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An adder is a digital circuit that performs addition of numbers. In many computers and other kinds of processors adders are used in the arithmetic logic units or ALU. They are also used in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators, and similar operations.

Addition is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division. The addition of two whole numbers is the total amount of those values combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 add 2 is equal to 5".

A computer is a machine that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of operations, called programs. These programs enable computers to perform an extremely wide range of tasks. A "complete" computer including the hardware, the operating system, and peripheral equipment required and used for "full" operation can be referred to as a computer system. This term may as well be used for a group of computers that are connected and work together, in particular a computer network or computer cluster.

A microprocessor is a computer processor that incorporates the functions of a central processing unit on a single integrated circuit (IC), or at most a few integrated circuits. The microprocessor is a multipurpose, clock driven, register based, digital integrated circuit that accepts binary data as input, processes it according to instructions stored in its memory and provides results as output. Microprocessors contain both combinational logic and sequential digital logic. Microprocessors operate on numbers and symbols represented in the binary number system.

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Although adders can be constructed for many number representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require more logic around the basic adder.

In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each decimal digit is represented by a fixed number of bits, usually four or eight. Special bit patterns are sometimes used for a sign or for other indications.

Excess-3, 3-excess or 10-excess-3 binary code or Stibitz code is a self-complementary binary-coded decimal (BCD) code and numeral system. It is a biased representation. Excess-3 code was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the 1970s, among other uses.

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one).

The half adder adds two single binary digits A and B. It has two outputs, sum (S) and carry (C). The carry signal represents an overflow into the next digit of a multi-digit addition. The value of the sum is 2C + S. The simplest half-adder design, pictured on the right, incorporates an XOR gate for S and an AND gate for C. The Boolean logic for the sum (in this case S) will be A′B + AB′ whereas for the carry (C) will be AB. With the addition of an OR gate to combine their carry outputs, two half adders can be combined to make a full adder. [1] The half adder adds two input bits and generates a carry and sum, which are the two outputs of a half adder. The input variables of a half adder are called the augend and addend bits. The output variables are the sum and carry. The truth table for the half adder is:

In computer programming, an integer overflow occurs when an arithmetic operation attempts to create a numeric value that is outside of the range that can be represented with a given number of digits – either larger than the maximum or lower than the minimum representable value.

XOR gate is a digital logic gate that gives a true output when the number of true inputs is odd. An XOR gate implements an exclusive or; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "one or the other but not both".

The AND gate is a basic digital logic gate that implements logical conjunction - it behaves according to the truth table to the right. A HIGH output (1) results only if all the inputs to the AND gate are HIGH (1). If none or not all inputs to the AND gate are HIGH, a LOW output results. The function can be extended to any number of inputs.

InputsOutputs
ABCS
0000
1001
0101
1110

A full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full-adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in from the previous less-significant stage. [2] The full adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers. The circuit produces a two-bit output. Output carry and sum typically represented by the signals Cout and S, where the sum equals 2Cout + S.

A full adder can be implemented in many different ways such as with a custom transistor-level circuit or composed of other gates. One example implementation is with S = ABCin and Cout = (AB) + (Cin ⋅ (AB)).

A transistor is a semiconductor device used to amplify or switch electronic signals and electrical power. It is composed of semiconductor material usually with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals controls the current through another pair of terminals. Because the controlled (output) power can be higher than the controlling (input) power, a transistor can amplify a signal. Today, some transistors are packaged individually, but many more are found embedded in integrated circuits.

In this implementation, the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. Using only two types of gates is convenient if the circuit is being implemented using simple integrated circuit chips which contain only one gate type per chip.

The OR gate is a digital logic gate that implements logical disjunction – it behaves according to the truth table to the right. A HIGH output (1) results if one or both the inputs to the gate are HIGH (1). If neither input is high, a LOW output (0) results. In another sense, the function of OR effectively finds the maximum between two binary digits, just as the complementary AND function finds the minimum.

An integrated circuit or monolithic integrated circuit is a set of electronic circuits on one small flat piece of semiconductor material that is normally silicon. The integration of large numbers of tiny MOS transistors into a small chip results in circuits that are orders of magnitude smaller, faster, and less expensive than those constructed of discrete electronic components. The IC's mass production capability, reliability, and building-block approach to circuit design has ensured the rapid adoption of standardized ICs in place of designs using discrete transistors. ICs are now used in virtually all electronic equipment and have revolutionized the world of electronics. Computers, mobile phones, and other digital home appliances are now inextricable parts of the structure of modern societies, made possible by the small size and low cost of ICs.

