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Arithmetizing Uniform NC

1991
*
Annals of Pure and Applied Logic
*

.,

doi:10.1016/0168-0072(91)90057-s
fatcat:enyupbcq4zb7tpucnef23ewzay
*Arithmetizing**Uniform**NC*, Annals of Pure and Applied Logic 53 (1991) l-50. ... We then define a fragment of bounded*arithmetic*, and, using our characterization of*Uniform**NC*, show that this fragment is capable of proving the totality of all of the functions in*Uniform**NC*. ... Remember, the main goal is to give a characterization of*Uniform**NC*in a theory of*arithmetic*.*Uniform**NC*contains functions which have polynomial growth rate. ...##
###
Boolean circuits versus arithmetic circuits

1991
*
Information and Computation
*

Over Q and finite fields, Boolean circuits can simulate

doi:10.1016/0890-5401(91)90078-g
fatcat:aydwnfiodfgy3bizzgauso2sei
*arithmetic*circuits efficiently with respect to size. ... We compare the two computational models of Boolean circuits and*arithmetic*circuits in cases where they both apply, namely the computation of polynomials over the rational numbers or over finite fields ...*ARITHMETIC*vs BOOLEAN CIRCUITS OVER 0 size. For the complexity class*NC*'(P-*uniform*), we also restrict the depth of the circuits to be O(log(inputsize)'), and finally*NC*(P-*uniform*) = 643!91! ...##
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Collapsing Exact Arithmetic Hierarchies
[chapter]

2014
*
Lecture Notes in Computer Science
*

We provide a

doi:10.1007/978-3-319-04657-0_26
fatcat:m4umv2v5kjda5jjszwjzq67dni
*uniform*framework for proving the collapse of the hierarchy,*NC*1 (C) for an exact*arithmetic*class C of polynomial degree. ... Our main collapsing exhibits are the classes C ∈ {C=*NC*1 , C=L, C=SAC 1 , C=P}.*NC*1 (C=L) and*NC*1 (C=P) are already known to collapse [1, 18, 19] . ... We provide a*uniform*framework for proving the collapse of the*NC*1 hierarchy over an exact*arithmetic*class. ...##
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Points not as hyperplane sections of projectively normal curves

1991
*
Proceedings of the American Mathematical Society
*

Thus, since C is linearly normal, its

doi:10.1090/s0002-9939-1991-1052870-9
fatcat:7jymvemgdbaihff6enqgowioea
*arithmetic*genus g is d-n-l . Let*Nc*be the normal sheaf (Ic/Ic)* to C in P"+ . ...*Uniform*position implies syzygetic*uniform*position for points in the plane by [CO] , but not in general. ...##
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Points not as Hyperplane Sections of Projectively Normal Curves

1991
*
Proceedings of the American Mathematical Society
*

Thus, since C is linearly normal, its

doi:10.2307/2048725
fatcat:qs4ir6cgsraopa543gh75nu6sa
*arithmetic*genus g is d-n-l . Let*Nc*be the normal sheaf (Ic/Ic)* to C in P"+ . ...*Uniform*position implies syzygetic*uniform*position for points in the plane by [CO] , but not in general. ...##
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Towards a tight hardness–randomness connection between permanent and arithmetic circuit identity testing

2012
*
Information Processing Letters
*

is strictly contained in

doi:10.1016/j.ipl.2012.08.001
fatcat:iyw4sp5iarbgza6xo7g4gkejoi
*uniform**NC*2 . ... that the Boolean circuit class*uniform*TC 0 is strictly contained in*uniform**NC*2 . ... The rank of an integer matrix can be computed in logspace*uniform**NC*2 , cf. [ABO99] . Due to [Ruz81] , logspace-*uniform**NC*2 is known to equal DLOGTIME-*uniform**NC*2 . ...##
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An Optimal Parallel Algorithm for Formula Evaluation

1992
*
SIAM journal on computing (Print)
*

This approach is then used to solve the more general problem of evaluating

doi:10.1137/0221046
fatcat:jc23c5v2jveivmsm3lxgqsoisu
*arithmetic*formulas using*arithmetic*circuits. ... A new approach to Buss's*NC*1 algorithm [Bus87] for evaluation of Boolean formulas is presented. This problem is shown to be complete for*NC*1 over AC 0 reductions. ... We use U E -*uniformity*in our definition of*NC*instead of the more common U E * -*uniformity*. Ruzzo shows that*NC*k (k ≥ 1) is the same under either definition. ...##
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Complexity of Regular Functions
[chapter]

