## Finite Limits and Monotone Computations: The Lower Bounds Criterion (1997)

Venue: | Proc. of the 12th IEEE Conference on Computational Complexity |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Jukna97finitelimits,

author = {Stasys Jukna},

title = {Finite Limits and Monotone Computations: The Lower Bounds Criterion},

booktitle = {Proc. of the 12th IEEE Conference on Computational Complexity},

year = {1997},

pages = {302--312}

}

### OpenURL

### Abstract

Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin non-decreasing real-valued functions as gates. The only requirement is their "locality ". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ; xm ) with small enough threshold value minfs; m \Gamma s + 1g, are simplest examples of local gates. The proof is relatively simple and direct, and combines the bottlenecks counting approach of Haken with the idea of finite limit due to Sipser. Apparently this is the first combinatorial lower bounds criterion for monotone computations. It is symmetric and yields (in a uniform and easy way) exponential lower bounds. 1. Introduction The question of determining how much economy the universal non-monotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. The The work was supported by a DFG grant Me 1077/10-1. Preliminary...

### Citations

138 | Lower bounds for resolution and cutting planes proofs and monotone computations
- Pudlák
- 1997
(Show Context)
Citation Context ... present the criterion in full generality. In the last section we apply the criteria to explicit Boolean functions and derive exponential lower bounds for them. The difference from known lower bounds =-=[21, 23, 3, 1, 31, 13, 22]-=- for monotone circuits is twofold. First, we achieve these lower bounds in a uniform and easy way: all we need is to compute several very simple combinatorial characteristics of a given function. Seco... |

134 |
bounds on the monotone complexity of some Boolean functions
- Razborov, Lower
- 1985
(Show Context)
Citation Context ...distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. breakthrough in the field was made by Razborov in his seminal paper =-=[23]-=- where the first super-polynomial lower bound was proved. Shortly after, such (and even exponential) lower bounds were obtained for different Boolean functions [24, 3, 1, 30, 4, 31], including those w... |

129 | The monotone circuit complexity of boolean functions
- Alon, Boppana
- 1987
(Show Context)
Citation Context ...de by Razborov in his seminal paper [23] where the first super-polynomial lower bound was proved. Shortly after, such (and even exponential) lower bounds were obtained for different Boolean functions =-=[24, 3, 1, 30, 4, 31]-=-, including those whose non-monotone circuits are polynomial [24, 30]. After this impressing progress one principal question still remained unclear: is there a tractable lower bounds criterion for mon... |

116 | Intersection theorems for systems of sets - Erdős, Rado - 1969 |

76 | Lower bounds for cutting planes proofs with small coefficients
- Bonet, Pitassi, et al.
- 1997
(Show Context)
Citation Context ...fiable sets of propositional clauses. They efficiently simulate resolution proofs, and in fact are known to provide exponentially shorter proofs on some examples (the pigeonhole clauses). Bonet et al =-=[6]-=- and Pudl'ak [22] reduced the problem to lower bounds for circuits with nondecreasing real functions of fanin 2 as gates. Thus, our general lower bound for such circuits (Theorem 2), as well as lower ... |

58 |
The gap between monotone and non-monotone circuit complexity is exponential
- Tardos
- 1987
(Show Context)
Citation Context ...de by Razborov in his seminal paper [23] where the first super-polynomial lower bound was proved. Shortly after, such (and even exponential) lower bounds were obtained for different Boolean functions =-=[24, 3, 1, 30, 4, 31]-=-, including those whose non-monotone circuits are polynomial [24, 30]. After this impressing progress one principal question still remained unclear: is there a tractable lower bounds criterion for mon... |

47 | Lower bounds for monotone span programs - Beimel, Gal, et al. - 1995 |

44 |
On a method for obtaining lower bounds for the complexity of individual monotone functions
- Andreev
- 1985
(Show Context)
Citation Context ...de by Razborov in his seminal paper [23] where the first super-polynomial lower bound was proved. Shortly after, such (and even exponential) lower bounds were obtained for different Boolean functions =-=[24, 3, 1, 30, 4, 31]-=-, including those whose non-monotone circuits are polynomial [24, 30]. After this impressing progress one principal question still remained unclear: is there a tractable lower bounds criterion for mon... |

27 | An exponential lower bound for the size of monotone real circuits
- Cook, Haken
- 1999
(Show Context)
Citation Context ... previous lower bounds were known only for AND/OR gates under additional restriction that circuits have constant-depth (cf. [21]). Our proof combines two ideas: the bottlenecks countingsidea of Haken =-=[11, 12, 13]-=- and Sipser's idea of finite limits [28, 29]. The resulting argument becomes extremely simple and is different from Razborov's method of approximations [23, 24, 26], although the general idea remains ... |

20 |
Circuits and local computation
- Yao
- 1989
(Show Context)
Citation Context |

13 |
On proving lower bounds for circuit size
- Karchmer
- 1993
(Show Context)
Citation Context ...in inputs from different parts f \Gamma1 (0) and f \Gamma1 (1): This makes the criterion easy to apply, but cannot handle negation gates, i.e. gates switching the role of 0 0 s and 1 0 s. Karchmer in =-=[19]-=- and Karchmer & Wigderson [20] established an interesting connection between Razborov's generalised method of approximations [26] and the ultraproduct construction in model theory. In this approach on... |

11 | Ulfberg: Symmetric approximation argument for monotone lower bounds without sunflowers
- Berg, S
(Show Context)
Citation Context ...tlenecks counting idea is not a new approach but rather an approximation method, although more symmetric and simpler. Quite recently, this connection between the two methods was made more explicit in =-=[7, 2, 27]-=- 4. The proof Just like in Razborov's method of approximations, our first goal is to reduce the lower bounds problem to an appropriate SET COVER problem. For this purpose we first recall Sipser's noti... |

