We continue the study of tradeoffs between space and length of resolution proofs and focus on two new results: begin{enumerate} item We show that length and space in resolution are uncorrelated. This is proved by exhibiting families of CNF formulas of size $O(n)$ that have proofs of length $O(n)$ but require space $Omega(n / log n)$. Our separation is the strongest possible since any proof of length $O(n)$ can always be transformed into a proof in space $O(n / log n)$, and improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and H{aa}stad 2008]. item We prove a number of trade-off results for space in the range from constant to $O(n / log n)$, most of them superpolynomial or even exponential. This is a dramatic improvement over previous results in [Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007]. end{enumerate} The key to our results is the following, somewhat surprising, theorem: Any CNF formula $F$ can be transformed by simple substitution transformation into a new formula $F'$ such that if $F$ has the right properties, $F'$ can be proven in resolution in essentially the same length as $F$ but the minimal space needed for $F'$ is lower-bounded by the number of variables that have to be mentioned simultaneously in any proof for $F$. Applying this theorem to so-called pebbling formulas defined in terms of pebble games over directed acyclic graphs and analyzing black-white pebbling on these graphs yields our results.
@InProceedings{bensasson_et_al:DagSemProc.08381.6, author = {Ben-Sasson, Eli and Nordstr\"{o}m, Jakob}, title = {{Understanding space in resolution: optimal lower bounds and exponential trade-offs}}, booktitle = {Computational Complexity of Discrete Problems}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.6}, URN = {urn:nbn:de:0030-drops-17815}, doi = {10.4230/DagSemProc.08381.6}, annote = {Keywords: Proof complexity, Resolution, Pebbling.} }
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