A full adder can also be constructed from two half adders by connecting A and B to the input of one half adder, then taking its sum-output S as one of the inputs to the second half adder and Cin as its other input, and finally the carry outputs from the two half-adders are connected to an OR gate. The sum-output from the second half adder is the final sum output (S) of the full adder and the output from the OR gate is the final carry output (Cout). The critical path of a full adder runs through both XOR gates and ends at the sum bit s. Assumed that an XOR gate takes 1 delays to complete, the delay imposed by the critical path of a full adder is equal to

${\displaystyle T_{\text{FA}}=2\cdot T_{\text{XOR}}=2D.}$

The critical path of a carry runs through one XOR gate in adder and through 2 gates (AND and OR) in carry-block and therefore, if AND or OR gates take 1 delay to complete, has a delay of

${\displaystyle T_{\text{c}}=T_{\text{XOR}}+T_{\text{AND}}+T_{\text{OR}}=D+D+D=3D.}$

The truth table for the full adder is:

InputsOutputs
ABCinCoutS
00000
00101
01001
01110
10001
10110
11010
11111

It is possible to create a logical circuit using multiple full adders to add N-bit numbers. Each full adder inputs a Cin, which is the Cout of the previous adder. This kind of adder is called a ripple-carry adder (RCA), since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder (under the assumption that Cin = 0).

The layout of a ripple-carry adder is simple, which allows fast design time; however, the ripple-carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder. The gate delay can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. In a 32-bit ripple-carry adder, there are 32 full adders, so the critical path (worst case) delay is 3 (from input to carry in first adder) + 31 × 2 (for carry propagation in latter adders) = 65 gate delays. [3] The general equation for the worst-case delay for a n-bit carry-ripple adder, accounting for both the sum and carry bits, is

${\displaystyle T_{\text{CRA}}(n)=T_{\text{HA}}+(n-1)\cdot T_{\text{c}}+T_{\text{s}}=T_{\text{FA}}+(n-1)\cdot T_{c}=3D+(n-1)\cdot 2D=(2n+1)\cdot D.}$

A design with alternating carry polarities and optimized AND-OR-Invert gates can be about twice as fast. [4]

To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry-lookahead adders (CLA). They work by creating two signals (P and G) for each bit position, based on whether a carry is propagated through from a less significant bit position (at least one input is a 1), generated in that bit position (both inputs are 1), or killed in that bit position (both inputs are 0). In most cases, P is simply the sum output of a half adder and G is the carry output of the same adder. After P and G are generated, the carries for every bit position are created. Some advanced carry-lookahead architectures are the Manchester carry chain, Brent–Kung adder (BKA), [5] and the Kogge–Stone adder (KSA). [6] [7]

Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the propagation delay of the circuits to optimize computation time. These block based adders include the carry-skip (or carry-bypass) adder which will determine P and G values for each block rather than each bit, and the carry-select adder which pre-generates the sum and carry values for either possible carry input (0 or 1) to the block, using multiplexers to select the appropriate result when the carry bit is known.

By combining multiple carry-lookahead adders, even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of LCUs.

If an adding circuit is to compute the sum of three or more numbers, it can be advantageous to not propagate the carry result. Instead, three-input adders are used, generating two results: a sum and a carry. The sum and the carry may be fed into two inputs of the subsequent 3-number adder without having to wait for propagation of a carry signal. After all stages of addition, however, a conventional adder (such as the ripple-carry or the lookahead) must be used to combine the final sum and carry results.

### 3:2 compressors

A full adder can be viewed as a 3:2 lossy compressor: it sums three one-bit inputs and returns the result as a single two-bit number; that is, it maps 8 input values to 4 output values. Thus, for example, a binary input of 101 results in an output of 1 + 0 + 1 = 10 (decimal number 2). The carry-out represents bit one of the result, while the sum represents bit zero. Likewise, a half adder can be used as a 2:2 lossy compressor, compressing four possible inputs into three possible outputs.

Such compressors can be used to speed up the summation of three or more addends. If the addends are exactly three, the layout is known as the carry-save adder. If the addends are four or more, more than one layer of compressors is necessary, and there are various possible design for the circuit: the most common are Dadda and Wallace trees. This kind of circuit is most notably used in multipliers, which is why these circuits are also known as Dadda and Wallace multipliers.

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In digital logic and computing, a counter is a device which stores the number of times a particular event or process has occurred, often in relationship to a clock signal. The most common type is a sequential digital logic circuit with an input line called the clock and multiple output lines. The values on the output lines represent a number in the binary or BCD number system. Each pulse applied to the clock input increments or decrements the number in the counter.

In digital circuits, an adder–subtractor is a circuit that is capable of adding or subtracting numbers. Below is a circuit that does adding or subtracting depending on a control signal. It is also possible to construct a circuit that performs both addition and subtraction at the same time.

In CPU design, the use of a sum-addressed decoder (SAD) or sum-addressed memory (SAM) decoder is a method of reducing the latency of the CPU cache access and address calculation. This is achieved by fusing the address generation sum operation with the decode operation in the cache SRAM.

