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Unbounded fan-in circuits and associative functions

1985
*
Journal of computer and system sciences (Print)
*

The computation of finite semigroups using

doi:10.1016/0022-0000(85)90015-7
fatcat:d6qdktcnobggto6y4n5xflxcki
*unbounded**fan*-*in**circuits*are considered. ... A consequence is that the same bounds apply for*circuits*computing the sum of two n-bit numbers. 0*UNBOUNDED**FAN*-*IN**CIRCUITS*223 At least two models of*unbounded**fan*-*in*parallelism have been proposed. ...*ASSOCIATIVE**FUNCTIONS*The main theorem proved*in*this section is a characterization of the*associative**functions*that can be computed by constant depth, polynomial size*circuits*. ...##
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The complexity of the parity function in unbounded fan-in, unbounded depth circuits

1991
*
Theoretical Computer Science
*

., The complexity of the parity

doi:10.1016/0304-3975(91)90052-4
fatcat:x4skzcjwdfgw7fyamfwgwup2zi
*function**in**unbounded**fan*-*in*,*unbounded*depth*circuits*, Theoretical Computer Science 85 (1991) 155-170. ... Almost everything is known on the complexity of the parity*function**in**fan*-*in*2*circuits*over various bases. ... Nevertheless, it is useful to investigate the complexity of the parity*functions**in**unbounded**fan*-*in*NOR*circuits*. ...##
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Lower bounds on monotone arithmetic circuits with restricted depths

1985
*
Computers and Mathematics with Applications
*

We consider monotone arithmetic

doi:10.1016/0898-1221(85)90103-8
fatcat:3fmwnigfwrb65piqxgqsm7gz5u
*circuits*with restrlcted depths to compute monotone multivariate polynomials such as the elementary symmetric*functions*. convolution of several vectors*and*raismg a matrix ... We develop general lower-*and*upper-bound techniques that seem to generate almost-matching bounds for all the*functions*consldered. ... Acknowledgemenr-This work was supported*in*part by the National Science Foundation under Grant MCS-X3-15980. ...##
###
Boolean complexity classes vs. their arithmetic analogs

1996
*
Random structures & algorithms (Print)
*

Recall that SAC 1 , the

doi:10.1002/(sici)1098-2418(199608/09)9:1/2<99::aid-rsa7>3.3.co;2-o
fatcat:vx7x7v7cj5dfvpj6m4d7cm3cr4
*circuit*analog of N L, is the class of Boolean*circuits*of depth O(log n) over {∨, ∧, ¬} with*unbounded**fan*-*in*∨*and*bounded*fan*-*in*∧ gates. ... The class ⊕SAC 1 , the*circuit*analog of ⊕L, is the class of arithmetic*circuits*of depth O(log n) over {⊕, ×}, with*unbounded**fan*-*in*⊕*and*bounded*fan*-*in*× gates. Clearly ⊕L/poly ⊆ ⊕SAC 1 . ... Acknowledgements Venkateswaran independently suggested a similar construction for the*circuit*result [V2]. ...##
###
Boolean complexity classes vs. their arithmetic analogs

1996
*
Random structures & algorithms (Print)
*

Recall that SAC 1 , the

doi:10.1002/(sici)1098-2418(199608/09)9:1/2<99::aid-rsa7>3.0.co;2-6
fatcat:whcehawyzndv7hd6eyuurt2a7a
*circuit*analog of N L, is the class of Boolean*circuits*of depth O(log n) over {∨, ∧, ¬} with*unbounded**fan*-*in*∨*and*bounded*fan*-*in*∧ gates. ... The class ⊕SAC 1 , the*circuit*analog of ⊕L, is the class of arithmetic*circuits*of depth O(log n) over {⊕, ×}, with*unbounded**fan*-*in*⊕*and*bounded*fan*-*in*× gates. Clearly ⊕L/poly ⊆ ⊕SAC 1 . ... Acknowledgements Venkateswaran independently suggested a similar construction for the*circuit*result [V2]. ...##
###
Page 2404 of Mathematical Reviews Vol. , Issue 96d
[page]

1996
*
Mathematical Reviews
*

It is shown that

*unbounded**fan*-*in**circuits*of linear size*and*depth, proportional to the inverse of Ackerman’s*function*, exist for computing prefixes. ...*In*Part Four the effects of bounded versus*unbounded**fan*-*in**and**fan*-out are examined. ...##
###
Unambiguity of circuits

1993
*
Theoretical Computer Science
*

*In*particular, we show CREW-TIME(logk n)= UnambAC" for each positive integer k. ing Turing machine,

*and*depth of a uniform

*circuit*of semi-

*unbounded*

*fan*-

*in*[18, 241,

*and*... Several classes of unambiguous

*circuit*families within the NC-hierarchy are introduced

*and*related to unambiguous automata

*and*to PRAMS with exclusive write access. ... the corresponding cell might be destroyed;

*and*assume the

*circuit*to avoid any multiple l-input to OR-gates of

*unbounded*

*fan*-

*in*(multiple O-input to

*AND*-gates of

*unbounded*

*fan*-

*in*). ...

