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# Questions tagged [circuits]

A computation model in which the computation is described via circuits of various logic gates.

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### Why is addition as fast as bit-wise operations in modern processors?

I know that bit-wise operations are so fast on modern processors, because they can operate on 32 or 64 bits on parallel, so bit-wise operations take only one clock cycle. However addition is a complex ...
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### Universality of the Toffoli gate

Regarding the quantum Toffoli gate: is it classicaly universal, and if so, why? is it quantumly universal, and why?
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### Why do all recent SAT solvers work on CNF instead of circuit SAT?

After the release of the AIGER library to handle and-inverter graphs sometime in 2006 (I think), some circuit SAT solvers were released in 2006-2008, and in a few SAT Races/Competitions there were AIG ...
213 views

### Which non-regular languages are in $AC^0$?

For example, I know that the non-regular language $a^nb^n$ is in $AC^0$. I would like to know more examples like this.
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### Why isn't P and P/poly trivially the same?

The definition of P is a language that can be decided by a polynomial time algorithm. The definition of P/poly can be taken to mean a language that can be decided by a polynomial-size circuit (see ...
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### How to understand the SR Latch

I can't wrap my head around how the SR Latch works. Seemingly, you plug an input line from R, and another from S, and you are supposed to get results in $Q$ and $Q'$. However, both R and S require ...
357 views

### Depth-2 circuits with OR and MOD gates are not universal?

It is well-known that every boolean function $f:\{0,1\}^n\to \{0,1\}$ can be realized using a boolean circuit of depth 2 (over the variables, their negation and constant values) containing AND gates ...
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### Isn't polynomial identity testing over arithmetic *expressions* trivial?

Polynomial identity testing is the standard example of a problem known to be in co-RP but not known to be in P. Over arithmetic circuits, it does indeed seem hard, since the degree of the polynomial ...
268 views

### Circuit size for “at least n inputs are true”

Say you have $m$ boolean inputs, and you are given a threshold $n$. You need to construct a boolean circuit that evaluates to true if at least $n$ of the inputs true. You may use AND, OR, NOT, or XOR ...
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### Combinational Logic Circuits and Theory of Computation

I'm trying to link Combinational Logic Circuits ( computers based on logical gates only ) with everything I have learned recently in Theory of Computation. I was wondering whether combinational ...
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### How to show that hard-to-compute Boolean functions exist?

How can one show that there exist Boolean functions on $n$ inputs which require at least $2^n/\log{n}$ logic gates to compute? This problem was originally stated in Exercise 3.16 of Nielsen & ...
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### Is there an intuitive proof for the existence of hard functions?

I am referring to the theorem on page 115 of the book by Arora and Barak, which states that, For every $n>1$, there exists a function $f:\{0,1\}^n \rightarrow \{0,1\}$ that cannot be computed by ...
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### What is the decidable language in $P/poly$ but not in $P$?

Except for the undecidable unaries I have no idea if there is anything in the gap between $P/poly$ and $P$
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### Difference between $\mathsf{SIZE}(n^k)$ vs $\mathsf{P/poly}$ and $\mathsf{SIZE}(n)$ vs linear size circuit?

In the Wikipedia page on the Karp–Lipton theorem it is mentioned that $$\Sigma_2\not\subseteq\mathsf{SIZE}(n^k)$$ (which is known) is not same as $$\Sigma_2\not\subseteq\mathsf{P/Poly}$$ (which ...
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### How does $\mathsf{NP} \subset \mathsf{P}/\mathsf{poly}$ imply these two inclusions?

In the proof of Theorem 1 in this paper by Chen, McKay, Murray, and Williams the authors assume $\mathsf{NP} \subset \mathsf{P}/\mathsf{poly}$ and (in different parts of the proof) state this implies ...
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### Lower bound of degree of polynomial approximating parity

Let $\text{MOD}_2 : \{0,1\}^n \rightarrow \{0,1\}$ be a parity function where $$\text{MOD}_2(x_1,\dots,x_n) = \sum_i x_i \bmod 2$$ It is known [See e.g. Lemma 5 of this lecture note] that any ...
221 views