2015
*
Lecture Notes in Computer Science
*

Let {D n } be the

doi:10.1007/978-3-319-15579-1_35
fatcat:7b67gh6a4bdm5jxzpne3xe6qx4
*uniform*family of*arithmetic*circuits, such that D n is the connected subcircuit of C n consisting only of*arithmetic*min and + gates. ... We remark that, for constant-depth classes such as AC 0 and TC 0 , U E -*uniformity*coincides with U D -*uniformity*, which is also frequently called DLOGTIME-*uniformity*.) ...##
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Inversion in finite fields using logarithmic depth

1990
*
Journal of symbolic computation
*

We note that their numerical approach can also be implemented purely algebraically, and that the resulting much simpler algorithm yields, also for large p, both

doi:10.1016/s0747-7171(08)80028-4
fatcat:gj25oy3ygvai3dcsf5lnm6gii4
*arithmetic*and Boolean reductions of inversion ...*uniform*Boolean*NC*2 if q is small. ... Even before the result of Fich gz Tompa (1988), it was known that a subresultant approach can reduce INV(K) to linear algebra over F and put it into*arithmetic**NC*~ (Borodin et al. 1982) and Boolean*NC*...##
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Parallel Construction of Irreducible Polynomials

1991
*
DAIMI Report Series
*

Let

doi:10.7146/dpb.v20i358.7955
fatcat:5oypuhztubenbktmb7ar6atqye
*arithmetic*pseudo-<strong>*NC</strong>*^k denote the problems that can be solved by log space*uniform**arithmetic*circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial ... We show that factor refinement of polynomials is in*arithmetic*<strong>*NC</strong>*^3. ... We define*arithmetic**NC*k to consist of those problems with domain F n that are solved by log space*uniform*circuits of depth O(log k (n)) and size n O(1) . 2. ...##
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Real functions, contraction mappings, and P-completeness

1991
*
Information and Computation
*

We introduce a

doi:10.1016/0890-5401(91)90027-y
fatcat:4hfu4w7blbfm5hyxiu3apxqxwm
*uniform*framework for describing what it means for a continuous real function to be computed by a Boolean circuit family, and we provide techniques for constructing such functions. ... equals the class of*NC*real functions. ... The family (y;} is P-*uniform*and approximates x. Thus under P-*uniformity*, the real number x is also an*NC*real. But x may not be an*NC*real under log-space*uniformity*. ...##
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Uniform Derandomization from Pathetic Lower Bounds
[chapter]

2010
*
Lecture Notes in Computer Science
*

-If there are no constant-depth

doi:10.1007/978-3-642-15369-3_29
fatcat:liqtsvpc4rd3nkbm6jlj3ya25m
*arithmetic*circuits of size n 1+e for the problem of multiplying a sequence of n 3 × 3 matrices, then, for every constant d, black-box identity testing for depth-d*arithmetic*... circuits can be solved in subexponential time (and, more strongly, can be accepted by a*uniform*family of deterministic constantdepth threshold circuits of subexponential size). ... The class of polynomials computed by polynomial size*arithmetic*formulae is known as*arithmetic**NC*1 . By [35] the polynomial IMM n is complete for this class. ...##
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Small-Space Analogues of Valiant's Classes
[chapter]

2009
*
Lecture Notes in Computer Science
*

In the

doi:10.1007/978-3-642-03409-1_23
fatcat:4kfukithbrf7nlpmuhec7hd5v4
*uniform*setting, we show that our definition coincides with that of VPSPACE at polynomial width. ... In the*uniform*circuit model of computation, the width of a boolean circuit exactly characterises the "space" complexity of the computed function. ... It is known that SC 0 equals*NC*1 ( [12] ) and*uniform*SC 1 equals L. An*arithmetic*(resp. ...##
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Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

2012
*
Symposium on Theoretical Aspects of Computer Science
*

) when restricted to input structures of bounded tree depth and (b) solvable by

doi:10.4230/lipics.stacs.2012.66
dblp:conf/stacs/ElberfeldJT12
fatcat:mznuhphuebcvvkul3vlpnedyti
*uniform*logarithmic-depth circuit families (*NC*1 for decision problems and #*NC*1 for counting problems) when a tree decomposition ... We contribute new algorithmic meta theorems, which state that mso-definable problems are (a) solvable by*uniform*constant-depth circuit families (AC 0 for decision problems and TC 0 for counting problems ... Buss [3] used pebbling-based strategies to evaluate Boolean sentences in*uniform**NC*1 . ...##
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Page 3567 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

Roman Murawski (PL-POZN)
92g:03064 03D15 03F30 68Q15
Allen, Bill (1-UCLA)

*Arithmetizing**uniform**NC*. Ann. Pure Appl. Logic 53 (1991), no. 1, 1-50. ... A function is in*uniform**NC*if it is in*NC*and there is a log-space algorithm to find C,,. In this paper several characterizations of*NC*are given. ...
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