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Threshold functions and bounded depth monotone circuits

Ravi B. Boppana
1986 Journal of computer and system sciences (Print)  
ACKNOWLEDGMENT I would like to thank my adviser Michael Sipser for many valuable discussions.  ...  Exponential lower bounds for monotone circuits have also been obtained by Klawe et al. [S] .  ...  Valiant [ 1 l] gave an exponential lower bound for monotone C, circuits which detect cliques in graphs.  ... 
doi:10.1016/0022-0000(86)90027-9 fatcat:l2wdtu73lrbxlcqmoxtqz4p3jq

Boolean Function Complexity Advances and Frontiers

Stasys Jukna
2014 Bulletin of the European Association for Theoretical Computer Science  
Acknowledgement I am thankful to Sasha Razborov for his comments on this summary.  ...  Strong (even exponential) lower bounds were only obtained for various restricted circuit models. Below I give a rough overview of the book's contents.  ...  Exponential lower bounds are only known for depth-2 circuits. Restrict the time, but allow omnipotent power In general circuits, arbitrary boolean functions are allowed as gates.  ... 
dblp:journals/eatcs/Jukna14 fatcat:x3y2bootivdd7mx4xswv7myrhy

Lower bounds on monotone arithmetic circuits with restricted depths

Joseph JáJá
1985 Computers and Mathematics with Applications  
These results imply exponential lower bounds for circuits of bounded depths which compute any of these functions. We also obtain several examples for which negation can reduce the size exponentially.  ...  We develop general lower-and upper-bound techniques that seem to generate almost-matching bounds for all the functions consldered.  ...  This bound almost matches all the lower bounds generated by our lower-bound technique. in particular we show the following bounds on the sizes S of monotone circuits of depth d (d' = (d/2]).  ... 
doi:10.1016/0898-1221(85)90103-8 fatcat:3fmwnigfwrb65piqxgqsm7gz5u

Explicit lower bounds on strong quantum simulation [article]

Cupjin Huang, Michael Newman, Mario Szegedy
2018 arXiv   pre-print
We consider the problem of strong (amplitude-wise) simulation of n-qubit quantum circuits, and identify a subclass of simulators we call monotone.  ...  We prove an unconditional (i.e. without relying on any complexity theoretic assumptions) and explicit (n-2)(2^n-3-1) lower bound on the running time of simulators within this subclass.  ...  Acknowledgments The authors would like to thank Jianxin Chen, Yaoyun Shi, Fang Zhang, and Avi Widgerson for helpful discussions.  ... 
arXiv:1804.10368v2 fatcat:5coshesm5bfb7ay56k6t77npei

Page 6631 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
but requires exponential size on monotone circuits [A.  ...  The presence of negations can lead to an exponential gain in size for non-monotone circuits computing certain monotone functions (e.g., bipartite perfect matching can be done in polynomial size with negations  ... 

An Exponential Lower Bound for the Size of Monotone Real Circuits

Armin Haken, Stephen A. Cook
1999 Journal of computer and system sciences (Print)  
We prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection.  ...  The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary real-valued functions (including AND and  ...  INTRODUCTION The RazborovÂAndreev [Raz85, AB87, And85] exponential lower bound on the size of monotone Boolean circuits which detect cliques represented a breakthrough in the theory of monotone circuit  ... 
doi:10.1006/jcss.1998.1617 fatcat:52inmph7cbfurof5nletxjygra

The Complexity of Finite Functions [chapter]

Ravi B. BOPPANA, Michael SIPSER
1990 Algorithms and Complexity  
These include bounded depth circuits, monotone circuits, and bounded width branching programs.  ...  The primary aim is to give accessible proofs of the more di cult theorems proving lower bounds on the complexity of speci c functions in restricted computational models.  ...  Zwick for extensive comments on an earlier draft of this paper. 62  ... 
doi:10.1016/b978-0-444-88071-0.50019-9 fatcat:5drwjnupi5ahrjmf4jvyzzqnna

