# Combinatorial Logic

Home * Hardware * Combinatorial Logic

A Combinatorial Logic (also Combinational Logic) is a digital circuit where one or more outputs are boolean functions of multiple inputs. The basic boolean operations conjunction, disjunction and logical negation are sufficient to derive all other boolean as well as arithmetical operations. Opposed to a sequential logic, outputs are not dependent on their history, that is a combinatorial logic does not require memory.

# Implementation

In hardware, combinatorial logic can either realized with hardwired gates of certain logic families or programmable logic devices. If the number of inputs is reasonable small, a once programmed ROM or LUT can act as combinatorial logic. The inputs are the address, while one output is associated with a data-pin. In software this is like performing ALU-operations versus a memory lookup with pre-calculated outputs for all relevant inputs, related to the space-time tradeoff.

# Basic Operations

Operator symbols, truth tables, IEC and ANSI circuit diagram symbols, as well as discrete and relay logic circuits are given.

## AND

An AND gate implements a logical conjunction.

```a ∧ b
```

a b a and b
0 0 0
0 1 0
1 0 0
1 1 1

## OR

An OR gate implements a logical disjunction.

```a ∨ b
```

a b a or b
0 0 0
0 1 1
1 0 1
1 1 1

## NOT

A NOT gate or Inverter implements a logical negation.

```¬a
```

a not a
0 1
1 0

# Derived Operations

Concrete electronic gates often combine AND and OR with trailing NOT for so called NAND and NOR gates. As application of De Morgan's laws a NAND can also be interpreted as OR of inverted inputs, and NOR as AND of inverted inputs.

## NAND

A NAND gate is the inversion of AND, NOT AND.

```a ⊼ b
```

a b not(a and b)
0 0 1
0 1 1
1 0 1
1 1 0

## NOR

A NOR gate is the inversion of OR, NOT OR.

```a ⊽ b
```

a b not(a or b)
0 0 1
0 1 0
1 0 0
1 1 0

## XOR

A XOR gate implements a exclusive disjunction, which might be derived from AND/OR/NOT, for instance from four NAND gates.

```a ⊻ b
```

a b a xor b
0 0 0
0 1 1
1 0 1
1 1 0

# DNF and CNF

Combinational logic can be visualized by truth tables and the construction is generally done using disjunctive (sum of products) or conjunctive normal form (products of sums), and using boolean algebra or Karnaugh maps to simplify the expression.

# ALU

Combinatorial logic is a huge part of the arithmetic logic unit (ALU) of processors, which provides accordant boolean logical instructions working on all bits of a register in parallel as mentioned in General Setwise Operations of Bitboards. Therefor each Combinatorial Logic can of course emulated in software.

A half adder performs an addition on two one-bit binary numbers. Output of an AND gate is the carry, while a XOR gate leaves the one-bit sum. A full adder with tad more gates adds three one-bit binary numbers, the third usually to feed in the carry from the previous digit, usually in carry look ahead architectures, such as Kogge-Stone adder, also mentioned as parallel prefix algorithm.

## Combinatorial Attacks

Assuming there are 13 times 64 digital inputs from a hardware wired chessboard. The 13 inputs per square has one exclusive "one" signal for either one of the twelve pieces or an empty signal. For each square a number of attacks/defend outputs may be defined to implement a huge Combinatorial Logic as a "zero cycle" attack table, i. e. output a8 is attacked from south by white rook as DNF (sum of products).

### C Syntax

With C-like operators, that is '&' for AND and '|' for OR, the DNF would look like this:

```southAttackByWhiteRook(a8) ::=
wrook(a7)
| ( empty(a7) & wrook(a6) )
| ( empty(a7) & empty(a6) & wrook(a5) )
| ( empty(a7) & empty(a6) & empty(a5) & wrook(a4) )
| ( empty(a7) & empty(a6) & empty(a5) & empty(a4) & wrook(a3) )
| ( empty(a7) & empty(a6) & empty(a5) & empty(a4) & empty(a3) & wrook(a2) )
| ( empty(a7) & empty(a6) & empty(a5) & empty(a4) & empty(a3) & empty(a2) & wrook(a1) )
```

### Circuit

The same sample as circuit f. i. in Diode logic with 34 diodes and 7 resistors:

``` Board bus
empty                    white rook
a1 a2 a3 a4 a5 a6 a7 a8  a1 a2 a3 a4 a5 a6 a7 a8      ANDs (MIN)         OR (MAX)
|  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |
|  |  |  |  |  |      |  |  |  |  |  |  |
o--|--|--|--|--|------|--|--|--|--|--|--|----------|<|-----o---| R1 |---o +Vcc
o--|--|--|--|------|--|--|--|--|--|--|----------|<|-----|
|  o--|--|--|------|--|--|--|--|--|--|----------|<|-----|  D1-D7
|  |  o--|--|------|--|--|--|--|--|--|----------|<|-----|
|  |  |  o--|------|--|--|--|--|--|--|----------|<|-----|
|  |  |  |  o------|--|--|--|--|--|--|----------|<|-----|
|  |  |  |  |      o--|--|--|--|--|--|----------|<|-----o----------|>|------o  D28
|  |  |  |  |         |  |  |  |  |  |                                      |
o--|--|--|--|---------|--|--|--|--|--|----------|<|-----o---| R2 |---o +Vcc |
o--|--|--|---------|--|--|--|--|--|----------|<|-----|                   |
|  o--|--|---------|--|--|--|--|--|----------|<|-----|  D8-D13           |
|  |  o--|---------|--|--|--|--|--|----------|<|-----|                   |
|  |  |  o---------|--|--|--|--|--|----------|<|-----|                   |
|  |  |  |         o--|--|--|--|--|----------|<|-----o----------|>|------o  D29
|  |  |  |            |  |  |  |  |                                      |
o--|--|--|------------|--|--|--|--|----------|<|-----o---| R3 |---o +Vcc |
o--|--|------------|--|--|--|--|----------|<|-----|                   |
|  o--|------------|--|--|--|--|----------|<|-----|  D14-18           |
|  |  o------------|--|--|--|--|----------|<|-----|                   |
|  |  |            o--|--|--|--|----------|<|-----o----------|>|------o  D30
|  |  |               |  |  |  |                                      |
o--|--|---------------|--|--|--|----------|<|-----o---| R4 |---o +Vcc |
o--|---------------|--|--|--|----------|<|-----|  D19-D22          |
|  o---------------|--|--|--|----------|<|-----|                   |
|  |               o--|--|--|----------|<|-----o----------|>|------o  D31
|  |                  |  |  |                                      |
o--|------------------|--|--|----------|<|-----o---| R5 |---o +Vcc |
o------------------|--|--|----------|<|-----|  D23-D25          |
|                  o--|--|----------|<|-----o----------|>|------o  D32
|                     |  |                                      |
o---------------------|--|----------|<|-----o---| R6 |---o +Vcc |
|  |                  |  D26-D27          |
o--|----------|<|-----o----------|>|------o  D33
|                                      |  D34
o------------------o----------|>|------o-->--o a8 attacked
|       by white rook
_       from south
| |  R7
|_|
|
--o--
---   gnd
-
```