Tutorials/Logic gates
This article makes use of diagrams in the MCRedstoneSim format for compactness and clarity. Some of the designs are more than two blocks high which is represented here by the layers being frames in an animated gif or labeled side by side. A full legend is on the Redstone schematics page.

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Contents
Adders[edit]
Version 1[edit]
 Full Adder
A full adder takes two inputs A and B and a Carry input and produces the Sum and Carry outputs. It relies on two XOR gates, two AND gates, and one OR gate. With some thought, these gates can be compressed (as both AND and XOR gates already exist in the game, and an OR gate can simply be a redstone wire).
A and B are the bit inputs and C' is the carry in. It produces a sum at S and a carry out at C. When full adder modules are tiled together C' and C will be connected, which allows the carry to propagate to the next module.
 Half Adder
The half adder is nearly identical to the full adder, except the second XOR gate is removed and the output from the first XOR gate becomes S. There is no carry in (C'), but the carry out (C) circuit is still on top of the first XOR gate and provides a carry to the first full adder. Some ALUs will not use a half adder for the first bit, to support INCREMENT (allow a carry in on the first bit).
InLine Adder[edit]
 Full Adder
This full adder is similar to the previous one, except for the fact that it is two wide and the inputs are aligned vertically. This design is great for minimizing horizontal space and can be built inline with two redstone buses, eliminating the space required to expand a bus to reach the inputs of a wider full adder.
Tutorial video (view on YouTube) 

Version 2[edit]
 Full Adder
Gates: XNOR (2), IMPLIES, NOT, OR, AND
Size: 6×12×5 (including I/O spaces)
This adder will take 2 bits and a carried over bit (actually C, rather than C, a value held in the redstone in the bottom left corner on layer 1) and add them all together, producing a sum (S) bit and a carry (actually C rather than C).
When using the gates above; mind the inputs and outputs. You may be wondering why there are so many inverted signals being used instead of the regular signal.
The adders shown here use XNOR gates rather than XOR gates because they are more compact, and as a result, implies gates must be used instead of AND gates, which also happen to be more compact.
Therefore, for the most compact adder, inverse signals must be used. These adders are too complex to be easily deciphered with 2 layers per square, so each single layer has been drawn separately to ease the building process.
 Half Adder
Gates: XNOR, IMPLIES
Size: 5×4×4
This adder will take 2 bits and add them together. The resulting bit will be the output of S (sum). If both bits are 1, there will be a carry over, and C will become 1 (C will become 0). This half adder can be modified to create a non inverted C output, but this configuration is used so that it can be implemented as the start of a chain of full adders.
Extension: for those new to advanced redstone like myself, it's easier to understand it like this: let's say output B (C) has a NOT gate that inverts the signal and it leads to an iron door or piston door etc. Output A (S) is connected to sticky pistons controlling the floor. Let's say for sake of argument that there is 1×1×1 block NOT affected by the sticky pistons, this is the safety block. When you activate input A, both the door will open and the floor will drop, if you're standing on the safety block, then you will not fall. Input B will control only the floor, but if input A is on the input B will control them both. When both are on, input A will only affect the floor. This means if you are off the server and want no one in, leave A and B on, when they deactivate A, the floor will drop, but the door will stay closed, so if they know the secret, they still cannot get in.
Version 3[edit]
 Full Adder
Size: 5×6×3
Carry input and output are aligned to easily connect many of these modules in series.
Fast Adding[edit]
When building advanced digital circuits like computers and multipliers, the adders used must be as fast as possible to ensure maximum running speed. Simple adders have one fundamental speed problem which numerous adder designs try to correct to speed up. The issue is carry propagation delay: delay caused by the way adders borrow carries. We can see this when we do the problem 1111 + 0001:
1111 0001  1110
This is the first step of the addition process, XORing the two inputs. Because there were two 1s in the least significant bit, the AND gate activates and carries to the next bit:
1 1111 0001  1100
But here is the issue: You now need to borrow a carry again, because, in the two's place, there are two ones. This is done by ANDing the output of the first halfadder with the carry from the previous bit and this is a huge issue. Because, for the next bit, you AND the borrowed carry again, and again. Each AND gate takes 2 ticks, so, in order to calculate all of the carries that need to be added up in the final step, it takes 2 ticks times 4 bits, or 8 ticks.
Imagine you see the problem 999 + 1. You don't sit around thinking "9 + 1 is 10, carry 1, so 9 + 1 is 10, carry the 1, so 9 + 1 is 10, so 1000." It's the same situation in an advanced circuit.
Real electrical engineers and creative redstoners have designed circuits that calculate adder carries faster than this sequential method.
Incidentally, adders that calculate carries one at a time in this fashion are called Ripple Carry adders.
Piston Adders[edit]
One of the simplest and most classic ways of solving the ripple carry problem is to use instant AND gates that use pistons. These adders are simple and fast, but are inconstant because they use pistons. When blocks are accidentally dropped, the entire circuit breaks. Pistons also have timing awkwardness that can be excruciatingly inconvenient when building an advanced circuit that relies heavily on timing.
Whenever a carry is created, it is sent through the wire with the lever on it, and, instead of going through an AND gate, the piston retracts and the carry can move on to the next bit which adds no carry propagation delay at all (until the signal strength runs out).
This video shows a straightforward implementation of the logic. The design is large and spread out, so it's easy to see each individual part of the adder and the carry logic.
4bit Adder[edit]
Gates: XNOR (7), IMPLIES (4), NOT (4), OR (3), AND (3)
Size: 23X12X5
Note! The least significant digit ("ones" digit) is on the left of the diagram so that the progression from half adder to the full adders can be seen more clearly. Reverse the diagram if you want a conventional left to right input.