The history and status of the P versus NP question

Michael Sipser
1992 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing - STOC '92  
Yao~a85] combined probabilistic restriction with a type of approximation to give an exponential lower bound for constant depth parity circuits.  ...  Take, for example Razborov's lower bound for the clique function. Consider any monotone circuit com- puting this function.  ...  from a generalization of the proof for the unsolvability of the Entscheidungsproblem; 2.) p(n) N Kn (or N Kn2) means, of course, simply that the number of steps vis-h-via dem Mossen Pro biereng can be  ... 
doi:10.1145/129712.129771 dblp:conf/stoc/Sipser92 fatcat:rcg35ekcnrejflhp4vct67qxby

Lower Bounds for DeMorgan Circuits of Bounded Negation Width

Stasys Jukna, Andrzej Lingas, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
Our motivation is that already circuits of moderate negation width w = n for an arbitrarily small constant > 0 can be even exponentially stronger than monotone circuits.  ...  Circuits of negation width w = 0 are equivalent to monotone Boolean circuits, while those of negation width w = n have no restrictions.  ...  While strong, even exponential, lower bounds for explicit monotone Boolean functions are already known for monotone Boolean {∨, ∧} circuits, we can currently prove only depressingly small (linear) lower  ... 
doi:10.4230/lipics.stacs.2019.41 dblp:conf/stacs/JuknaL19 fatcat:m34haekq4nd2nhur4ubevzuya4

On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems

Maria Luisa Bonet, Juan Luis Esteban, Nicola Galesi, Jan Johannsen
2000 SIAM journal on computing (Print)  
An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refutations is proved.  ...  In both cases only superpolynomial separations were known [In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [Combinatorica, 19 (1999), pp.  ...  We would like to thank Ran Raz for reading a previous version of this work and discovering an error, Andreas Goerdt for sending us copies of his papers, Sam Buss for helpful discussions, and, finally,  ... 
doi:10.1137/s0097539799352474 fatcat:xiyrppe6vvc6zakntfetsrjcdu

Page 5694 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
bound for monotone arithmetic circuits computing 0-1] permanent.  ...  In this model, Jerrum and Snir have proved exponential lower bounds for the size of circuits computing the permanent for general inputs.  ... 

Page 4494 of Mathematical Reviews Vol. , Issue 87h [page]

1987 Mathematical Reviews  
Almost optimal exponential lower bounds on the monotone com- plexity of several functions with respect to arithmetic circuits with restricted depth are proved.  ...  Alexander Leitsch (Newark, Del.) 87h:68048 J’a J’a, Joseph (1-MD-E) 87h:68049 Lower bounds on monotone arithmetic circuits with restricted depths. Comput. Math. Appl. 11 (1985), no. 12, 1155-1164.  ... 

Page 7072 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
One restricts to monotonicity because we do not know any method for proving nontrivial lower bounds on the size of gen- eral Boolean circuits (combinational complexity).  ...  lower bound on the depth of any polynomial-size monotone circuit computing connectivity. This improves the previous result of A. C.-C.  ... 

Observations on Symmetric Circuits [article]

Christian Engels
2020 arXiv   pre-print
Their result showed an exponential lower bound of the permanent computed by symmetric circuits.  ...  We also show super-polynomial lower bounds for smaller groups.  ...  An approach to this question is to prove lower bounds in restricted models and try to extend these.  ... 
arXiv:2007.07496v3 fatcat:r2nl6u3de5hd3mduu4pbtxo2va

Random Formulas, Monotone Circuits, and Interpolation

Pavel Hrubes, Pavel Pudlak
2017 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)  
We give a superpolynomial lower bound on monotone real circuits that approximately decide the satisfiability of k-CNFs, where k = ω(1). For k ≈ log n, the lower bound is exponential.  ...  Specifically, we prove exponential lower bounds for random k-CNFs, where k is the logarithm of the number of variables, and for the Weak Bit Pigeon Hole Principle.  ...  ACKNOWLEDGEMENT We thank Neil Thapen and Nicola Galesi for useful discussions. This research was supported by ERC grant FEALORA 339691.  ... 
doi:10.1109/focs.2017.20 dblp:conf/focs/HrubesP17 fatcat:zywld4dapvac7kpshlxghggvae
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