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Collapsing Exact Arithmetic Hierarchies
[chapter]

2014
*
Lecture Notes in Computer Science
*

We provide a uniform framework for proving the

doi:10.1007/978-3-319-04657-0_26
fatcat:m4umv2v5kjda5jjszwjzq67dni
*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. ...##
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Recursion theoretic characterizations of complexity classes of counting functions

1996
*
Theoretical Computer Science
*

in the context of this

doi:10.1016/0304-3975(95)00237-5
fatcat:3esk7336bbh3rpznmclz6enr7a
*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] . ...##
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The bounded arithmetic hierarchy

1978
*
Information and Control
*

In analogy with Kleene's

doi:10.1016/s0019-9958(78)90257-7
fatcat:aslt6dyy7fbk3l6xnzmzva4zk4
*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 ...##
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Arithmetic theories for computational complexity problems

1986
*
Information and Control
*

This paper considers a number of

doi:10.1016/s0019-9958(86)80041-9
fatcat:mp344yvwefeitja4jhkenj3n7q
*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. ...##
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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 ...##
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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. ...##
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Efficient Learning Algorithms Yield Circuit Lower Bounds
[chapter]

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

doi:10.1007/11776420_27
fatcat:2doc46s7mba7bk4dzhjxaznsti
*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 ). ...##
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Efficient learning algorithms yield circuit lower bounds

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

doi:10.1016/j.jcss.2008.07.006
fatcat:tb56jbg74zazfcb2vfviik4wgy
*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 ). ...##
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Counting classes and the fine structure between NC1 and L

2012
*
Theoretical Computer Science
*

In particular, the constant-depth oracle

doi:10.1016/j.tcs.2011.05.050
fatcat:wy7i2fwrurapnphsvrz5bufvjq
*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. ...##
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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*. ...##
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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*...##
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Counting Classes and the Fine Structure between NC 1 and L
[chapter]

2010
*
Lecture Notes in Computer Science
*

In particular, the constant-depth oracle

doi:10.1007/978-3-642-15155-2_28
fatcat:hpneqo5t3rgtfm3ss56kegtnmq
*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). ...##
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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. ...##
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Interpolating Arithmetic Read-Once Formulas in Parallel

1998
*
SIAM journal on computing (Print)
*

We present a randomized (Las Vegas) parallel algorithm for the

doi:10.1137/s009753979528812x
fatcat:fn3yngysbfapflyl2f3sn5ha7q
*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 ? ...##
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Industrial application of exact Boolean operations for meshes

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

doi:10.1145/1925059.1925089
dblp:conf/sccg/SchifkoJK10
fatcat:4wlk3k766vbddddv74pz2qmsmm
*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*. ...
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