when one considers "random" functions. Unfortunately, the powerful Shannon counting arguments =-=(see e.g. [23]) used for-=- dealing with random functions are hardly relevant to proving lower bounds for "explicit" functions. For this reason it is customary in the modern Boolean complexity to distance oneself from... |

4 |
Ob odnom metode poluqeni�� ni��nih ocenok slo��nosti \Pishem. Matem. zametki, t
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Citation Context ...on the complexity of explicitly given Boolean functions is one of the most challenging tasks in computational complexity. This theory met with remarkable success at least twice: in the 60's (see e.g. =-=[36, 31, 32, 37, 38]-=-) and in more recent time ([9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]). A nice survey of many major results known in Boolean complexity at present can be found in [5]. Both times, howeve... |

3 |
S i 3 and ffi V i 2
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Citation Context ...ese methods in a natural, "straightforward" way. We carefully present both formal and informal arguments supporting the claim that the desired fragment is V 1 1 . Note that, due to RSUV - is=-=omorphism [25, 26, 20]-=-, this system is equivalent to S 1 2 , the latter being considered as the most important among various fragments of Bounded Arithmetic. For several reasons, however, it is more natural and elegant to ... |

3 |
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Citation Context ...on the complexity of explicitly given Boolean functions is one of the most challenging tasks in computational complexity. This theory met with remarkable success at least twice: in the 60's (see e.g. =-=[36, 31, 32, 37, 38]-=-) and in more recent time ([9, 1, 28, 11, 33, 34, 29, 3, 27, 30, 35, 24, 4, 12, 18]). A nice survey of many major results known in Boolean complexity at present can be found in [5]. Both times, howeve... |

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