Circuit complexity: Difference between revisions

Revision as of 18:47, 7 July 2023

Revision as of 08:16, 24 July 2023

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==Relation to time complexity==

If a certain language, <math>A</math>, belongs to the [[Complexity class|time-complexity class]] <math>\text{TIME}(t(n))</math> for some function <math>t:\mathbb{N}\to\mathbb{N}</math>, then <math>A</math> has circuit complexity <math>\mathcal{O}(t(n) \log t(n))</math>. If the Turing Machine that accepts the language is [[Turing machine equivalents|oblivious]] (meaning that it reads and writes the same memory cells regardless of input), then <math>A</math> has circuit complexity <math>\mathcal{O}(t(n))</math>.<ref>{{cite journal|firstfirst1=Nicholas|lastlast1=Pippenger|authorlink=Nick Pippenger|first2=Michael J.|last2=Fischer|authorlink2=Michael J. Fischer|title=Relations Among Complexity Measures|journal=[[Journal of the ACM]]|year=1979|volume=26|issue=3|pages=361–381|doi=10.1145/322123.322138|s2cid=2432526 }}

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<ref name="Ajtai-Komlós-Szemerédi_1983">{{cite book |author-last1=Ajtai |author-first1=Miklós |author-link1=Miklós Ajtai |author-last2=Komlós |author-first2=János |author-last3=Szemerédi |author-first3=Endre |title=An 0(n log n) sorting network |journal=STOC '83 Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing |pages=1–9 |date=1983 |isbn=978-0-89791-099-6}}</ref>

<ref name="Ajtai_1983">{{cite journal |author-first=Miklós |author-last=Ajtai |author-link=Miklós Ajtai |title=<math>\Sigma^1_1</math>-formulae on finite structures |journal=Annals of Pure and Applied Logic |date=1983 |volume=24 |pages=1–24 |doi=10.1016/0168-0072(83)90038-6|doi-access=free }}</ref>

<ref name="Furst-Saxe-Sipser_1984">{{cite journal |author-last1=Furst |author-first1=Merrick L. |author-last2=Saxe |author-first2=James Benjamin |author-link2=James Benjamin Saxe |author-last3=Sipser |author-first3=Michael Fredric |author-link3=Michael Fredric Sipser |doi=10.1007/BF01744431 |issue=1 |journal=[[Mathematical Systems Theory]] |mr=738749 |pages=13–27 |title=Parity, circuits, and the polynomial-time hierarchy |volume=17 |date=1984|s2cid=6306235 }}</ref>

<ref name="Santhanam_2007">{{cite conference |author-last=Santhanam |author-first=Rahul |url=http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.1811 |title=Circuit lower bounds for Merlin-Arthur classes |book-title=STOC 2007: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing |date=2007 |pages=275–283 |doi=10.1145/1250790.1250832 |citeseerx=10.1.1.92.4422}}</ref>

<ref name="Williams_2011">{{cite conference |author-last=Williams |author-first=Richard Ryan |author-link=Richard Ryan Williams |title=Non-Uniform ACC Circuit Lower Bounds |url=http://www.stanford.edu/~rrwill/acc-lbs.pdf |doi=10.1109/CCC.2011.36 |date=2011 |book-title=CCC 2011: Proceedings of the 26th Annual IEEE Conference on Computational Complexity |pages=115–125}}</ref>

<ref name="Kabanets-Impagliazzo_2004">{{cite journal |author-last1=Kabanets |author-first1=Valentine |author-last2=Impagliazzo |author-first2=Russell Graham |author-link3=Russell Graham Impagliazzo |journal=Computational Complexity |doi=10.1007/s00037-004-0182-6 |title=Derandomizing polynomial identity tests means proving circuit lower bounds |pages=1–46 |volume=13 |number=1 |date=2004|s2cid=12451799 }}</ref>

<ref name="Razborov-Rudich_1997">{{cite news |author-first1=Aleksandr Aleksandrovich |author-last1=Razborov |author-link1=Aleksandr Aleksandrovich Razborov |author-first2=Steven |author-last2=Rudich |author-link2=Steven Rudich |title=Natural proofs |journal=[[Journal of Computer and System Sciences]] |volume=55 |pages=24–35 |date=1997}}</ref>

<ref name="Carmosino-Impagliazzo-Kabanets-Kolokolova_2016">{{cite news |author-first1=Marco |author-last1=Carmosino |author-first2=Russell Graham |author-last2=Impagliazzo |author-link2=Russell Graham Impagliazzo |author-first3=Valentine |author-last3=Kabanets |author-first4=Antonina |author-last4=Kolokolova |title=Learning algorithms from natural proofs |journal=Computational Complexity Conference |date=2016}}</ref>

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<ref name="Rossman_2008">{{cite conference |author-first=Benjamin E. |author-last=Rossman |author-link=Benjamin E. Rossman |title=On the constant-depth complexity of k-clique |date=2008 |pages=721–730 |book-title=STOC 2008: Proceedings of the 40th annual ACM symposium on Theory of computing |publisher=[[Association for Computing Machinery]] |doi=10.1145/1374376.1374480}}</ref>

<ref name="Smolensky_1987">{{cite conference |author-first=Roman |author-last=Smolensky |title=Algebraic methods in the theory of lower bounds for Boolean circuit complexity |date=1987 |pages=77–82 |book-title=Proceedings of the 19th Annual ACM Symposium on Theory of Computing |publisher=[[Association for Computing Machinery]] |doi=10.1145/28395.28404|doi-access=free }}</ref>

<ref name="Alon-Boppana_1987">{{cite journal |author-first1=Noga |author-last1=Alon |author-link1=Noga Alon |author-first2=Ravi B. |author-last2=Boppana |title=The monotone circuit complexity of Boolean functions |journal=[[Combinatorica]] |volume=7 |date=1987 |number=1 |pages=1–22 |doi=10.1007/bf02579196 |citeseerx=10.1.1.300.9623|s2cid=17397273 }}</ref>

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