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Deterministic approximate counting of depth-2 circuits
<i title="IEEE">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rl24zg6tg5g2hdek7nnlgicgja" style="color: black;">[1993] The 2nd Israel Symposium on Theory and Computing Systems</a>
</i>
We describe deterministic algorithms which for a given depth-2 circuit F approximate the probability that on a random input F outputs a speci c value . ...
Our approach gives an algorithm which for a given GF 2] multivariate polynomial p and given > 0 approximates the number of zeros of p within a multiplicative factor 1+ . ...
Introduction This paper deals with the problem of approximating the accepting probability of general depth-2 boolean circuits. ...
<span class="external-identifiers">
<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/istcs.1993.253488">doi:10.1109/istcs.1993.253488</a>
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Deterministic simulation of probabilistic constant depth circuits
<span title="">1985</span>
<i title="IEEE">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v4uclqixrrhlnk5pbw4wii6cfq" style="color: black;">26th Annual Symposium on Foundations of Computer Science (sfcs 1985)</a>
</i>
Moreover, the functions $f_{n,\varepsilon}$ themselves can be computed by uniform polynomial size, constant depth circuits. Some (interrelated) consequences of this result are given below. ...
property: for a random seed, the pseudorandom output "looks random" to any polynomial size, constant depth, unbounded fan-in circuit. ...
We also present two results for the special case of depth 2 circuits. They deal, respectively, with finding an assignment and approximately counting the number of assignments. ...
<span class="external-identifiers">
<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/sfcs.1985.19">doi:10.1109/sfcs.1985.19</a>
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Floating point representations in quantum circuit synthesis
<span title="2013-09-27">2013</span>
<i title="IOP Publishing">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bt7vbjky6zbevclechrlkkahae" style="color: black;">New Journal of Physics</a>
</i>
We provide a non-deterministic quantum protocol that approximates the single qubit rotations R_x(2a^2 b^2) using R_x(2a) and R_x(2b) and a constant number of Clifford and T operations. ...
We also discuss the T-depth of our method and see that the vast majority of the cost of the resultant circuits can be shifted to parallel computation paths. ...
The first component is a non-deterministic circuit that can combine two rotations R x (α) and R x (β) to approximate R x (α 2 β 2 ). ...
<span class="external-identifiers">
<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1367-2630/15/9/093041">doi:10.1088/1367-2630/15/9/093041</a>
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Self-reducible with easy decision version counting problems admit additive error approximation. Connections to counting complexity, exponential time complexity, and circuit lower bounds
[article]
<span title="2016-11-05">2016</span>
<i >
arXiv
</i>
<span class="release-stage" >pre-print</span>
error ϵ· 2^n', ∀ϵ>0, which for many of those problems (when n'=n+constant) implies additive approximation to the fraction f(x)/2^n. ...
of all circuits for which the problems of counting either satisfying or unsatisfying assignments belong to TotP (which is the Karp-closure of self reducible problems with easy decision version). ...
Acknowledgements I want to thank Manolis Zampetakis for mentioning to me the relationship between these results and circuit lower bounds. ...
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<a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1611.01706v1">arXiv:1611.01706v1</a>
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Randomness Buys Depth for Approximate Counting
<span title="2014-01-08">2014</span>
<i title="Springer Nature">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t46w5lpc3ngnfmzil33zdjjrp4" style="color: black;">Computational Complexity</a>
</i>
In this work we focus on the trade-off between the approximation parameter and the depth of the polynomial-size circuits that count approximately. ...
fan-in) poly(n)-size depth-d circuits with error ≤ 1/3, but cannot be solved by deterministic poly(n)-size depth-(d + 1) circuits, for every d ≥ 2; and the depth of both is tight. ...
We thank Kord Eickmeyer for helpful initial discussions on this problem, and Joshua Brody and Elad Verbin for sending us a preliminary version of [BV10] . ...
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<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00037-013-0076-6">doi:10.1007/s00037-013-0076-6</a>
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Randomness Buys Depth for Approximate Counting
<span title="">2011</span>
<i title="IEEE">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v4uclqixrrhlnk5pbw4wii6cfq" style="color: black;">2011 IEEE 52nd Annual Symposium on Foundations of Computer Science</a>
</i>
In this work we focus on the trade-off between the approximation parameter and the depth of the polynomial-size circuits that count approximately. ...
fan-in) poly(n)-size depth-d circuits with error ≤ 1/3, but cannot be solved by deterministic poly(n)-size depth-(d + 1) circuits, for every d ≥ 2; and the depth of both is tight. ...
We thank Kord Eickmeyer for helpful initial discussions on this problem, and Joshua Brody and Elad Verbin for sending us a preliminary version of [BV10] . ...
<span class="external-identifiers">
<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/focs.2011.19">doi:10.1109/focs.2011.19</a>
<a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/focs/Viola11a.html">dblp:conf/focs/Viola11a</a>
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Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds
<span title="2018-08-20">2018</span>
<i >
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5b35w7jagjckfngb6pj76nbam4" style="color: black;">International Symposium on Mathematical Foundations of Computer Science</a>
</i>
We give a deterministic algorithm for counting the number of satisfying assignments of any AC 0 [⊕] circuit C of size s and depth d over n variables in time 2 n−f (n,s,d) , where As a consequence, we get ...
that for each d, there is a language in E NP that does not have AC 0 [⊕] circuits of size 2 o(n 1/(d+1) ) . ...
