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Lowness and the complexity of sparse and tally descriptions

Vikraman Arvind, Johannes Köbler, Martin Mundhenk, Universität Ulm
2016
We investigate the complexity of obtaining sparse descriptions for sets in various reduction classes to sparse sets. Let A be a set in a certain reduction class R(SPARSE) to the class of sparse sets.  ...  This implies that the class IC[log,poly] of sets with low instance complexity is contained in first level of the extended low hierarchy.  ...  Acknowledgments The rst author is grateful to Uwe S c h oning for his hospitality and for the research environment provided at Universit at Ulm during the year 1991-92.  ... 
doi:10.18725/oparu-1059 fatcat:4u4m53tqjndbbdqfnk3o7djhxu

Page 7322 of Mathematical Reviews Vol. , Issue 90M [page]

1990 Mathematical Reviews  
The class of sets with polynomial-size circuits, often de- noted P/poly, plays an important role in structural complexity the- ory, and it is known that P/poly = P,,(SPARSE) = P;(SPARSE) = Py (TALLY) =  ...  We show that, for a slightly modified Turing machine model, the low level determin- istic and nondeterministic space bounded complexity classes are different.  ... 

Monotonous and randomized reductions to sparse sets

V. Arvind, J. Köbler, M. Mundhenk
1996 RAIRO - Theoretical Informatics and Applications  
We study the conséquences of the existence of sparse hard sets for different complexity classes under monotonous and randomized réductions.  ...  We prove trade-ojfs between the randomized time complexity of NP sets that reduce to a set B via such réductions and the density ofB as well as the number ofqueries made by the monotonous réduction.  ...  In order to prove the lowness of a set A that reduces to a sparse set, first a suitable bound for the complexity of a sparse description for A is derived.  ... 
doi:10.1051/ita/1996300201551 fatcat:yhc536jqwjfabj62fwgsdllv44

Page 2575 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews  
A consequence of this fact is that if a set A and its complement conjunctively reduce to a tally set, then A is in the EL*-level of the extended low hierarchy.  ...  The paper deals with comparative analysis of three parameters of tasks over finite or infinite check systems: the complexity of a task description, the minimal complexity of a decision tree solving this  ... 

Locating P/poly optimally in the extended low hierarchy [chapter]

Johannes Köbler
1993 Lecture Notes in Computer Science  
The location of P=poly in EL P 3 is optimal since, as shown by Allender and Hemachandra (1992) , there exist sparse sets that are not contained in the next lower level EL P 2 .  ...  A s a consequence of our result, all NP sets in P=poly are relocated from the third -level L P 3 (Ko and Sch oning, 1985) to the third -level L P 3 of the low hierarchy.  ...  Sheu, and R. Schuler. The author also thanks the two referees for their valuable suggestions.  ... 
doi:10.1007/3-540-56503-5_5 fatcat:wh7pgteoyfajbp5pbyyetjk4wa

Page 6613 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews  
Two important particular cases are investigated in detail: the identification of finite automata and the program synthesis of Post machines from operational descriptions of the computation for a finite  ...  With respect to the time complexity of strategies, it is shown that the strategies performing an optimal number of mind-changes are extremely time-consuming.  ... 

Kolmogorov complexity and degrees of tally sets

Eric Allender, Osamu Watanabe
1990 Information and Computation  
We show that either EP,(TALLY) = E;"(TALLY) or EP,  ...  ACKNOWLEDGMENTS We thank the referees for their careful work and for their many helpful comments, We also thank Ron Book for making us aware of this problem area.  ...  Also, note that S consists only of strings of low Kolmogorov complexity, since any element (x,, x2, . . . . xk) of S can be constructed from the descriptions of the x,.  ... 
doi:10.1016/0890-5401(90)90052-j fatcat:f5x6apq4afdpdabhdca2bt5ssm

In memoriam Ronald V. Book

D.-Z. Du, K.-I. Ko
1998 Theoretical Computer Science  
He originated studies of lowness of sparse sets, of characterizations of sparse sets by Kolmogorov complexity, and of reducibility and equivalence to sparse sets.  ...  This work has strongly influenced the direction of research in relativization. He was among the first to recognize the importance of tally sets, sparse sets, and other sets of succinct descriptions.  ... 
doi:10.1016/s0304-3975(98)90025-1 fatcat:icedtc3tencjbgl7ujm5wxdkn4

