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Arithmetizing Uniform NC
1991
Annals of Pure and Applied Logic
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., 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
1991
Information and Computation
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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]
2014
Lecture Notes in Computer Science
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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
1991
Proceedings of the American Mathematical Society
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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
1991
Proceedings of the American Mathematical Society
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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
2012
Information Processing Letters
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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
1992
SIAM journal on computing (Print)
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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]
2015
Lecture Notes in Computer Science
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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
1990
Journal of symbolic computation
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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
1991
DAIMI Report Series
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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
1991
Information and Computation
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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]
2010
Lecture Notes in Computer Science
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-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]
2009
Lecture Notes in Computer Science
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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
2012
Symposium on Theoretical Aspects of Computer Science
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) 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
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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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