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Lowness and the complexity of sparse and tally descriptions

2016

We investigate

doi:10.18725/oparu-1059
fatcat:4u4m53tqjndbbdqfnk3o7djhxu
*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. ...##
###
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

1996
*
RAIRO - Theoretical Informatics and Applications
*

We study

doi:10.1051/ita/1996300201551
fatcat:yhc536jqwjfabj62fwgsdllv44
*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. ...##
###
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]

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. ...

##
###
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

1990
*
Information and Computation
*

We show that either EP,(

doi:10.1016/0890-5401(90)90052-j
fatcat:f5x6apq4afdpdabhdca2bt5ssm
*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,. ...##
###
In memoriam Ronald V. Book

1998
*
Theoretical Computer Science
*

He originated studies

doi:10.1016/s0304-3975(98)90025-1
fatcat:icedtc3tencjbgl7ujm5wxdkn4
*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*. ...##
###
Locating P/poly optimally in the extended low hierarchy

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}. ...

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

2020
*
arXiv
*
pre-print

We develop a polynomial method on finite fields to amplify

arXiv:2003.00669v2
fatcat:m2en6mpxynenzlztisv7u7lq4y
*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 . ...##
###
Sparse Sets and Collapse of Complexity Classes

2001
*
Information and Computation
*

It is applied to prove that for a class

doi:10.1006/inco.2001.3056
fatcat:uzeattp6jrb33lmjtasqc7gtda
*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. ...##
###
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]

1995
*
Lecture Notes in Computer Science
*

Upper

doi:10.1007/3-540-59119-2_172
fatcat:lum6nk4mrve6xgzwk6ghtemfau
*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 ...##
###
On the Existence of Complete Disjoint NP-Pairs

2009
*
2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
*

Disjoint NP-pairs are an interesting model

doi:10.1109/synasc.2009.9
dblp:conf/synasc/Beyersdorff09
fatcat:tpujekk3uvgytiwd4q6ekudd5i
*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. ...##
###
Relating Equivalence and Reducibility to Sparse Sets

1992
*
SIAM journal on computing (Print)
*

Additionally, this paper studies

doi:10.1137/0221034
fatcat:cpyverduyfektmdm56fjjmpg5i
*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. ...
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