The Razborov-Smolensky approximation method yields size lower bounds of the form 2 Ω(n 1/2(d−1) ) for Majority against AC 0 [⊕] circuits of depth d over n variables. ...
<span class="external-identifiers">
<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.mfcs.2018.78">doi:10.4230/lipics.mfcs.2018.78</a>
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On the complexity of counting in the polynomial hierarchy
<span title="">1991</span>
<i title="Elsevier BV">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a>
</i>
., On the complexity of counting in the polynomial hierarchy, Theoretical Computer Science 81 (1991) 77-95. ...
We investigate the time complexity of the following counting problem: for a given set of words A, a length n and a number k, check whether A contains exactly k words of length n (in symbol, if cr,( n) ...
It suffices to consider a depth k circuit as a depth k + 1 circuit of bottom fan-in 1 and choose D < D,, + , . Then for sufficiently large n we have 2 18" < 2 I:? ...
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<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(91)90317-u">doi:10.1016/0304-3975(91)90317-u</a>
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On the Computational Power of Radio Channels
<span title="2019-10-11">2019</span>
<i >
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2pcluxp32rfojgwekdrryxoaqu" style="color: black;">International Symposium on Distributed Computing</a>
</i>
Next, we use the technique of random restrictions, used to prove AC 0 lower bounds, to prove a tight lower bound of Ω(1/ 2 ) on computing a (1 ± )-approximation to the sum of the nodes' inputs. ...
Using techniques from circuit complexity, we show that in many cases, the answer is "no". ...
The circuit C P is given by C P = t∈T V t , a depth-3 circuit of size O(n • r • 2 r ) = 2 O(r+log n) , since |T | ≤ 2 r and the size of each sub-circuit V t is O(n • r). ...
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Pseudo-Derandomizing Learning and Approximation
<span title="2018-08-02">2018</span>
<i >
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gdc7gjvgv5hd7mr5o3uwxvngwu" style="color: black;">International Workshop on Approximation Algorithms for Combinatorial Optimization</a>
</i>
Finally, we investigate the notion of approximate canonization of Boolean circuits. ...
We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser [7]. ...
If BPE requires circuits of size 2 n Ω(1) almost everywhere, then AC 0 [p] circuits can be approximately canonized in pseudo-deterministic quasi-polynomial time. ...
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<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.approx-random.2018.55">doi:10.4230/lipics.approx-random.2018.55</a>
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Algorithms for Circuits and Circuits for Algorithms
<span title="">2014</span>
<i title="IEEE">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fvcoh2vdcnbppmicpce3jet6r4" style="color: black;">2014 IEEE 29th Conference on Computational Complexity (CCC)</a>
</i>
and proofs of circuit size lower bounds. ...
Algorithms for circuits refers to the design of interesting algorithms which can perform non-trivial circuit analysis of some kind, on either a circuit or a Boolean function given as a truth table. ...
Approximate Circuit Analysis A different form of circuit analysis is that of (additive) approximate counting; that is, approximating the fraction of satisfying assignments to a given circuit: Circuit Approximation ...
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<a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ccc.2014.33">doi:10.1109/ccc.2014.33</a>
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On derandomizing algorithms that err extremely rarely
<span title="">2014</span>
<i title="ACM Press">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14</a>
</i>
For the latter we present a deterministic polynomial-time hitting set generator for the class of n-variate polynomials of degree d over GF(2) that evaluate to 0 on at most an O(2 −d ) fraction of their ...
, B(n) = n log n or B(n) = exp(n 0.99 ))), given an n-input circuit C from C that evaluates to 1 on all but at most B(n) of its inputs, find (in deterministic polynomial-time) an input x such that C(x) ...
Specifically, the known deterministic algorithms for approximate counting for AC 0 run in quasipolynomial time. ...
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More on AC^0[oplus] and Variants of the Majority Function
<span title="2019-12-10">2019</span>
<i >
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yjwtenvm3vc4tjdmag7doffpby" style="color: black;">Foundations of Software Technology and Theoretical Computer Science</a>
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Formally, we show that for any fixed constant d ≥ 2, there is a family of Boolean functions that has polynomialsized randomized uniform AC 0 circuits of depth d but no polynomial-sized (deterministic) ...
AC 0 [⊕] circuits of depth d. ...
Ajtai and Ben-Or [3] showed that for any randomized depth-d AC 0 circuit of size at most s, there is deterministic AC 0 circuit of depth d + 2 and size at most poly(s) that computes the same function ...
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My favorite ten complexity theorems of the past decade
[chapter]
<span title="">1994</span>
<i title="Springer Berlin Heidelberg">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a>
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We use each of the theorems as a springboard to discuss work done in various areas of complexity theory. ...
Their proof uses a new idea of approximating bounded-depth circuits by low-degree polynomials over a nite eld. ...
Instead of the restriction method used by H astad (Theorem 2), Razborov uses an approximation method. Razborov shows how to approximate each AND and OR with approximate AND and approximate OR. ...
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Page 5582 of Mathematical Reviews Vol. , Issue 96i
[page]
<span title="">1996</span>
<i title="American Mathematical Society">
<a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a>
</i>
Every counting problem in #Zo is computable in deterministic polynomial time, and every prob- lem in #2; has a fully polynomial-time randomized approximation scheme. ...
/A,\-circuits of depth 2d that cannot be computed by a small monotone perceptron of depth d. ...
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