# Formulas versus circuits for small distance connectivity

Conference Paper

© 2018 Society for Industrial and Applied Mathematics We prove an nΩ (log k) lower bound on the AC0formula size of Distance k(n) Connectivity for all k(n) ≤ log log n and formulas up to depth log n/(log log n)O(1). This lower bound strongly separates the power of bounded-depth formulas versus circuits, since Distance k(n) Connectivity is solvable by polynomial-size AC0circuits of depth O(log k). For all d(n) ≤ log log log n, it follows that polynomial-size depth-d circuits-which are a semantic subclass of nO(d)size depth-d formulas-are not a semantic subclass of no(d)-size formulas of much higher depth log n/(log log n)O(1). Our lower bound technique probabilistically associates each gate in an AC0 formula with an object called a pathset. We show that with high probability these random pathsets satisfy a family of density constraints called smallness, a property akin to low average sensitivity. We then study a complexity measure on small pathsets, which lower bounds the AC0 formula size of Distance k(n) Connectivity. The heart of our technique is an nΩ (log k) lower bound on this pathset complexity measure.

### Full Text

### Duke Authors

### Cited Authors

- Rossman, B

### Published Date

- January 1, 2018

### Published In

### Volume / Issue

- 47 / 5

### Start / End Page

- 1986 - 2028

### Electronic International Standard Serial Number (EISSN)

- 1095-7111

### International Standard Serial Number (ISSN)

- 0097-5397

### Digital Object Identifier (DOI)

- 10.1137/15M1027310

### Citation Source

- Scopus