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The complexity of central slice functions

Paul E. Dunne
1986 Theoretical Computer Science  
large combinational complexity, then some slice off must have large monotone complexity.  ...  Berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of fk imply lower bounds of the same order on the combinational complexity of f, and that iff has  ...  Unfortunately, none of these results imply superlinear lower bounds on the combinational complexity of any function or set of functions since no efficienl simulation of combinational networks by monotone  ... 
doi:10.1016/0304-3975(86)90122-2 fatcat:rjbhxuvatzbitph77th2733yvi

The complexity of negation-limited networks — A brief survey [chapter]

Michael J. Fischer
1975 Lecture Notes in Computer Science  
combinational complexity.  ...  Unfortunately, few techniques exist for proving lower hounds on the combinational complexity of specific functions of interest, even though the following theorem shows that "most" functions are hard. 2  ... 
doi:10.1007/3-540-07407-4_9 fatcat:tdbkmyttjndqvez3of7u6djibm

On monotone simulations of nonmonotone networks

Paul E. Dunne
1989 Theoretical Computer Science  
= the set of all n-argument monotone Boolean functions, fk (X,) = k-slice of some function f (X,,) in Mn 9 c r a = i = 1 ai, C(f) = combinational complexity of J; C"(f) = monotone network complexity  ...  Recent work of Berkowitz [3] , Wegener [is, 151 and Dunne [S, 6) has considered the problem of relating the combinational and monotone network complexity of monotone Boolean functions via slice functions  ...  The result of Section 3 shows that the last relation of Corollary 5 is in fact the best possible, since there it was proved that C"*(J) = O(&'/log n) for any function in this final class. References  ... 
doi:10.1016/0304-3975(89)90142-4 fatcat:duhyw4abxrb3rmao2ggv2tio5q

The gap between monotone and non-monotone circuit complexity is exponential

É. Tardos
1988 Combinatorica  
Razborov has shown that there exists a polynomial time computable monotone Booleanfunction whose monotone circuit complexity is at least n clos ".  ...  The proof is immediate by combining the Alon-Boppana version of another argument of Razborov with results of Grotschel-Lovasz-Schrijver on the Lovasz -capacity, 9 of a graph.  ...  I would like to thank Laszlo Babai for many helpful discussions, and Laszlo Lovasz for pointing out an error in an earlier version of this note.  ... 
doi:10.1007/bf02122563 fatcat:vkdl2vzfifdjpjz7b5knnp57am

On the complexity of slice functions

Ingo Wegener
1985 Theoretical Computer Science  
By a result of Berkowitz (1982) , the monotone circuit complexity of slice functions cannot be much larger than the circuit (combinational) complexity of these functions for arbitrary complete bases.  ...  All the main methods known for proving lower bounds on the monotone complexity of Boolean functions fail to work in their present form for slice functions.  ...  For the computation of Boolean functions we consider Boolean circuits (for the definition and elementary properties, see [6] ) and the circuit complexity (combinational complexity) either over the complete  ... 
doi:10.1016/0304-3975(85)90209-9 fatcat:zmdf2oxlcngf5gg4htxa4s3qui

On the Negation-Limited Circuit Complexity of Merging [chapter]

Kazuyuki Amano, Akira Maruoka, Jun Tarui
1999 Lecture Notes in Computer Science  
A negation-limited circuit is a combinational circuit that consists of AND, OR gates and a limited number of NOT gates. In this paper, we investigate the complexity of negation-limited circuits.  ...  We prove that the size complexity of the (n; n) merging function with t = (log 2 log 2 n − a) NOT gates is (2 a n). ?  ...  Combining this and the fact that the monotone circuit complexity of MERGE(n=2 t ; n=2 t ) is ((n=2 t )log(n=2 t )) = (2 a n) to obtain the lower bound for the size of C.  ... 
doi:10.1007/3-540-48686-0_20 fatcat:ub6elrmuofac7k4vu32d7r2x2u

On the negation-limited circuit complexity of merging

Kazuyuki Amano, Akira Maruoka, Jun Tarui
2003 Discrete Applied Mathematics  
A negation-limited circuit is a combinational circuit that consists of AND, OR gates and a limited number of NOT gates. In this paper, we investigate the complexity of negation-limited circuits.  ...  We prove that the size complexity of the (n; n) merging function with t = (log 2 log 2 n − a) NOT gates is (2 a n). ?  ...  Combining this and the fact that the monotone circuit complexity of MERGE(n=2 t ; n=2 t ) is ((n=2 t )log(n=2 t )) = (2 a n) to obtain the lower bound for the size of C.  ... 
doi:10.1016/s0166-218x(02)00215-9 fatcat:hgx6dakwazbrvjhqopu3ohqauy

