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Logical Circuits

• Fundamentals
  • Propositional calculus
  • Semiconductors
  • MOSFETs
  • CMOS
• Circuits
  • Identity
  • NOT
  • AND
  • NAND
  • OR
  • XOR
  • XNOR
• Credits
• Sources

Propositional calculus / logic

Subjects of the propositional calculus are statements and their connections. Due to a formal representation of such statements it is possible to determine whether or not another proposition is true. The first people using some kind of logic were philosophers in the ancient Greece. Nowadays the propositional calculus is a fundmental part of digital devices.

Propositions

As mentioned earlier propositions are basic parts of the propositional calculus. Propositions are atomic, i.e. they cannot be splitted. Each proposition can be either true or false. In formal descriptions these values can also be represented by 1/0 or high Voltage / low Voltage. The meaning of these representations is equal, so one can use any of them (isomorphy).

The following sentences are examples of propositions:

• Dresden is located in Brandenburg.
• Dresden has at least 100,000 citizens.
• Cities with at least 100,000 citizens are called major cities.
• Dresden is a major city.

Connection with sentential connectives

Propositions can be connected. This can be achieved with connectives or operators. Logical operators work exactly in the same way mathematical ones like "+" or "√" do: The operation gives a result that depends on its inputs. There are operators that work on a single input like "√" (unary operator) and operators that work on multiple inputs. For example "+" works on two inputs (binary operator). That's exactly the same with logical operators. The only difference is that input and output values of logical connectives are logical values and therefore cannot only be true or false. Connected Repositions are called formulas. Every reposition is also a formula. Formulas can contain subformulas.

Some logical connectives are explained in the subchapters in the circuits-chapter. Due to that the following examples are only expressed in every day language. Repositions are underlined for reasons of clarity:

• Dresden is located in Brandenburg
  This formula's result is obviously "false"
• NOT(Dresden is located in Brandenburg)
  The unary NOT-Operator inverts its operand, so the result is"true"
• IF (Dresden has at least 100,000 citizens AND Cities with at least 100,000 citizens are called major cities) THEN Dresden is a major city
  The operator IF ... THEN ... connects a subformula with a reposition. In the subformula the operator AND is used. Both connected repositions are true. So the result of the subformula is true. So the formula is reduced to IF "true" THEN "true". This formula is obviously true..

Inference

If a formula and the value of some subformulas are given, it may be possible to determine the value of the other subformulas. In the following animation you can see how to do that.
Given formula:

• IF (Dresden has at least 1,000,000 citizens AND cities with at least 1,000,000 citizens are called megacities) THEN Dresden is a megacity

Given values:

• The entire formula is "true"
• Dresden is a megacity is "false"
• cities with at least 1,000,000 citizens are called megacities ist "true"

Example from digital technology

As mentioned in the introduction the repositional logic is a fundamental part of digital technology. So the addition of two binary numbers of length one can be performed. Reminder:

• 0 + 0 = 00
• 0 + 1 = 01
• 1 + 0 = 01
• 1 + 1 = 10

The first position of the result is exactly then 1, if both inputs are 1:
Res₁ = In₁ AND In₂
The second position of the result is exactly then 1, if both inputs are different:
Res₂ = In₁ NOTEQUAL In₂
So every functionality in a computer can be represented with the propositional calculus.

Logical circuits

The purpose of logical circuits is to calculate logical functions. Being mechanical in the first computers nowadays electronical implementations are used. A electric potential that corresponds to the input values is applied to the component. The output pin then has the potential that corresponds to the result.