The 74181 ALU (arithmetic/logic unit) chip powered many of the minicomputers of the 1970s: it provided fast 4-bit arithmetic and logic functions, and could be combined to handle larger words, making it a key part of many CPUs. But if you look at the chip more closely, there are a few mysteries. It implements addition, subtraction, and the Boolean functions you’d expect, but why does it provide several bizarre functions such as “A plus (A and not B)”? And if you look at the circuit diagram (below), why does it look like a random pile of gates rather than being built from standard full adder circuits. In this article, I explain that the 74181’s set of functions isn’t arbitrary but has a logical explanation. And I show how the 74181 implements carry lookahead for high speed, resulting in its complex gate structure.

The 74181 chip is important because of its key role in minicomputer history. Before the microprocessor era, minicomputers built their processors from boards of individual chips. A key part of the processor was the arithmetic/logic unit (ALU), which performed arithmetic operations (addition, subtraction) and logical operations (AND, OR, XOR). Early minicomputers built ALUs out of a large number of simple gates. But in March 1970, Texas Instruments introduced the 74181 Arithmetic / Logic Unit (ALU) chip, which put a full 4-bit ALU on one fast TTL chip. This chip provided 32 arithmetic and logic functions, as well as carry lookahead for high performance. Using the 74181 chip simplified the design of a minicomputer processor and made it more compact, so it was used in many minicomputers. Computers using the 74181 ranged from the popular PDP-11 and Xerox Alto minicomputers to the powerful VAX-11/780 “superminicomputer“. The 74181 is still used today in retro hacker projects.1

The 74181 implements a 4-bit ALU providing 16 logic functions and 16 arithmetic functions, as the datasheet (below) shows. As well as the expected addition, subtraction, and Boolean operations, there are some bizarre functions such as “(A + B) PLUS AB“.

So how is the 74181 implemented and why does it include such strange operations? Is there any reason behind the 74181’s operations, or did they just randomly throw things in? And why are the logic functions and arithmetic functions in any particular row apparently unrelated? I investigated the chip to find out.

## The 16 Boolean logic functions

There’s actually a system behind the 74181’s set of functions: the logic functions are the 16 possible Boolean functions f(A,B). Why are there 16 possible functions? If you have a Boolean function f(A,B) on one-bit inputs, there are 4 rows in the truth table. Each row can output 0 or 1. So there are 2^4 = 16 possible functions. Extend these to 4 bits, and these are exactly the 16 logic functions of the 74181, from trivial 0 and 1 to expected logic like A AND B to contrived operations like NOT A AND B. These 16 functions are selected by the S0-S3 select inputs.

## Arithmetic functions

The 74181’s arithmetic operations are a combination of addition, subtraction, logic operations, and strange combinations such as “A PLUS AB PLUS 1”. It turns out that there is a rational system behind the operation set: they are simply the 16 logic functions added to A along with the carry-in.2 That is, the arithmetic functions are: A PLUS f(A,B) PLUS carry-in. For example, If f(A,B)=B, you get simple addition: A PLUS B PLUS carry-in. If f(A,B) = NOT B, you get A PLUS NOT B PLUS carry-in, which in two’s-complement logic turns into subtraction: A MINUS B MINUS 1 PLUS carry-in.

Other arithmetic functions take a bit more analysis. Suppose f(A,B) = NOT (A OR B). Then each bit of A PLUS f(A,B) will always be 1 except in the case where A is 0 and B is 1, so the result of the sum is A OR NOT B. Even though you’re doing addition, the result is a logical function since no carry can be generated. The other strange arithmetic functions can be understood similarly.3

Thus, the 16 arithmetic functions of the 74181 are a consequence of combining addition with one of the 16 Boolean functions. Even though many of the functions are strange and probably useless, there’s a reason for them. (The Boolean logic functions for arithmetic are in a different order than for logical operations, explaining why there’s no obvious connection between the arithmetic and logical functions.)

## Carry lookahead: how to do fast binary addition

The straightforward but slow way to build an adder is to use a simple one-bit full adders for each bit, with the carry out of one adder going into the next adder. The result is kind of like doing long addition by hand: in decimal if you add 9999 + 1, you have to carry the 1 from each column to the next, which is slow. This “ripple carry” makes addition a serial operation instead of a parallel operation, harming the processor’s performance. To avoid this, the 74181 computes the carries first and then adds all four bits in parallel, avoiding the delay of ripple carry. This may seem impossible: how can you determine if there’s a carry before you do the addition? The answer is carry lookahead.

Carry lookahead uses “Generate” and “Propagate” signals to determine if each bit position will always generate a carry or can potentially generate a carry. For instance, if you’re adding 0+0+C (where C is the carry-in), there’s no way to get a carry out from that addition, regardless of what C is. On the other hand, if you’re adding 1+1+C, there will always be a carry out generated, regardless of C. This is called the Generate case. Finally, for 0+1+C (or 1+0+C), there will be a carry out if there is a carry in. This is called the Propagate case since if there is a carry-in, it is propagated to the carry out.4 Putting this all together, for each bit position you create a G (generate) signal if both bits are 1, and a P (propagate) signal unless both bits are 0.

The carry from each bit position can be computed from the P and G signals by determining which combinations can produce a carry. For instance, there will be a carry from bit 0 to bit 1 if P_{0} is set (i.e. a carry is generated or propagated) and there is either a carry-in or a generated carry. So C_{1} = P_{0} AND (C_{in} OR G_{0}).

Higher-order carries have more cases and are progressively more complicated. For example, consider the carry in to bit 2. First, P_{1} must be set for a carry out from bit 1. In addition, a carry either was generated by bit 1 or propagated from bit 0. Finally, the first carry must have come from somewhere: either carry-in, generated from bit 0 or generated from bit 1. Putting this all together produces the function used by the 74181: C_{2} = P_{1} AND (G_{1} OR P_{0}) AND (C_{0} OR G_{0} OR G_{1}).

As you can see, the carry logic gets more complicated for higher-order bits, but the point is that each carry can be computed from G and P terms and the carry-in. Thus, the carries can be computed in parallel, before the addition takes place.5

## Creating P and G with an arbitrary Boolean function

The previous section showed how the P (propagate) and G (generate) signals can be used when adding two values. The next step is to examine how P and G are created when adding an arbitrary Boolean function f(A, B), as in the 74181. The table below shows P and G when computing “A PLUS f(A,B)”. For instance, when A=0 there can’t be a Generate, and Propagate depends on the value of f. And when A=1, there must be a Propagate, while Generate depends on the value of f.

A | B | A PLUS f(a,b) | P | G |

0 | 0 | 0+f(0,0) | f(0,0) | 0 |

0 | 1 | 0+f(0,1) | f(0,1) | 0 |

1 | 0 | 1+f(1,0) | 1 | f(1,0) |

1 | 1 | 1+f(1,1) | 1 | f(1,1) |

In the 74181, the four f values are supplied directly by the four Select (S pin) values, resulting in the following table:6

A | B | A PLUS f | P | G |

0 | 0 | 0 | S1 | 0 |

0 | 1 | 1 | S0 | 0 |

1 | 0 | 1 | 1 | S2 |

1 | 1 | 10 | 1 | S3 |

The chip uses the logic block below (repeated four times) to compute P and G for each bit. It is straightforward to verify that it implements the table above. For instance, G will be set if A is 1, B is 1 and S3 is 1, or if A is 1, B is 0 and S2 is set.

Read more: Inside the vintage 74181 ALU chip