Is it possible to build an AND gate from a NOT gate?

Question

I'm reading through an old physics book and the author is speculating how could he build logic gates using black holes. Taking aside the "black holes" thing, I have stumbled upon his definition for how to build an AND gate that uses a NOT gate first for one of the inputs.

It doesn't make sense to me how to build it with his procedure. Did the author make a mistake or am I missing something?

To make an AND gate using black holes, we start with the concept of the NOT gate constructed earlier. The first input stream of black holes is passed through a NOT gate so that the output stream of that interaction is “the opposite” of the original input stream. This processed stream is then put into a collision course with the second input stream. The remaining portion of the processed stream, after the collisional interaction with the second input stream, becomes the output stream of the whole AND gate (see the figure on page 141).

Let’s see how this gate works. Consider a given position in the stream. If input stream number one has a black hole in that position, then its processed opposite has a space at that position. This space then interacts with the second input stream. If the second input stream also contains ablack hole, the output stream will have a black hole. Both streams must thus have a black hole in the given position in order for the output stream to have a black hole. Although these operations are simple, with enough logic gates a computing machine of enormous complexity can be built.

In practice, a black hole computer constructed from these logic elements will be compromised [...] - The Five Ages of the Universe: Inside the Physics of Eternity, by Fred C. Adams, Greg Laughlin

P.S. The figure mentioned on page 141 is just a normal picture of the AND, NOT and OR gates.

Answer 1

No, it's not possible.

But you missed the context of the book.

At the most basic level, digital computers are built from three fundamental logic elements, which are usually called NOT, AND, and OR gates. By combining large numbers of these simple gates, which perform the basic operations of logic, a computer of virtually unlimited complexity can be constructed.

Before the author mentions the construction of AND gates using NOT gates, the construction of an OR gate is already given in page 141. Similarly, in the following pages, the author also described the construction of a NOT gate.

In order to enforce this operation using strings of black holes to represent numbers, we must build a gravitational potential well (or field of force) which channels the two streams of black holes together side by side. As the two strings of black holes become closer, gravitational attraction takes hold. When the distance between the two streams becomes much less than the distance between successive spaces within each stream, the black holes (if present in a given space) merge and form new black holes. We have thus made an OR gate. If either of the two input streams has a black hole at a given position, the output stream also has a black hole at that position.

The author claims that it's theoretically possible to implement an OR gate by putting two black holes in a course of collision, since black hole + black hole = new black hole, black hole + space = black hole, and space + space = space. This is basically the principle behind Wired-OR: connect two wires together, the output will be energized if either one wire is energized, it only needs a simple junction, and no electrical component is needed (in real circuits, diodes are often required to isolate both inputs).

Given an OR gate and a NOT gate, it's certainly possible to construct all logic gates since a NOR gate is functional complete. It's basically the idea behind a DTL (diode-transistor logic) circuit - two diodes are used to implement Wired-OR, and a transistor is used as an inverter to implement NOT.

However, it cannot work as the way described by the author, it seems to be an oversight.

In the construction of the AND gate, the authors said, "the first input stream is inverted and put onto "a collision course" to the second input stream". In other words, (NOT a) OR b. But this expression is always 1, and it's absolutely useless. To build a proper AND gate, one needs to invert both input, OR them, and invert them again.

• NOT ((NOT a) OR (NOT b))

I assume it's an oversight in the book (the author wants to avoid NAND or NOR and use AND and OR, but eventually mixed up their truth tables). Nevertheless, the idea of building AND gates from NOT gates and OR gates is valid.

Answer 2

Here is one way to use inverters (NOT) to yield the AND function:

^{simulate this circuit – Schematic created using CircuitLab}

If both inputs are high, then the inputs to the second inverters are both low and the output is high. If either input goes low, one of the second inverters (or both if both inputs are low) get an input high and pulls the output low.

Answer 3

No, because a pure NOT gate has one input which gets inverted to output. There is no way that combining mathematical or ideal NOT gates will make an AND gate. However if the physical implementation of the NOT gate allow gate outputs to be connected together, which is not mentioned in the text, then it is possible.

Answer 4

Without seeing the rest of it, diagrams and all, it is hard to tell. But here is one way it could work.

The logic gate is not the NOT. It is the collision that does the logic.

Imagine a sequence of “ball” or “no ball” firing from the second jet towards the detector. Let’s take the case where nothing is coming from the inverted first jet (that is, the input before inversion is all 1s). Look at it from downstream of the collision zone. If a ball is fired in the second jet, you see the ball. If a ball isn’t fired in the second jet, you don’t see the ball.

To summarise so far: when the input to the inverter is 1, you see, at the output, whatever pattern is coming in on the second jet.

Now suppose that the first-jet inverter has an input of 0. So the inverter is Now shooting balls.

If the second jet shoots a ball, it collides with the output of the inverter. It explodes or annihilates or goes charging off in another direction. Never mind which: your detector will see nothing. Its output is 0.

If the second jet doesn’t shoot a ball, the output of the first jet goes flying in whatever direction it was flying in: presumably not at the detector. Again, nothing is detected.

Summary: first jet 1, inverter output 0, detected result = second jet input. First jet 0, inverter output 1, detected result = 0.

It is all pretty fatuous and over-elaborate and pointless, but not actually nonsense.