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Arithmetizing Uniform NC

Bill Allen
1991 Annals of Pure and Applied Logic  
., 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.  ... 
doi:10.1016/0168-0072(91)90057-s fatcat:enyupbcq4zb7tpucnef23ewzay

Boolean circuits versus arithmetic circuits

Joachim von zur Gathen, Gadiel Seroussi
1991 Information and Computation  
Over Q and finite fields, Boolean circuits can simulate 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!  ... 
doi:10.1016/0890-5401(91)90078-g fatcat:aydwnfiodfgy3bizzgauso2sei

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 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.  ... 
doi:10.1007/978-3-319-04657-0_26 fatcat:m4umv2v5kjda5jjszwjzq67dni

Points not as hyperplane sections of projectively normal curves

Edoardo Ballico
1991 Proceedings of the American Mathematical Society  
Thus, since C is linearly normal, its 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.  ... 
doi:10.1090/s0002-9939-1991-1052870-9 fatcat:7jymvemgdbaihff6enqgowioea

Points not as Hyperplane Sections of Projectively Normal Curves

Edoardo Ballico
1991 Proceedings of the American Mathematical Society  
Thus, since C is linearly normal, its 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.  ... 
doi:10.2307/2048725 fatcat:qs4ir6cgsraopa543gh75nu6sa

Towards a tight hardness–randomness connection between permanent and arithmetic circuit identity testing

Maurice Jansen
2012 Information Processing Letters  
is strictly contained in 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 .  ... 
doi:10.1016/j.ipl.2012.08.001 fatcat:iyw4sp5iarbgza6xo7g4gkejoi

An Optimal Parallel Algorithm for Formula Evaluation

S. Buss, S. Cook, A. Gupta, V. Ramachandran
1992 SIAM journal on computing (Print)  
This approach is then used to solve the more general problem of evaluating 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.  ... 
doi:10.1137/0221046 fatcat:jc23c5v2jveivmsm3lxgqsoisu

Complexity of Regular Functions [chapter]

Eric Allender, Ian Mertz
2015 Lecture Notes in Computer Science  
Let {D n } be the 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.)  ... 
doi:10.1007/978-3-319-15579-1_35 fatcat:7b67gh6a4bdm5jxzpne3xe6qx4

Inversion in finite fields using logarithmic depth

Joachim von zur Gathen
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 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  ... 
doi:10.1016/s0747-7171(08)80028-4 fatcat:gj25oy3ygvai3dcsf5lnm6gii4

Parallel Construction of Irreducible Polynomials

Gudmund Skovbjerg Frandsen
1991 DAIMI Report Series  
Let 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.  ... 
doi:10.7146/dpb.v20i358.7955 fatcat:5oypuhztubenbktmb7ar6atqye

Real functions, contraction mappings, and P-completeness

H.James Hoover
1991 Information and Computation  
We introduce a 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.  ... 
doi:10.1016/0890-5401(91)90027-y fatcat:4hfu4w7blbfm5hyxiu3apxqxwm

Uniform Derandomization from Pathetic Lower Bounds [chapter]

Eric Allender, V. Arvind, Fengming Wang
2010 Lecture Notes in Computer Science  
-If there are no constant-depth 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.  ... 
doi:10.1007/978-3-642-15369-3_29 fatcat:liqtsvpc4rd3nkbm6jlj3ya25m

Small-Space Analogues of Valiant's Classes [chapter]

Meena Mahajan, B. V. Raghavendra Rao
2009 Lecture Notes in Computer Science  
In the 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.  ... 
doi:10.1007/978-3-642-03409-1_23 fatcat:4kfukithbrf7nlpmuhec7hd5v4

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

Michael Elberfeld, Andreas Jakoby, Till Tantau, Marc Herbstritt
2012 Symposium on Theoretical Aspects of Computer Science  
) when restricted to input structures of bounded tree depth and (b) solvable by 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 .  ... 
doi:10.4230/lipics.stacs.2012.66 dblp:conf/stacs/ElberfeldJT12 fatcat:mznuhphuebcvvkul3vlpnedyti

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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