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22
A Switching Lemma for Small Restrictions and Lower Bounds for kDNF Resolution (Extended Abstract)
 SIAM J. Comput
, 2002
"... We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of cla ..."
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Cited by 45 (7 self)
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We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of clauses. We also obtain an exponential separation between depth d circuits of k + 1.
On the Complexity of Resolution with Bounded Conjunctions
 IN 29TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2004
"... We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Kraj'icek in [21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to ..."
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Cited by 28 (4 self)
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We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Kraj'icek in [21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreover
On approximate majority and probabilistic time
 in Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and ..."
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Cited by 18 (6 self)
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We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�size depth3 circuits for Approximate Majority on n bits have bottom fanin Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomialsize circuits of depth 3. Prior to our work, the only known polynomialsize depth3 circuits for Approximate Majority were nonuniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a randomaccess input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the
Narrow proofs may be spacious: Separating space and width in resolution (Extended Abstract)
 REVISION 02, ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2005
"... The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously ..."
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Cited by 13 (7 self)
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The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of kCNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.
Is P versus NP formally independent
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy! ..."
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Cited by 8 (0 self)
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I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!
The provable total search problems of bounded arithmetic
, 2007
"... We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument use ..."
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Cited by 8 (4 self)
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We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a nonlogical question about multiparty communication protocols.
PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when compari ..."
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Cited by 7 (4 self)
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Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and tradeoffs between proof size and proof space.
Separation Results for the Size of ConstantDepth Propositional Proofs
, 2004
"... This paper proves exponential separations between depth d  LK and depth (d + 2 )  LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d  LK and depth (d+1)  LK for d N . We investigate the relationship between ..."
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Cited by 5 (3 self)
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This paper proves exponential separations between depth d  LK and depth (d + 2 )  LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d  LK and depth (d+1)  LK for d N . We investigate the relationship between the sequencesize, treesize and height of depth d  LKderivations for d 2 N , and describe transformations between them.
On Minimal Unsatisfiability and TimeSpace Tradeoffs for kDNF Resolution
, 2009
"... In the context of proving lower bounds on proof space in kDNF resolution, [BenSasson and Nordström 2009] introduced the concept of minimally unsatisfiable sets of kDNF formulas and proved that a minimally unsatisfiable kDNF set with m formulas can have at most O ( (mk) k+1) variables. They also ..."
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Cited by 3 (2 self)
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In the context of proving lower bounds on proof space in kDNF resolution, [BenSasson and Nordström 2009] introduced the concept of minimally unsatisfiable sets of kDNF formulas and proved that a minimally unsatisfiable kDNF set with m formulas can have at most O ( (mk) k+1) variables. They also gave an example of such sets with Ω(mk 2) variables. In this paper we significantly improve the lower bound to Ω(m) k, which almost matches the upper bound above. Furthermore, we show that this implies that the analysis of their technique for proving timespace separations and tradeoffs for kDNF resolution is almost tight. This means that although it is possible, or even plausible, that stronger results than in [BenSasson and Nordström 2009] should hold, a fundamentally different approach would be needed to obtain such results.
A Space Hierarchy for kDNF Resolution ∗
, 2009
"... The kDNF resolution proof systems are a family of systems indexed by the integer k, where the k th member is restricted to operating with formulas in disjunctive normal form with all terms of bounded arity k (kDNF formulas). This family was introduced in [Krajíček 2001] as an extension of the well ..."
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Cited by 2 (1 self)
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The kDNF resolution proof systems are a family of systems indexed by the integer k, where the k th member is restricted to operating with formulas in disjunctive normal form with all terms of bounded arity k (kDNF formulas). This family was introduced in [Krajíček 2001] as an extension of the wellstudied resolution proof system. A number of lower bounds have been proven on kDNF resolution proof length and space, and it has also been shown that (k+1)DNF resolution is exponentially more powerful than kDNF resolution for all k with respect to length. For proof space, however, no corresponding hierarchy has been known except for the (very weak) subsystems restricted to treelike proofs. In this work, we establish a strict space hierarchy for the general, unrestricted kDNF resolution proof systems. 1