## Abstract

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., mathrm{P} nsubseteq text{NC}{1}). Karchmer, Raz, and Wigderson [13] suggested to approach this problem by proving that depth complexity behaves'as expected' with respect to the composition of functions f diamond g. They showed that the validity of this conjecture would imply that mathrm{P} nsubseteq text{NC}{1}. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function f.

Original language | English |
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Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |

Publisher | IEEE Computer Society |

Pages | 43-49 |

Number of pages | 7 |

ISBN (Electronic) | 9781728196213 |

DOIs | |

State | Published - Nov 2020 |

Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2020-November |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
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Country/Territory | United States |

City | Virtual, Durham |

Period | 16/11/20 → 19/11/20 |

### Bibliographical note

Funding Information:Susanna F. de Rezende is supported by the European Research Council under the European Union’s Seventh Frame-work Programme (FP7/2007–2013) ERC grant agreement no. 279611, as well as by the Knut and Alice Wallenberg grants KAW 2016.0066 and KAW 2018.0371. Or Meir is supported by the Israel Science Foundation (grant No. 1445/16). Jakob Nordström is supported by the Swedish Research Council grant 2016-00782, the Knut and Alice Wallenberg grant KAW 2016.006, and the Independent Research Fund Denmark grant 9040-00389B. Toniann Pitassi is supported by NSERC and by NSF CCF grant 1900460. This research was performed while Robert Robere was a postdoctoral researcher at DIMACS and the Institute for Advanced Study. Robert Robere was supported by NSERC, the Charles Simonyi Endowment, and indirectly supported by the National Science Foundation Grant No. CCF-1900460. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Funding Information:

This work was partly carried out while the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation.

Publisher Copyright:

© 2020 IEEE.

## Keywords

- KRW
- KW relations
- Karchmer-Wigderson relations
- Lifting
- Simulation
- circuit complexity
- circuit lower bounds
- communication complexity
- depth complexity
- depth lower bounds
- formula complexity
- formula lower bounds

## ASJC Scopus subject areas

- Computer Science (all)