## 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 n^{1/2-∈} 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 |
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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 |

Pages | 850-866 |

Number of pages | 17 |

ISBN (Electronic) | 9783939897897 |

DOIs | |

State | Published - 1 Aug 2015 |

Externally published | Yes |

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 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 40 |

ISSN (Print) | 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 |
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Country/Territory | United States |

City | Princeton |

Period | 24/08/15 → 26/08/15 |

### Bibliographical note

Publisher Copyright:© Siyao Guo and Ilan Komargodski.

## Keywords

- De Morgan formulas
- Negation complexity
- Shrinkage