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Collapsing Exact Arithmetic Hierarchies [chapter]

Nikhil Balaji, Samir Datta
2014 Lecture Notes in Computer Science  
We provide a uniform framework for proving the collapse of the hierarchy, NC 1 (C) for an exact arithmetic class C of polynomial degree.  ...  Our proof generalizes a proof from [8] where it is used to prove the collapse of the AC 0 (C=NC 1 ) hierarchy.  ...  The most well known amongst the exact arithmetic circuit hierarchy collapses is the collapse of the NC 1 hierarchy over C = L.  ... 
doi:10.1007/978-3-319-04657-0_26 fatcat:m4umv2v5kjda5jjszwjzq67dni

Recursion theoretic characterizations of complexity classes of counting functions

Heribert Vollmer, Klaus W. Wagner
1996 Theoretical Computer Science  
in the context of this hierarchy the operation of modified subtraction is as powerful as substitution.  ...  This leads us to a number of consequences concerning closure of #P under certain arithmetical operations.  ...  equivalent to a collapse of the counting hierarchy to UP [22] .  ... 
doi:10.1016/0304-3975(95)00237-5 fatcat:3esk7336bbh3rpznmclz6enr7a

The bounded arithmetic hierarchy

Keith Harrow
1978 Information and Control  
In analogy with Kleene's arithmetic hierarchy, there is a bounded arithmetic hierarchy of predicate classes within B-d, based on the number of alternations of bounded quantifiers.  ...  Although the existence of a strict hierarchy is not established, necessary and sufficient conditions for the hierarchy to be strict are shown.  ...  ACKNOWLEDGMENTS The author would like to thank Martin Davis for originally suggesting the question of a bounded arithmetic hierarchy, Sheldon Finkelstein for his careful reading of the manuscript and simplification  ... 
doi:10.1016/s0019-9958(78)90257-7 fatcat:aslt6dyy7fbk3l6xnzmzva4zk4

Arithmetic theories for computational complexity problems

Steve Homer, John Reif
1986 Information and Control  
This paper considers a number of arithmetic theories and shows how the strength of these theories relates to certain open problems in complexity theory concerning the polynomial-time hierarchy.  ...  These results are proved quite generally and hold for a wide class of subrecursive hierarchies. They can be used to characterize certain properties of functions provable in these theories.  ...  The results give an exact correspondence between lower bound proofs and independence of particular true sentences from the fragments of arithmetic herein considered.  ... 
doi:10.1016/s0019-9958(86)80041-9 fatcat:mp344yvwefeitja4jhkenj3n7q

Page 5 of Mathematical Reviews Vol. 26, Issue 1 [page]

1963 Mathematical Reviews  
Anderson [ibid. 8 (1961), 587] has since established an exact con- nection between the ‘collapse’ of Kleene’s hierarchy and the possibility of defining arbitrary REC functions by means of ordinal recursions  ...  Notices 8 (1961), 276-277] that for quite arbitrary expanding REC hierarchies of REC functions one has non-uniqueness at w*+ 1 (though the connections between collapse and non- uniqueness may be complicated  ... 

Page 6352 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews  
A wide range of computational complexity hierarchies are shown to collapse to a low level.  ...  Summary: “A refinement of the Hausdorff hierarchy generated by NP is investigated. The new classes allow an exact complexity classification of certain counting problems.  ... 

Efficient Learning Algorithms Yield Circuit Lower Bounds [chapter]

Lance Fortnow, Adam R. Klivans
2006 Lecture Notes in Computer Science  
We prove that the existence of an efficient learning algorithm for a circuit class C in Angluin's model of exact learning from membership and equivalence queries or in Valiant's PAC model yields a lower  ...  More specifically, we prove that any subexponential time, deterministic exact learning algorithm for C (from membership and equivalence queries) implies the existence of a function f in EXP NP such that  ...  Perhaps the most well-known collapse is due to Karp and Lipton, stating that if NP ⊆ P/poly then the polynomial-time hierarchy collapses (PH = Σ 2 ).  ... 
doi:10.1007/11776420_27 fatcat:2doc46s7mba7bk4dzhjxaznsti