##
###
Semantical Counting Circuits

2003
*
Theory of Computing Systems
*

*In*the

*circuit*based model, it was done for #P , but for low-level complexity classes such as #AC 0

*and*#NC 1 only the syntactical definitions were considered. ... We also consider semantically defined probabilistic complexity classes corresponding to AC 0

*and*NC 1

*and*prove that

*in*the case of

*unbounded*error, they are identical to their syntactical counterparts ... An AC 0

*circuit*family is a uniform,

*unbounded*

*fan*-

*in*

*circuit*family of polynomial size

*and*constant depth. ...

##
###
Semantical Counting Circuits
[chapter]

2000
*
Lecture Notes in Computer Science
*

*In*the

*circuit*based model, it was done for #P , but for low-level complexity classes such as #AC 0

*and*#NC 1 only the syntactical definitions were considered. ... We also consider semantically defined probabilistic complexity classes corresponding to AC 0

*and*NC 1

*and*prove that

*in*the case of

*unbounded*error, they are identical to their syntactical counterparts ... An AC 0

*circuit*family is a uniform,

*unbounded*

*fan*-

*in*

*circuit*family of polynomial size

*and*constant depth. ...

##
###
Optimal bounds for decision problems on the CRCW PRAM

1987
*
Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87
*

Furthermore, we show that almost all Boolean

doi:10.1145/28395.28405
dblp:conf/stoc/BeameH87
fatcat:kepq6msierfdtixy4vvshmnzlm
*functions*of n bits require logn -log log n + Q( 1) time when the number of processors is at most polynomial*in*n. ... We prove optimal R(log n/log log n) lower bounds on the time for CRCW PRAM's with polynomially bounded numbers of processors or memory cells to compute parity*and*a number of related problems. ... log n) for*unbounded**fan*-*in**circuits*given by Chandra, Stockmeyer*and*Vishkin [CSV] . ...##
###
Optimal bounds for decision problems on the CRCW PRAM

1989
*
Journal of the ACM
*

Furthermore, we show that almost all Boolean

doi:10.1145/65950.65958
fatcat:tja3jvt6yrckncrkqzouzi7vxi
*functions*of n bits require logn -log log n + Q( 1) time when the number of processors is at most polynomial*in*n. ... We prove optimal R(log n/log log n) lower bounds on the time for CRCW PRAM's with polynomially bounded numbers of processors or memory cells to compute parity*and*a number of related problems. ... log n) for*unbounded**fan*-*in**circuits*given by Chandra, Stockmeyer*and*Vishkin [CSV] . ...##
###
Power of uninitialized qubits in shallow quantum circuits

2020
*
Theoretical Computer Science
*

First, we show that such a

doi:10.1016/j.tcs.2020.11.039
fatcat:ofbjnujnzve2vo375uafbixb34
*circuit*can compute any symmetric*function*on n bits that is classically computable*in*polynomial time. ... Lastly, to understand the limitations of uninitialized ancillary qubits, we focus on near-logarithmic-depth quantum*circuits*with them*and*show the impossibility of computing the parity*function*on n bits ... it consists of the gates*in*G,*unbounded**fan*-out gates,*and**unbounded*Z gates. ...##
###
Power of Uninitialized Qubits in Shallow Quantum Circuits
[article]

2017
*
arXiv
*
pre-print

First, we show that such a

arXiv:1608.07020v2
fatcat:5xf2mupb2vb4bmptyn7r4demm4
*circuit*can compute any symmetric*function*on n bits that is classically computable*in*polynomial time. ... Lastly, to understand the limitations of uninitialized ancillary qubits, we focus on near-logarithmic-depth quantum*circuits*with them*and*show the impossibility of computing the parity*function*on n bits ... of the OR reduction*circuit*,*and*then compute g m (s) = f n (x), where g m is the Boolean*function**associated*with f n described above. ...##
###
Nonlinear lower bounds on the number of processors of circuits with sublinear separators

1991
*
Information and Computation
*

The above stated result holds also for

doi:10.1016/0890-5401(91)90041-y
fatcat:tqmihtdv3rcgnk7auok7y7svsq
*unbounded**fan*-*in*,*fan*-out Boolean*circuits*. ... A nonlinear lower bound on the number of processors is achieved also for planar VLSI*circuits*computing some one-output Boolean*functions**in*time O(nb) for b<;. 'f=. ... Let f: { 0, 1 }" -+ (0, 1 } be a Boolean*function*with the set of input variables X. Let S be an*unbounded**fan*-*in*Boolean*circuit*computing f. ...##
###
Nonlinear lower bounds on the number of processors of circuits with sublinear separators
[chapter]

1991
*
Lecture Notes in Computer Science
*

The above stated result holds also for

doi:10.1007/3-540-54458-5_68
fatcat:xq3qnvriyrbedhdkjk7pnlrk2q
*unbounded**fan*-*in*,*fan*-out Boolean*circuits*. ... A nonlinear lower bound on the number of processors is achieved also for planar VLSI*circuits*computing some one-output Boolean*functions**in*time O(nb) for b<;. 'f=. ... Let f: { 0, 1 }" -+ (0, 1 } be a Boolean*function*with the set of input variables X. Let S be an*unbounded**fan*-*in*Boolean*circuit*computing f. ...
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