Negation-limited formulas

Siyao Guo; Ilan Komargodski

doi:10.4230/LIPIcs.APPROX-RANDOM.2015.850

Negation-limited formulas

Siyao Guo
, Ilan Komargodski

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-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 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	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
Pages	850-866
Number of pages	17
ISBN (Electronic)	9783939897897
DOIs	https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.850
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
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
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

Access to Document

10.4230/LIPIcs.APPROX-RANDOM.2015.850

Cite this

Guo, S., & Komargodski, I. (2015). Negation-limited formulas. In N. Garg, K. Jansen, A. Rao, & J. D. P. Rolim (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015 (pp. 850-866). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 40). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.850

Guo, Siyao ; Komargodski, Ilan. / Negation-limited formulas. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015. editor / Naveen Garg ; Klaus Jansen ; Anup Rao ; Jose D. P. Rolim. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2015. pp. 850-866 (Leibniz International Proceedings in Informatics, LIPIcs).

@inproceedings{e32de8568a484356aecc844601ffbba1,

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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.",

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author = "Siyao Guo and Ilan Komargodski",

note = "Publisher Copyright: {\textcopyright} Siyao Guo and Ilan Komargodski.; 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 ; Conference date: 24-08-2015 Through 26-08-2015",

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Guo, S & Komargodski, I 2015, Negation-limited formulas. in N Garg, K Jansen, A Rao & JDP Rolim (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015. Leibniz International Proceedings in Informatics, LIPIcs, vol. 40, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 850-866, 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, 24/08/15. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.850

Negation-limited formulas. / Guo, Siyao; Komargodski, Ilan.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015. ed. / Naveen Garg; Klaus Jansen; Anup Rao; Jose D. P. Rolim. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2015. p. 850-866 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 40).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Komargodski, Ilan

N1 - Publisher Copyright: © Siyao Guo and Ilan Komargodski.

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N2 - 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.

AB - 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.

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KW - Shrinkage

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Guo S, Komargodski I. Negation-limited formulas. In Garg N, Jansen K, Rao A, Rolim JDP, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2015. p. 850-866. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX-RANDOM.2015.850

Negation-limited formulas

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Keywords

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