10 | Potential of the Approximation Method
- Amano, Maruoka
- 1996
(Show Context)
Citation Context ...tlenecks counting idea is not a new approach but rather an approximation method, although more symmetric and simpler. Quite recently, this connection between the two methods was made more explicit in =-=[7, 2, 27]-=- 4. The proof Just like in Razborov's method of approximations, our first goal is to reduce the lower bounds problem to an appropriate SET COVER problem. For this purpose we first recall Sipser's noti... |

10 | Characterizing non-deterministic circuit size
- Karchmer, Wigderson
- 1993
(Show Context)
Citation Context ... f \Gamma1 (0) and f \Gamma1 (1): This makes the criterion easy to apply, but cannot handle negation gates, i.e. gates switching the role of 0 0 s and 1 0 s. Karchmer in [19] and Karchmer & Wigderson =-=[20]-=- established an interesting connection between Razborov's generalised method of approximations [26] and the ultraproduct construction in model theory. In this approach one uses ultra-filters (or filte... |

10 |
On monotone formulae with restricted depth
- Klawe, Paul, et al.
- 1984
(Show Context)
Citation Context ...r the complete basis f; ; :g. Moreover, in the case of unbounded fanin gates previous lower bounds were known only for AND/OR gates under additional restriction that circuits have constant-depth (cf. =-=[21]-=-). Our proof combines two ideas: the bottlenecks countingsidea of Haken [11, 12, 13] and Sipser's idea of finite limits [28, 29]. The resulting argument becomes extremely simple and is different from ... |

10 |
Lower bounds on the monotone complexity of boolean functions
- Razborov
- 1985
(Show Context)
Citation Context ...ne principal question still remained unclear: is there a tractable lower bounds criterion for monotone circuits? Razborov raised this problem as a candidate for a "final chord" in that direc=-=tion (see [25]-=-, Problem 4). The point is that the combinatorial parts of all the above mentioned lower bounds proofs depend heavily on specific properties of concrete Boolean functions, and it was unclear if there ... |

10 |
A topological view of some problems in complexity theory, in
- Sipser
- 1984
(Show Context)
Citation Context ...R gates under additional restriction that circuits have constant-depth (cf. [21]). Our proof combines two ideas: the bottlenecks countingsidea of Haken [11, 12, 13] and Sipser's idea of finite limits =-=[28, 29]-=-. The resulting argument becomes extremely simple and is different from Razborov's method of approximations [23, 24, 26], although the general idea remains the same: we try to map a large set of input... |

9 |
The intractability of resolution. Theor
- Haken
(Show Context)
Citation Context ... previous lower bounds were known only for AND/OR gates under additional restriction that circuits have constant-depth (cf. [21]). Our proof combines two ideas: the bottlenecks countingsidea of Haken =-=[11, 12, 13]-=- and Sipser's idea of finite limits [28, 29]. The resulting argument becomes extremely simple and is different from Razborov's method of approximations [23, 24, 26], although the general idea remains ... |

8 | A note on the bottleneck counting argument
- Simon, Tsai
- 1997
(Show Context)
Citation Context ...tlenecks counting idea is not a new approach but rather an approximation method, although more symmetric and simpler. Quite recently, this connection between the two methods was made more explicit in =-=[7, 2, 27]-=- 4. The proof Just like in Razborov's method of approximations, our first goal is to reduce the lower bounds problem to an appropriate SET COVER problem. For this purpose we first recall Sipser's noti... |

6 |
On the complexity of cutting plane proofs
- Turan
- 1987
(Show Context)
Citation Context ...or CLIQUE m;k would consist of just one gate (with minterms corresponding to k-cliques). Finally, let us mention that the results in the present paper have also an application to cutting plane proofs =-=[8]-=- in the propositional calculus. Cutting plane proofs provide a complete refutation system for unsatisfiable sets of propositional clauses. They efficiently simulate resolution proofs, and in fact are ... |

5 | A criterion for monotone circuit complexity, manuscript
- Jukna
- 1991
(Show Context)
Citation Context ... Remark 1. The criterion, stated in the introduction is a special instance of Theorem 1 for the case when inputs x and y are uniformly distributed in A and B, respectively. Remark 2. We have shown in =-=[15] that-=- (at least for the case of fanin-2 AND/OR circuits) similar criterion can be derived using Razborov's method of approximations. This criterion was also based on Max a [x; "] (although notation is... |

4 | Top-down lower bounds for depththree circuits
- H˚astad, Jukna, et al.
- 1995
(Show Context)
Citation Context ...lower bound. 7. Conclusion and open problems Finite limits have already been shown to provide a convenient framework in which to prove lower bounds for different models of computation: AC 0 -circuits =-=[14]-=-, depth-three threshold circuits [17], multi-party protocols and syntactic read-k-times branching programs [16]. All these applications are based on an appropriatesLimit Lemma about the existence of i... |

2 | bounds of monotone complexity of the logical permanent function - Lower - 1985 |

1 |
Some results on monotone real circuits
- Cook, Rosenbloom
- 1996
(Show Context)
Citation Context ...strate this in Section 6). The simplicity of the whole argument (as well as of the criterion itself) may be somewhat surprising because the model we are dealing with is quite powerful: it is shown in =-=[9]-=- that every slice function has a linear size circuit with fanin-2 non-decreasing real gates, whereas easy counting yields that most of slice functions require exponential size Boolean circuits over th... |