### Modulo operation in monotone complexity

Given $x\in\Bbb N$, I would like to find $x\bmod N$, where $N$ is composite. For example $N=35$, $x=53$ and $x\bmod N=18$. Is this operation considered monotone in circuit/algebraic complexity ...
884 views

### Formulas vs Circuits

In boolean circuit complexity, a circuit is just defined by a Directed Acyclic Graphs with designated input and output nodes, where the intermediate nodes compute a specific boolean function. A ...
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### Finding minimal and complete test sets for circuits

I have been playing around with analysis of circuits and am trying to generate test vectors. In order to exercise the circuit in the manner I require, I need a vector that includes every change in the ...
109 views

### Size of constant depth circuit for digital comparator?

Is a lower bound of $\Omega(n^2)$ known for the size of any constant depth circuit expressing a digital comparator for two $n$-bit numbers? Two $n$-bit binary numbers can be compared using a digital ...
180 views

### Amortizing or batching circuit evaluation for many different inputs?

Suppose that I have a boolean function of size $k$ with $n$ inputs. I would expect to be able to evaluate it on all possible inputs in time $O(k*2^n)$ simply by calculating all the intermediate values ...
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### On relation between FFT and polynomial multiplication

Is it known that if polynomial multiplication of degree $n$ polynomials and coefficient size bounded by $M$ can be done in $O(n)$ arithmetic operations on $O(\log n+\log M)$ bit sized words then $FFT$ ...
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### What is the relation between arithmetic circuits and straight line programs?

One definition of arithmetic circuits is as follows: An arithmetic circuit $\Phi$ over the field $\mathbb F$ and the set of variables $X$ usually, $X = \{x_1, \dots , x_n\}$) is a directed acyclic ...
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### Is it possible to determine if C=A+B faster than adding A+B in logical circuits

With an adder circuit where A+B=C, I am trying to have a method to determine when C will be valid based on a change in A or B. I know that it is possible to just determine the longest the circuit ...
245 views

### Creating a logical circuit

Task: Design a 2 bit comparator. Input: 2x 2 bit (I take it as 2 2-bit values, let them be unsigned for simplicity) Output: 1 if result input1>input2 is true, 0 otherwise Develop truth table and ...
340 views

### Universality of NOT and CNOT

I'm trying to figure out why NOT and CNOT gates are not sufficient to create all bijective functions in classical circuits. I have been struggling on this for hours, and just can't make sense of it. ...
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### Is it possible to construct a C^5(U) with V^2=U and no work qubits (Nielsen & Chuang Exercise 4.28)

My question is related to the exercise 4.28 in the book of Nielsen and Chuang (Quantum Computation and Quantum Information). Here is the exercise For $U=V^2$ with $V$ unitary, construct a $C^5(U)$ ...
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### PARITY using depth one TC0 circuit

I need to disprove that a PARITY gate can be simulated using a single MAJORITY gate, or even a ...
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### Function that cannot be computed by a Boolean circuit of size $2^n/2n$

Show that, for sufficiently large $n$, there is a function $f\colon\{0,1\}^n \to \{0,1\}$ that cannot be computed by a Boolean circuit with fan-in $2$ with $\frac{2^n}{2n}$ gates. Please give me a ...
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### Classical Computation without NOT

Is it possible to do universal classical computation using bits and 2-bit gates when you cannot perform a NOT operation on a single bit (hence cant do CNOT and so on). If yes, what are the possible ...
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### Can a Boolean circuit be considered an algorithm?

Can a Boolean circuit by itself be considered an algorithm (a single step algorithm if you like)? For instance say you have a simple tree circuit with two AND gates as the input gates feeding a single ...
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### Weaker, but similar conditions to Turing completeness?

A model of computation is called Turing complete if it can simulate any Turing machine. This rules out for example a combinational logic circuit. However, there is a sense in which combinational ...
### An AC$^1$ circuit for 2-SAT
We know that $NC^1 \subseteq NL \subseteq AC^1$ and that 2-SAT is complete for $NL$. How does one construct an $AC^1$ circuit for 2-SAT? Recall that $AC^1$ circuits have $O(\log n)$ depth where $n$...
There are $2^{2^n}$ possible functions that have $n$ boolean inputs and a single boolean output. Some of these functions have very short boolean logic circuits. Some have longer circuits. A classic ...