# Tag Info

Accepted

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

Addition is fast because CPU designers have put in the circuitry needed to make it fast. It does take significantly more gates than bitwise operations, but it is frequent enough that CPU designers ...
• 140k

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

There are several aspects. The relative cost of a bitwise operation and an addition. A naive adder will have a gate-depth which depend linearly of the width of the word. There are alternative ...
• 2,994

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

CPUs operate in cycles. At each cycle, something happens. Usually, an instruction takes more cycles to execute, but multiple instructions are executed at the same time, in different states. For ...
• 508
Accepted

### Why aren't P and P/poly trivially the same?

The point about circuits is that a circuit has a fixed number of inputs. This means that, to define a language, we need a family of circuits $C_0, C_1, C_2, \dots$ such that the circuit $C_i$ ...
• 80.1k

### Does there exist an equivalent arithmetic circuit for each computable function?

Any computable boolean function with a fixed-length input can be computed by an arithmetic circuit. Consider any boolean function $f:\{0,1\}^n \to \{0,1\}$. Then there exists a multivariate ...
• 140k

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

Processors are clocked, so even if some instructions can clearly be done faster than others, they may well take the same number of cycles. You'll probably find that the circuitry required to ...
• 296

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

Addition is important enough to not have it wait for a carry bit to ripple through a 64-bit accumulator: the term for that is a carry-lookahead adder and they are basically part of 8-bit CPUs (and ...
• 121
Accepted

### Combinational Logic Circuits and Theory of Computation

Logic circuits are common in complexity theory, where they go by the name circuits. There is a big difference between circuits and models of computation such as the Turing machine: each circuit can ...
• 269k
Accepted

• 269k
Accepted

### Is there an intuitive proof for the existence of hard functions?

As Pål GD mentions in his comment, the proof is actually very simple: there are $2^{2^n}$ functions, but only $C_S = S^{O(S)}$ circuits of size at most $S \geq n$. The exact constant in the exponent ...
• 269k

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

At the gate level, you are correct that it takes more work to do addition, and thus takes longer. However, that cost is sufficiently trivial that doesn't matter. Modern processors are clocked. You ...
• 3,025
Accepted

### What is the decidable language in $P/poly$ but not in $P$?

Take a language $L$ which is not in $\mathsf{E} = \bigcup_{c=1}^\infty \mathsf{TIME}(2^{cn})$. Now consider the language $L' = \{1^m : m \in L\}$. Then $L'$ is clearly in $\mathsf{P/poly}$, but it's ...
• 269k
Accepted

### Is Green's the best 16-input sorting network so far?

No, a lower bound means that somebody has proved that anything smaller than 53 is impossible. That doesn't mean that a 53-gate network is known or even necessarily possible; just that there cannot be ...
• 80.1k

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

Modern processors are clocked: Every operation takes some integral number of clock cycles. The designers of the processor determine the length of a clock cycle. There are two considerations there: One,...
• 24.9k
Accepted

### Implications of the $\Omega(\frac{2^n}{n})$ circuit lower bound being tight

This has been proved by Muller as early as 1956. Here is the construction. Let $k$ be a parameter. We first compute all possible functions on the first $k$ inputs in size $O(2^{2^k})$ (see below). We ...
• 269k
For a univariate polynomial $p(x)$, yes, it's that easy. For a multivariate polynomial $p(x_1,x_2,\dots,x_k)$, no, no such algorithm works. In particular, when you write "a polynomial of degree $d$ ...