The blue pebbling cost and the space in tree-like and negative Resolution

Abstract

The red-blue pebble game is a well known two-player game on graphs that has been used in the past as a tool to analyze complexity measures in several computation models as well as proof systems. We define a new way to measure the cost of the game, the blue cost, which only counts the number of pebbles that are colored blue during the game. This new measure characterizes exactly several space bounds in tree-like and negative Resolution. In particular we prove that for any unsatiafiable formula $F$, the clause space requirements of the formula in tree-like Resolution, exactly coincide with the minimum blue pebbling cost of the game played on a refutation graph of $F$ (not necessarily a tree). This exactly parallels the known result for general Resolution in terms of the standard black pebble game, and improves the existing approximated characterization of tree-like space in terms of reversible pebbling. We show that the blue pebbling cost is also well suited for analyzing the space requirements of the lifted pebbling formulas $Peb_G[\vee]$ and $Peb_G[\oplus]$ in the two Resolution restrictions. In the case of tree-like Resolution, the clause space of $Peb_G[\vee]$ asymptotically coincides with the blue cost of the underlying graph $G$. For the case of negative Resolution, we obtain almost matching upper and lower bounds for the space in the two classes of lifted formulas, similar to the ones existing for general Resolution. We also prove a close to optimal space separation between tree-like and negative Resolution, presenting a class of formulas with $n$ variables that require clause space $Ω(\frac{n}{\log n})$ in negative Resolution, but have constant space tree-like refutations. This contrasts with the fact that negative Resolution can simulate tree-like Resolution with only a small increase in size.