Abstract
We show that there are CNF formulas which can be refuted in resolution in both small space and small width, but for which any small-width proof must have space exceeding by far the linear worst-case upper bound. This significantly strengthens the space-width trade-offs in [E. Ben-Sasson, SIAM J. Comput., 38 (2009), pp. 2511-2525], and provides one more example of trade-offs in the supercritical regime above worst case recently identified by [A.A. Razborov, J. ACM, 63 (2016), 16]. We obtain our results by using Razborov's new hardness condensation technique and combining it with the space lower bounds in [E. Ben-Sasson and J. Nordstrom, Short proofs may be spacious: An optimal separation of space and length in resolution, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS '08), 2008, pp. 709-718]. (Less)
Highlights
Propositional proof complexity studies the problem of how to provide concise, polynomialtime checkable certificates that formulas in conjunctive normal form (CNF) are unsatisfiable
Answering Razborov’s call in [32] for more examples of the type of trade-offs discussed above, we prove a supercritical trade-off between space and width in resolution
We show that from any refutation in width O(w) of this new, XORified formula it is possible to recover a refutation of the original formula with comparable space complexity
Summary
Propositional proof complexity studies the problem of how to provide concise, polynomialtime checkable certificates that formulas in conjunctive normal form (CNF) are unsatisfiable. Linear space lower bounds matching the worst-case upper bound up to constant factors were obtained for a number of formula families in [1, 9, 23] These space lower bounds matched known lower bounds on width, and in a strikingly simple and beautiful result Atserias and Dalmau [2] showed that the resolution width of refuting a k-CNF formula F provides a lower bound for the clause space required. Since the number of variables provides a worst-case upper bound on space (independently of formula size), measured in terms of variables it seems fair to say that the trade-off result in Theorem 1.1 is fairly dramatic We omit some of the proofs in this extended abstract, referring the reader to the upcoming full-length version for the missing details
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