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 (CCC'15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.
@InProceedings{guo_et_al:LIPIcs.APPROX-RANDOM.2015.850, author = {Guo, Siyao and Komargodski, Ilan}, title = {{Negation-Limited Formulas}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {850--866}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.850}, URN = {urn:nbn:de:0030-drops-53400}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.850}, annote = {Keywords: Negation complexity, De Morgan formulas, Shrinkage} }
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