Fast submodular maximization subject to k-extendible system constraints [article]

Teng Li and Hyo-Sang Shin and Antonios Tsourdos
2018 arXiv   pre-print
submodular functions and of (p(1-p)-ϵ) for non-monotone cases with expected computational complexity of only O(pn/ϵr/ϵ), where r is the largest size of the feasible solutions, 0<p ≤1/1+k is the sampling  ...  Combining the sampling process and the decreasing threshold strategy, our algorithm Sample Decreasing Threshold Greedy Algorithm (SDTGA) obtains an expected approximation guarantee of (p-ϵ) for monotone  ...  For non-monotone submodular functions, Gupta et al. [25] provided an approximation ratio of k (k+1)(3k+3) with time complexity of O(nrk).  ... 
arXiv:1811.07673v1 fatcat:ib3virxnsjgkvozfwvylqlum6q

Lower Bounds for Testing Properties of Functions over Hypergrid Domains

Eric Blais, Sofya Raskhodnikova, Grigory Yaroslavtsev
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
We use this method to prove strong, and in many cases optimal, lower bounds on the query complexity of testing fundamental properties of functions f : {1, . . . , n} d → R over hypergrid domains: monotonicity  ...  We show how the communication complexity method introduced in (Blais, Brody, Matulef 2012) can be used to prove lower bounds on the number of queries required to test properties of functions with non-hypercube  ...  We also thank Joshua Brody and Oded Goldreich for insightful conversations about the communication complexity method.  ... 
doi:10.1109/ccc.2014.38 dblp:conf/coco/BlaisRY14 fatcat:dj4ouaxx25akhbinvxpnfo7hoi

Combining testing and proof to gain high assurance in software: A case study

Peter Bishop, Robin Bloomfield, Lukasz Cyra
2013 2013 IEEE 24th International Symposium on Software Reliability Engineering (ISSRE)  
This approach combines a monotonicity analysis with a defined set of tests to demonstrate the accuracy of a software function over its entire input range.  ...  This paper presents a specific example of this approach applied to the verification of continuous monotonic functions.  ...  The combination of static and dynamic analysis comprises: • a formal proof of some property P of program f(⋅) • a specific set of test inputs T drawn from the input domain I that meet some success criterion  ... 
doi:10.1109/issre.2013.6698924 dblp:conf/issre/BishopBC13 fatcat:qwnfbtig3jdurggzria3vhn7b4

Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

Marc Roth, Johannes Schmitt
2020 Algorithmica  
Euler characteristic of the associated simplicial (graph) complex of Φ is non-zero.  ...  monotone property Φ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for k > 2 .  ...  Furthermore we wish to point out that Sloane's OEIS [1] was very useful in the course of this work.  ... 
doi:10.1007/s00453-020-00676-9 pmid:32801408 pmcid:PMC7416760 fatcat:s7i4a6yrtjd43okvw3r4cmreqe

Page 5428 of Mathematical Reviews Vol. , Issue 82m [page]

1982 Mathematical Reviews  
The algorithm makes use of the Boolean difference technique.” Tosi¢, Ratko 82m:94062 An optimal identification algorithm for some subclasses of monotone Boolean functions.  ...  The author assumes that Boolean functions belong to some class of monotone ones and gives an identification procedure. W. Greblicki (Wroclaw) Wu, X.; Hurst, S.  ... 

Page 2651 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews  
complexity of f implies large monotone complexity of some k-slice of f which is not effectively specified.  ...  Further, it has been shown that large lower bounds on the monotone complexity of some k-slice of f imply lower bounds of the same order on the com- binational complexity of {, and conversely, large combinational  ... 

Lower bounds for monotone span programs

A. Beimel, A. Gal, M. Paterson
1995 Proceedings of IEEE 36th Annual Foundations of Computer Science SFCS-95  
The main result proved here yields quadratic lower bounds for the size of monotone span programs, improving on the largest previously known bounds for explicit functions.  ...  The bound is asymptotically tight for the function corresponding to a class of 4-cliques.  ...  Next we present a monotone span program of linear size (exactly m) for a function on m variables, that is known to have ((m= log m) 3=2 ) monotone circuit complexity.  ... 
doi:10.1109/sfcs.1995.492669 dblp:conf/focs/BeimelGP95 fatcat:vc4la34uarh5ji53esx6wsueya

Page 6983 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
Since for slice functions the combinational complexity and the monotone complexity are closely related, it was assumed that the investiga- tion of the monotone complexity of slice functions would yield  ...  an important approach for proving nonlinear lower bounds on the combinational complexity of Boolean functions.  ... 
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