Efficient learning algorithms yield circuit lower bounds

Lance Fortnow, Adam R. Klivans
2009 Journal of computer and system sciences (Print)  
We prove that the existence of an efficient learning algorithm for a circuit class C in Angluin's model of exact learning from membership and equivalence queries or in Valiant's PAC model yields a lower  ...  More specifically, we prove that any subexponential time, deterministic exact learning algorithm for C (from membership and equivalence queries) implies the existence of a function f in EXP NP such that  ...  Perhaps the most well-known collapse is due to Karp and Lipton, stating that if NP ⊆ P/poly then the polynomial-time hierarchy collapses (PH = Σ 2 ).  ... 
doi:10.1016/j.jcss.2008.07.006 fatcat:tb56jbg74zazfcb2vfviik4wgy

Counting classes and the fine structure between NC1 and L

Samir Datta, Meena Mahajan, B.V. Raghavendra Rao, Michael Thomas, Heribert Vollmer
2012 Theoretical Computer Science  
In particular, the constant-depth oracle hierarchy over PNC 1 collapses to its first level PNC 1 , and the constant-depth oracle hierarchy over C = NC 1 collapses to its second level.  ...  We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses.  ...  The PNC 1 hierarchy collapses In this section we show that the constant-depth PNC 1 hierarchy, AC 0 (PNC 1 ), collapses to the base level. Theorem 19. AC 0 (PNC 1 ) = PNC 1 . Proof.  ... 
doi:10.1016/j.tcs.2011.05.050 fatcat:wy7i2fwrurapnphsvrz5bufvjq

Page 902 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
The main result is a generalization to the arithmetical hierarchy of the theo- rem by A. B.  ...  The authors suggest a general definition of the notion of a com- putable numeration which allows them to study computable nu- merations of the classes of the arithmetical hierarchy.  ... 

Page 2779 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
(D-WRZB-TI; Wiirzburg) Arithmetic circuits and polynomial replacement systems.  ...  We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses  ... 

Counting Classes and the Fine Structure between NC 1 and L [chapter]

Samir Datta, Meena Mahajan, B. V. Raghavendra Rao, Michael Thomas, Heribert Vollmer
2010 Lecture Notes in Computer Science  
In particular, the constant-depth oracle hierarchy over PNC 1 collapses to its first level PNC 1 , and the constant-depth oracle hierarchy over C=NC 1 collapses to its second level.  ...  We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses.  ...  Next, we show that the constant-depth hierarchy over PNC 1 (and hence also the Boolean hierarchy) collapses to PNC 1 (Theorem 4.1).  ... 
doi:10.1007/978-3-642-15155-2_28 fatcat:hpneqo5t3rgtfm3ss56kegtnmq

Page 1793 of Mathematical Reviews Vol. , Issue 85e [page]

1985 Mathematical Reviews  
Thus, no weakly P-selective set can be <f-hard for NP unless the polynomial-time hierarchy (PH) collapses to LP.  ...  Some remarkable formulas are obtained which give exact connections between this and the other two hierarchies of number-theoretic functions. The exposition is self-contained and crystal clear. W.  ... 

Interpolating Arithmetic Read-Once Formulas in Parallel

Nader H. Bshouty, Richard Cleve
1998 SIAM journal on computing (Print)  
We present a randomized (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over su ciently large elds.  ...  An arithmetic formula is one in which the operations are addition, subtraction, multiplication, and division (and constants are allowed).  ...  We present a (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over su ciently large elds. For elds of size at least 3(n 2 + 3n ?  ... 
doi:10.1137/s009753979528812x fatcat:fn3yngysbfapflyl2f3sn5ha7q

Industrial application of exact Boolean operations for meshes

Martin Schifko, Bert Jüttler, Bernhard Kornberger
2010 Proceedings of the 26th Spring Conference on Computer Graphics - SCCG '10  
In order to avoid potential robustness problems, which may be caused by (almost) degenerate triangles or by intersections of nearly co-planar triangles, we use filtered exact arithmetic, based on the libraries  ...  CGAL and GNU Multi Precision Arithmetic Library.  ...  This method relies on exact arithmetic to avoid the well known problems with floatingpoint arithmetic.  ... 
doi:10.1145/1925059.1925089 dblp:conf/sccg/SchifkoJK10 fatcat:4wlk3k766vbddddv74pz2qmsmm
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