# Pebble Games and Complexity

## THIS REPORT HAS BEEN WITHDRAWN

### Siu Man Chan

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2013-120

June 6, 2013

We study the connection between pebble games and complexity. First, we derive complexity results using pebble games. It is shown that three pebble games used for studying computational complexity are equivalent: namely, the two-person pebble game of Dymond–Tompa, the two-person pebble game of Raz–McKenzie, and the one-person reversible pebble game of Bennett have the same pebble costs over any directed acyclic graph. The three pebble games have been used for studying parallel complexity and for proving lower bounds under restricted settings, and we show one more such lower bound on circuit-depth. Second, the pebble costs are applied to proof complexity. Concerning a family of unsatisfiable CNFs called pebbling contradictions, the pebble cost in any of the pebble games controls the scaling of some parameters of resolution refutations. Namely, the pebble cost controls the minimum depth of resolution refutations and the minimum size of tree-like resolution refutations. Finally, we study the space complexity of computing the pebble costs and of computing the minimum depth of resolution refutations. It is PSPACE-complete to compute the pebble cost in any of the three pebble games, and to compute the minimum depth of resolution refutations.

**Advisor:** Elchanan Mossel

**Author Comments:** An updated version, which fixes some statements in the introduction on known complexity results about complete problems, is accessible at [ http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-145.pdf ].