In Boolean algebra, any Boolean function can be put into the canonical disjunctive normal form (CDNF) or minterm canonical form and its dual canonical conjunctive normal form (CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form, and the algebraic normal form.

The Fredkin gate is a computational circuit suitable for reversible computing, invented by Edward Fredkin. It is universal, which means that any logical or arithmetic operation can be constructed entirely of Fredkin gates. The Fredkin gate is a circuit or device with three inputs and three outputs that transmits the first bit unchanged and swaps the last two bits if, and only if, the first bit is 1.

A carry-lookahead adder (CLA) or fast adder is a type of adder used in digital logic. A carry-look ahead adder improves speed by reducing the amount of time required to determine carry bits. It can be contrasted with the simpler, but usually slower, ripple-carry adder (RCA), for which the carry bit is calculated alongside the sum bit, and each stage must wait until the previous carry bit has been calculated to begin calculating its own sum bit and carry bit. The carry-lookahead adder calculates one or more carry bits before the sum, which reduces the wait time to calculate the result of the larger-value bits of the adder. The Kogge–Stone adder (KSA) and Brent–Kung adder (BKA) are examples of this type of adder.

A carry-skip adder is an adder implementation that improves on the delay of a ripple-carry adder with little effort compared to other adders. The improvement of the worst-case delay is achieved by using several carry-skip adders to form a block-carry-skip adder.

In electronics, a carry-select adder is a particular way to implement an adder, which is a logic element that computes the -bit sum of two -bit numbers. The carry-select adder is simple but rather fast, having a gate level depth of .

The Brent–Kung adder, proposed in 1982, is an advanced binary adder design, having a gate level depth of .

Early completion is a property of some classes of asynchronous circuit. It means that the output of a circuit may be available as soon as sufficient inputs have arrived to allow it to be determined. For example, if all of the inputs to a mux have arrived, and all are the same, but the select line has not yet arrived, the circuit can still produce an output. Since all the inputs are identical, the select line is irrelevant.

A carry-save adder is a type of digital adder, used in computer microarchitecture to compute the sum of three or more n-bit numbers in binary. It differs from other digital adders in that it outputs two numbers of the same dimensions as the inputs, one which is a sequence of partial sum bits and another which is a sequence of carry bits.

In electronics, a subtractor can be designed using the same approach as that of an adder. The binary subtraction process is summarized below. As with an adder, in the general case of calculations on multi-bit numbers, three bits are involved in performing the subtraction for each bit of the difference: the minuend, subtrahend, and a borrow in from the previous bit order position. The outputs are the difference bit and borrow bit . The subtractor is best understood by considering that the subtrahend and both borrow bits have negative weights, whereas the X and D bits are positive. The operation performed by the subtractor is to rewrite as the sum .

The serial binary adder or bit-serial adder is a digital circuit that performs binary addition bit by bit. The serial full adder has three single-bit inputs for the numbers to be added and the carry in. There are two single-bit outputs for the sum and carry out. The carry-in signal is the previously calculated carry-out signal. The addition is performed by adding each bit, lowest to highest, one per clock cycle.

A negative base may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base b is equal to −r for some natural number r.

A redundant binary representation (RBR) is a numeral system that uses more bits than needed to represent a single binary digit so that most numbers have several representations. An RBR is unlike usual binary numeral systems, including two's complement, which use a single bit for each digit. Many of an RBR's properties differ from those of regular binary representation systems. Most importantly, an RBR allows addition without using a typical carry. When compared to non-redundant representation, an RBR makes bitwise logical operation slower, but arithmetic operations are faster when a greater bit width is used. Usually, each digit has its own sign that is not necessarily the same as the sign of the number represented. When digits have signs, that RBR is also a signed-digit representation.

An arithmetic logic unit (ALU) is a combinational digital electronic circuit that performs arithmetic and bitwise operations on integer binary numbers. This is in contrast to a floating-point unit (FPU), which operates on floating point numbers. An ALU is a fundamental building block of many types of computing circuits, including the central processing unit (CPU) of computers, FPUs, and graphics processing units (GPUs). A single CPU, FPU or GPU may contain multiple ALUs.

Garbled circuit is a cryptographic protocol that enables two-party secure computation in which two mistrusting parties can jointly evaluate a function over their private inputs without the presence of a trusted third party. In the garbled circuit protocol, the function has to be described as a Boolean circuit.

Open Quantum Assembly Language is an intermediate representation for quantum instructions. The language was first described in a paper published in July 2017, and source code was released as part of IBM's Quantum Information Software Kit (Qiskit) for use with their IBM Q Experience cloud quantum computing platform. The language has similar qualities to traditional hardware description languages such as Verilog.

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