On minimal unsatisfiability and time-space trade-offs for k-DNF resolution
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On minimal unsatisfiability and time-space trade-offs for k-DNF resolution. / Nordström, Jakob; Razborov, Alexander.
Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings. PART 1. ed. 2011. p. 642-653 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); No. PART 1, Vol. 6755 LNCS).Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review
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TY - GEN
T1 - On minimal unsatisfiability and time-space trade-offs for k-DNF resolution
AU - Nordström, Jakob
AU - Razborov, Alexander
PY - 2011
Y1 - 2011
N2 - A well-known theorem by Tarsi states that a minimally unsatisfiable CNF formula with m clauses can have at most m - 1 variables, and this bound is exact. In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordström 2009] extended the concept of minimal unsatisfiability to sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formulas can have at most (mk) k + 1 variables. This result is far from tight, however, since they could only present explicit constructions of minimally unsatisfiable sets with Ω(mk 2) variables. In the current paper, we revisit this combinatorial problem and significantly improve the lower bound to (Ω(m)) k , which almost matches the upper bound above. Furthermore, using similar ideas we show that the analysis of the technique in [Ben-Sasson and Nordström 2009] for proving time-space separations and trade-offs for k-DNF resolution is almost tight. This means that although it is possible, or even plausible, that stronger results than in [Ben-Sasson and Nordström 2009] should hold, a fundamentally different approach would be needed to obtain such results.
AB - A well-known theorem by Tarsi states that a minimally unsatisfiable CNF formula with m clauses can have at most m - 1 variables, and this bound is exact. In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordström 2009] extended the concept of minimal unsatisfiability to sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formulas can have at most (mk) k + 1 variables. This result is far from tight, however, since they could only present explicit constructions of minimally unsatisfiable sets with Ω(mk 2) variables. In the current paper, we revisit this combinatorial problem and significantly improve the lower bound to (Ω(m)) k , which almost matches the upper bound above. Furthermore, using similar ideas we show that the analysis of the technique in [Ben-Sasson and Nordström 2009] for proving time-space separations and trade-offs for k-DNF resolution is almost tight. This means that although it is possible, or even plausible, that stronger results than in [Ben-Sasson and Nordström 2009] should hold, a fundamentally different approach would be needed to obtain such results.
UR - http://www.scopus.com/inward/record.url?scp=79959927201&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-22006-7_54
DO - 10.1007/978-3-642-22006-7_54
M3 - Article in proceedings
AN - SCOPUS:79959927201
SN - 9783642220050
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 642
EP - 653
BT - Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings
T2 - 38th International Colloquium on Automata, Languages and Programming, ICALP 2011
Y2 - 4 July 2011 through 8 July 2011
ER -
ID: 251870868