# Negation-limited formulas

Siyao Guo, Ilan Komargodski

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

## Abstract

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 ([33]), 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 Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015 Naveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 850-866 17 9783939897897 https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.850 Published - 1 Aug 2015 Yes 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United StatesDuration: 24 Aug 2015 → 26 Aug 2015

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 40 1868-8969

### Conference

Conference 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 United States Princeton 24/08/15 → 26/08/15