## Abstract

One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., P ⊈ NC^{1}). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [Comput. Complexity, 5 (1995), pp. 191-204] suggested approaching this problem by proving the following conjecture: given two Boolean functions f and g, the depth complexity of the composed function g ⋄f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ⊈ NC_{1}. As a starting point for studying the composition of functions, they introduced a relation called "the universal relation" and suggested studying the composition of universal relations. This suggestion proved fruitful, and an analogue of the Karchmer-Raz-Wigderson (KRW) conjecture for the universal relation was proved by Edmonds et al. [Comput. Complexity, 10 (2001), pp. 210-246]. An alternative proof was given later by Håstad and Wigderson [in Advances in Computational Complexity Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 13, AMS, Providence, RI, 1993, pp. 119-134]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still an open question. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation.

Original language | American English |
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Pages (from-to) | 114-131 |

Number of pages | 18 |

Journal | SIAM Journal on Computing |

Volume | 46 |

Issue number | 1 |

DOIs | |

State | Published - 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017 Society for Industrial and Applied Mathematics.

## Keywords

- Communication complexity
- Formula
- Information complexity
- KRW conjecture
- Karchmer-Wigderson relations
- Lower bounds