We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications: We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log t negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n1/2-∈ negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.
|Original language||American English|
|Title of host publication||Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015|
|Editors||Naveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|Number of pages||17|
|State||Published - 1 Aug 2015|
|Event||18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United States|
Duration: 24 Aug 2015 → 26 Aug 2015
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015|
|Period||24/08/15 → 26/08/15|
Bibliographical notePublisher Copyright:
© Siyao Guo and Ilan Komargodski.
- De Morgan formulas
- Negation complexity