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Questions tagged [circuit-families]

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-3
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1answer
188 views

Some questions about the depth hierarchy for threshold circuits [closed]

(I am hugely editing the question. My initial question was if lowerbounds on threshold circuits say anything about P/NP and it seems that they dont. Irrespective of P/NP its an independently true fact ...
3
votes
1answer
84 views

About the sign-rank of the Minsky-Pappert function

Apologies this might be a very trivial thing I am getting confused by! Firstly in corollary 1.1 (page 3) in this paper, https://eccc.weizmann.ac.il/report/2016/075/ the authors claim that they have ...
5
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0answers
73 views

About Boolean functions with a high sign-rank

Recently in this beautiful paper, https://arxiv.org/pdf/1705.02397.pdf it has been shown that there is an explicit $Th \circ Th$ function with sign-rank scaling exponentially in dimension. I wanted to ...
6
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0answers
84 views

Using epsilon biased sets for circuit lower bounds

I have seen instances of how the technique of epsilon biased sets can be used to construct hard functions against a circuit class - like how in the recent paper of Kane-Williams this was used to ...
14
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1answer
294 views

Small circuits for circuit evaluation problem

Let $\mathsf{CircuitEval}_{s, n}$ be the function which maps an $s$-gate circuit $C$ on $n$ bits and an $n$-bit string $x$ to $C(x)$. Assume that circuits are encoded as an acyclic sequence of ...
6
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1answer
352 views

OR-weft Hierarchy

Say that a node of a circuit is small if it has fan-in at most 2 and large if it has fan-in greater than 2. The weft of a circuit is the maximum number large nodes in any path from an input node to an ...
5
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0answers
339 views

About the ``recent" paper by Razborov in the Annals of Mathematics

Recently this paper on complexity theory was published at the Annals of Mathematics by Razborov, http://annals.math.princeton.edu/2015/181-2/p01. Curiously this seems to have been submitted to the ...
11
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1answer
136 views

Is there a finite unitary gate set which can exactly realise all QFTs of order $2^n$?

I am considering ideas about exact quantum algorithms. In particular, I am considering likely limitations of $\mathsf{EQP}$, which consists of languages exactly decideable by polytime-uniform quantum ...
22
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1answer
566 views

Classifying reversible gates

Post's lattice, described by Emil Post in 1941, is basically a complete inclusion diagram of sets of Boolean functions that are closed under composition: for example, the monotone functions, the ...
7
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1answer
311 views

May Boolean circuits be exponentially more concise than Boolean formulae?

Consider a family $(f_n)_{1 \leq n}$ of Boolean functions, where $f_n$ is a function on $n$ variables. Consider for every $n$ the smallest Boolean formula $F_n$ describing $f_n$, and the smallest ...
0
votes
1answer
90 views

DC uniform circuit and parity-P

This is a question from the lecture about Toda's theorem: http://www.cs.princeton.edu/courses/archive/spring01/cs522/lecnotes/lec8.ps The lecture uses theorem 1 and theorem 2 but not include proof. ...
13
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3answers
2k views

Circuit complexity of Majority function

Let $f: \{0,1\}^n \to \{0,1\}$ be the majority function, i.e. $f(x) = 1$ if and only if $\sum_{i = 1}^n x_i > n/2$. I was wondering if there was a simple proof of the following fact (by "simple" I ...
2
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0answers
135 views

Satisfiability of circuits with infinite input

As we all know, satisfiability of Boolean circuits is NP-complete. I am wondering if there are any studies of circuits with infinite inputs? That is, suppose the input is from the set $\{0,1\}^\omega$...
27
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2answers
693 views

Deciding whether an NC${}^0_3$ circuit computes a permutation or not

I would like to ask about a special case of the question “Deciding if a given NC0 circuit computes a permutation” by QiCheng that has been left unanswered. A Boolean circuit is called an NC0k circuit ...
5
votes
1answer
195 views

Other types of uniformity for circuits (incl. by small modifications)

I've seen poly-time and logspace uniformity for circuit families, typically defined as the existence of a poly-time/logspace Turing machine "generator" that outputs the correctly sized circuit $C_n$ ...
2
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2answers
252 views

Reversible Logic Circuit Synthesis [closed]

I am about to choose a project regarding "Reversible logic circuit synthesis". I've studied well about "Switching circuits and Logic design" and I found it very intriguing but I've got no idea about ...
1
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0answers
178 views

Problems or issues with a proposed circuit class?

I'm looking to use something close to the following as a definition for a circuit class. This is obviously semi-informal. I am curious if any one sees any potential problems with it, or where ...
5
votes
2answers
364 views

Is the Balanced Boolean Formula problem solvable in sublogarithmic space if the input has a tree structure?

Suppose that instead of the usual linear work tape the input is given in a binary tree structure with n leaves and log n depth, the initial position being the root. At every node, we can step to its ...
2
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2answers
251 views

Nonuniform circuit families - don't have to specify for arbitrarily large, but finite, input lengths?

This is a question about nonuniform circuit families that's kind of bothering me. Let $\lbrace C_n \rbrace$ be a family of circuits for a language $L$ such that for inputs $x$ of length $n$, $C_n(x) = ...
39
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3answers
2k views

Why are mod_m gates interesting?

Ryan Williams just posted his lower bound on ACC, the class of problems that have constant depth circuits with unbounded fan-in and gates AND, OR, NOT and MOD_m for all possible m's. What's so ...
4
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1answer
1k views

Simulating Turing machines (output included) with circuits

A Turing machine with input alphabet {0,1} computes a partial or total function $f \colon \{0,1\}^* \to \{0,1\}^*$. Is it possible to construct a circuit family $\{C_n\}$ such that for an input $x$ of ...
40
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3answers
2k views

Circuit lower bounds over arbitrary sets of gates

In the 1980s, Razborov famously showed that there are explicit monotone Boolean functions (such as the CLIQUE function) that require exponentially many AND and OR gates to compute. However, the basis ...
2
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1answer
274 views

Complexity of advice language?

Let $L$ be a language in P/poly. There is then a deterministic polynomial-time Turing machine $M$ with polynomial-sized advice that decides $L$. Consider the language $A(M)$ of all advice strings ...
13
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1answer
590 views

Circuits with oracles vs. Turing Machines with oracles

Put simply: what is the correspondance between Turing machines with oracles, and uniform circuit families with oracles? How are the latter defined in order to obtain the same computational model, for ...