This adder will take 2, 4 bit numbers (A and B) and add them together, producing a sum (S) bit for each bit added and a carry (C) for the whole sum. The sum bits are in the same order as the input bits, which on the diagram means that the leftmost S output is the least significant digit of the answer. This is just an example of a string of adders; adders can be strung in this way to add bigger numbers as well.
Alternate 4bit Adder[edit]
The same function but a different design with 4 full adders instead of 1 half adder and 3 full adders
NOTE: switches are inputs A and B (top switch C input)
Subtracting[edit]
Subtracting and adding are the same thing when reduced down to the idea that, for example, 32 = 3 + (2) = 1. Since we already have the framework in place to add bits, it is fairly simple to subtract by just adding the negative bit. The problem lies in the representation of negative numbers.
We are all familiar with the elementary school concept of "borrowing" in subtraction from the next column like this:
5623  128 
We are not capable of taking 8 from three, so we "borrow" a 1 from the next decimal place to allow us to subtract 8 from 13 instead, resulting in 5
1 5623  128  5
Computers are not capable of assumptions, so when a computer needs to find a negative it does not (and cannot) put a negative sign in front of the input. It just subtracts from zero "borrowing" from the next column like so:
000000  3  999997
This is the same in binary. Let us, for example use a 4 bit binary number for the example:
1 11 111 1111 0000 0000 0000 0000 0011 0011 0011 0011      1  01  101 1101
We could repeat this forever, but that would be useless. This is about what a 4 bit register does: it truncates after 4 bits worth of data. So after we truncate the number (which I kindly did for you in the example, otherwise the number would have an infinite number of 1's to the left). Thanks to this little perk, we can do whatever we want to the 0's after the four of them, including (which will prove to be fantastically useful later) adding a single 1 in front of them.
10000 0011  1101 < NOTE: This number is positive! Success!
Remember how we said that our redstone had no special way of designating a negative from a positive? We just created a way. If the most significant (first) bit of a number is 1 that means that it is a negative number. This fantastic perk of binary numbers is a theorem called "Two's Complement".
Formally, Two's Complement is defined as:
The negative of a number b with bit length n is equal to 2^(n+1)  b
Essentially what this is saying is that b is just the inversion of b (exchange 1's for 0's and 0's for 1's) plus 1.
What we have done is turn the first bit into a "negative sign" if it is on, but if you have been reading this you realize it is not that simple. Numbers that have a negative sign like this are commonly referred to as signed integers. Numbers like in a normal adder, where two's compliment is not taken into effect are called unsigned integers. Unsigned integers can go to a higher value, but cannot go below zero where as signed integers can only go half as high, but they can go equally as far below zero. This means that the two numbers have the same range, it is just in a different location like so (this is with an 8 bit number):
Unsigned: 0255 Signed 128127
It should be noted that some strange effects can take place when using the lowest signed value (in this case 128) so this should be avoided.
Now that we have a positive way of representing our negative numbers it is very trivial to implement this into an adder. Currently our adder solves
A + B
We want it to solve
A  B
or
A + (B)
Therefore, if we enter the two's complement of B, our adder becomes a subtractor. This is easily implemented by using the Carryin bit of the least significant (first) bit as the "+1" and then all that is left is to invert B.
There is one important thing to note when implementing this. Because it is possible to get a two's complement number out, when subtracting the most significant digit must be inverted. This is usually the Carry out of the last adder.
This can all be implemented into an adder like so:
A control bit is added to the circuit such that when it is on, the unit subtracts, and when it is off the unit adds. After this, add XOR gates between the control bit and each B input. Route the output of each XOR to the B input of each adder. Finally, to make the unit Two's compliment compatible, a final XOR gate must be added between the control bit and the carry out of the most significant bit.
This is the simplest way to implement negatives and subtraction in a CPU, as it will add gracefully and store well in registers. If this is to be implemented in a calculator, simply subtract 1 from the output and then invert all the outputs except the most significant one. The most significant bit will be on if the number is negative.
In order to make a subtracter, simply invert one of the binary inputs (the 1st or 2nd number). If the number is negative, the answer comes out inverted. In real computers, the first bit (also called the sign) decides whether the number is positive or negative, if you include this (applying the same inverting rule) you can detect whether the number is negative, or if it is just a big number.
Logic units[edit]
In circuits, it might be useful to have a logic unit that will, based on the input, decide which output is to be chosen. Such a unit can then be used for more complex circuits, such as an ALU.
This is an example of a 2bit logic unit that will have four states depending on the input.
The outputs are in top row, with 11, 00, 01, 10 order (input order: first first, bottom second).
This is another example of a simplified version using Gray codes. The output appears at the torches at the end of the top rows. This design can be extended to any number of bits, but practical limitations due to timing considerations restrict the use of more than a byte or so. The outputs are triggered by the inputs 11, 01, 00, 10, respectively.
Arithmetic logic unit[edit]
The Arithmetic logic unit (ALU) is the central part of the CPU. It does calculations and logical processing and then passes this information to a register. The ALU, on basis of the input, selects a specific function, performs it, and then gives the result.
The ALU shown below is a 1bit ALU with the functions: ADD, AND, XOR. It takes the A and B inputs and then performs the selected functions. Read about the adders to see how the ADD function works. XOR and AND are basic functions that are explained on the logic circuits page. There can be more functions added to an ALU, like multiplication, division, OR, NAND... etc. These functions could, with some modifications, be added to this 1 bit ALU.
This 1bit ALU can be linked to each other to create an as many bit ALU as possible. Just like adders you need to connect the Carry out (Cout) to the Carry in (Cin) of the next ALU