## Abstract

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P ⊈ NC1). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested approaching this problem by proving that depth complexity of a composition of functions f ⋄ g is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ⊈ N C1. The intuition that underlies the KRW conjecture is that the composition f ⋄ g should behave like a 'direct-sum problem', in a certain sense, and therefore the depth complexity of f ⋄ g should be the sum of the individual depth complexities. Nevertheless, there are two obstacles toward turning this intuition into a proof: first, we do not know how to prove that f ⋄ g must behave like a direct-sum problem; second, we do not know how to prove that the complexity of the latter direct-sum problem is indeed the sum of the individual complexities. In this work, we focus on the second obstacle. To this end, we study a notion called 'strong composition', which is the same as f ⋄ g except that it is forced to behave like a direct-sum problem. We prove a variant of the KRW conjecture for strong composition, thus overcoming the above second obstacle. This result demonstrates that the first obstacle above is the crucial barrier toward resolving the KRW conjecture. Along the way, we develop some general techniques that might be of independent interest.

Original language | English |
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Title of host publication | Proceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023 |

Publisher | IEEE Computer Society |

Pages | 1056-1081 |

Number of pages | 26 |

ISBN (Electronic) | 9798350318944 |

DOIs | |

State | Published - 2023 |

Event | 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, United States Duration: 6 Nov 2023 → 9 Nov 2023 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Conference

Conference | 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 |
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Country/Territory | United States |

City | Santa Cruz |

Period | 6/11/23 → 9/11/23 |

### Bibliographical note

Publisher Copyright:© 2023 IEEE.

## Keywords

- KRW conjecture
- KW relations
- Karchmer-Wigderson relations
- circuit complexity
- communication complexity
- depth complexity
- formula complexity
- formulas

## ASJC Scopus subject areas

- General Computer Science