Locating P/poly optimally in the extended low hierarchy

J. Köbler
1994 Theoretical Computer Science  
The location of P/poly in EL3 ',@ is optimal since, as shown by Allender and Hemachandra (1992) , there exist sparse sets that are not contained in the next lower level ELt,x.  ...  As a consequence of our result, all NP sets in P/poly are relocated from the third X-level L, '9x (Ko and &honing, 1985) to the third O-level Ly,@ of the low hierarchy.  ...  TALLY denotes the class of all tally sets, and SPARSE denotes the class of all sparse sets. The join of two sets A and B is A @ B= {Ox IxEA} u {lx 1 XEB}.  ... 
doi:10.1016/0304-3975(94)00016-6 fatcat:dheas3gv4zb6tb425eh4gtxvwq

Hardness of Sparse Sets and Minimal Circuit Size Problem [article]

Bin Fu
2020 arXiv   pre-print
We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model.  ...  One of our results shows that if there exists a 2^n^o(1)-sparse set in NTIME(2^n^o(1)) that does not have any randomized streaming algorithm with n^o(1) updating time, and n^o(1) space, then NEXP≠BPP,  ...  Our magnification result has a flexible trade off between the spareness and time complexity. We use two functions d(n) and g(n) to control the sparseness of a tally set T .  ... 
arXiv:2003.00669v2 fatcat:m2en6mpxynenzlztisv7u7lq4y

Sparse Sets and Collapse of Complexity Classes

Vladimı́r Glasnák
2001 Information and Computation  
It is applied to prove that for a class of functions F a separation DTIME(F) = NTIME(F) can be characterized by the existence of (not only polynomially) sparse sets in certain complexity classes.  ...  Lett. 16, 55-60) is obtained: There is an Further we prove that there is an n O(log n) -sparse set in NP The technique also allows us to characterize the existence of sets of different densities in NP  ...  The importance of Theorem 6.6 is in the connection of tally sets and P-sparse sets.  ... 
doi:10.1006/inco.2001.3056 fatcat:uzeattp6jrb33lmjtasqc7gtda

Page 6549 of Mathematical Reviews Vol. , Issue 87k [page]

1987 Mathematical Reviews  
Chapters 5 and 6 deal with low and high hierarchies (measuring the amount of information in sets in NP) and relativization results about the P and NP problem.  ...  The following results are established. (1) A set has small generalized Kolmogorov complexity if and only if it is ‘semi-isomorphic’ to a tally set. (2) The class of sets with small generalized Kolmogorov  ... 

Learnability of Kolmogorov-easy circuit expressions via queries [chapter]

José L. Balcázar, Harry Buhrman, Montserrat Hermo
1995 Lecture Notes in Computer Science  
Upper and lower bounds on the learnability of circuit expressions are known. We study here the case in which the circuit expressions are of low (time-bounded) Kolmogorov complexity.  ...  The extension of the results to various Kolmogorov complexity bounds is discussed.* *  ...  We study here the case in which the circuit expressions are of low timebounded Kolmogorov complexity; speci cally, the case in which they have logarithmically long descriptions, from which the (polynomial  ... 
doi:10.1007/3-540-59119-2_172 fatcat:lum6nk4mrve6xgzwk6ghtemfau

On the Existence of Complete Disjoint NP-Pairs

Olaf Beyersdorff
2009 2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity.  ...  The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field.  ...  ACKNOWLEDGMENTS I thank Jan Krajíček and Zenon Sadowski for helpful discussions on the topic of this paper.  ... 
doi:10.1109/synasc.2009.9 dblp:conf/synasc/Beyersdorff09 fatcat:tpujekk3uvgytiwd4q6ekudd5i

Relating Equivalence and Reducibility to Sparse Sets

Eric Allender, Lane A. Hemachandra, Mitsunori Ogiwara, Osamu Watanabe
1992 SIAM journal on computing (Print)  
Additionally, this paper studies the relative power of different notions of reducibility, and proves that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting  ...  and reducibility to sparse sets provably differ.  ...  We thank the Tokyo Institute of Technology for hosting a workshop on computational complexity, in August 1990, at which this work was done in part.  ... 
doi:10.1137/0221034 fatcat:cpyverduyfektmdm56fjjmpg5i
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