205 lines
7.5 KiB
Markdown
205 lines
7.5 KiB
Markdown
---
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tags: [logic-gates, binary]
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---
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# Logic gates
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> [A logic gate consists in] three connections where there may or may not be
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> some electricity. Two of those connections are places where electricity may be
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> put into the device, and the third connection is a place where electricity may
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> come out of the device.
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[J.C. Scott. 2009. **But How Do It Know? The Basics of Computers for Everyone**,
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21]
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Logic gates are the basic building blocks of digital computing. **A logic gate
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is an electrical circuit that has one or more than one input and only one
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output.** The input and output points of the gate are
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[pins](Integrated_circuits.md) The
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input controls the output and the logic determining which types of input
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(on/off) lead to specific outputs (on/off) is isomorphic with the
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truth-conditions of the
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[Boolean connectives](Truth-functional_connectives.md) specifiable in
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terms of [truth tables](Truth-tables.md).
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Physically, what 'travels through' the gates is electrical current and what
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constitutes the 'gate' is a
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[transistor](Transistors.md)
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responding to the current. Going up a level of abstraction, the current/ charge
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is identified with a
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[bit](Binary_units_of_measurement.md#binary-units-of-measurement).
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It is bits that go into the gate and bits which come out: binary information
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that may be either 1 or 0.
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## Elementary and composite gates
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We distinguish elementary from composite logic gates. An elementary gate is a
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single gate embodying a single logical connective. It cannot be reduced any
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lower as a logical abstraction. A composite gate is a gate made up of more than
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one elemen>tary gate and/or other composite gates.
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An example of a composite gate would be a three-way AND. An AND with three
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inputs rather than the standard two that furnish the elementary AND gate. This
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gate would output 1 when all three gates have the value 1 and 0 otherwise.
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[Adders](Half_adder_and_full_adder.md)
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and [latche>s](Latches.md) whilst
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being
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[integrated circuits](Integrated_circuits.md)
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are also, technically speaking, composite gates.
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## Gate interface / gate implementation
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The gate _interface_ is an abstraction that the enables the user to think of the
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gate simply in terms of inputs and outputs, without being conc>erned with the
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technical details of how this is achieved. How it is achieved is the gate
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_implementation_.
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We can demonstrate this with the earlier example of a three-way AND. The diagram
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below represents the gate as an interface:
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// Add: Interface diagram
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Whereas this diagram presents the implementation of the gate: it shows the
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specific combination of gates which creates the enables the behaviour
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represented in the interface diagram.
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// Add: Implementation diagram
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> Importantly, a single interface may be implemented in a variety of ways. There
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> is a one-to-many relationship at work here. From the point of view of the user
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> interface these differences should not be detectable. This is another example
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> of
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> [hardware abstraction](Hardware_abstraction_and_modularity.md)
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## NOT gate
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> The NOT gate inverts the value of whatever input it receives
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### Truth conditions
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| $P$ | $\lnot P$ |
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| --- | --------- |
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| 1 | 0 |
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| 0 | 1 |
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### Interactive circuit
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<iframe src="https://circuitverse.org/simulator/embed/not-gate-aeb5f9e5-9f58-4883-b8e5-d70f6d023185?theme=default&display_title=false&clock_time=true&fullscreen=true&zoom_in_out=true" style="border-width:; border-style: solid; border-color:;" name="myiframe" id="projectPreview" scrolling="no" frameborder="1" marginhe
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### Truth conditions
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tion) truth functional connective
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### Symbol
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### Truth conditions
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### Truth conditions
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| 1 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |
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### Interactive circuit
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<iframe src="https://circuitverse.org/simulator/embed/and-gate-b6937338-e83f-474b-af79-854b39c151a3?theme=default&display_title=false&clock_time=true&fullscreen=true&zoom_in_out=true" style="border-width:; border-style: solid; border-color:;" name="myiframe" id="projectPreview" scrolling="no" frameborder="1" marginheight="0px" marginwidth="0px" height="250" width="500" allowFullScreen></iframe>
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## NAND gate
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> The NAND gate inverts the truth conditions of AND.
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### Symbol
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### Truth conditions
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| $P$ | $Q$ | $\lnot(P \land Q)$ |
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| --- | --- | ------------------ |
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| 1 | 1 | 0 |
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| 1 | 0 | 0 |
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| 0 | 1 | 0 |
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| 0 | 0 | 1 |
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### Interactive circuit
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<iframe src="https://circuitverse.org/simulator/embed/nand-gate-60613ab9-9562-445e-9883-c4ea1920e206?theme=default&display_title=false&clock_time=true&fullscreen=true&zoom_in_out=true" style="border-width:; border-style: solid; border-color:;" name="myiframe" id="projectPreview" scrolling="no" frameborder="1" marginheight="0px" marginwidth="0px" height="250" width="500" allowFullScreen></iframe>
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NAND is a **universal logic gate**: equipped with just a NAND we can represent
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every other possible logical condition. In practice with circuits, it is more
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efficient to use specific dedicated gates (i.e OR, AND, NOT etc) for the other
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Boolean connectives but in principle the same output can be achieved through
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NANDs alone.
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## OR gate
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> The OR gate represents the truth conditions of the
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> [disjunction](Truth-functional_connectives### Truth
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> conditions.md#disjunction) truth functional connective
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### Symbol
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### Truth condition
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### Truth conditions
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s
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| $P$ | $Q$ | $P \lor Q$ |
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| --- | --- | ---------- |
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| 1 | 1 | 1 |
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| 1 | 0 | 1 |
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| 0 | 1 | 1 |
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| 0 | 0 | 0 |
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### Interactive circui
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### Truth conditions
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t
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<iframe src="https://circuitverse.org/simulator/embed/or-gate-087e4933-7963-482d-b4bf-9e130ef05706?theme=default&display_title=false&clock_time=true&fullscreen=true&zoom_in_out=true" style="border-width:; border-style: solid; border-color:;" name="myiframe" id="projectPreview" scrolling="no" frameborder="1" marginheight="0px" marginwidth="0px" height="250" width="500" allowFullScreen></iframe>
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## XOR gate
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> The OR gate represents the truth conditions of the exclusive OR
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### Symbol
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### Truth conditions
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| $P$ | $Q$ | $\lnot(P \Leftrightarrow Q)$ |
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| --- | --- | ---------------------------- |
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| 1 | 1 | 0 |
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| 1 | 0 | 1 |
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| 0 | 1 | 1 |
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| 0 | 0 | 0 |
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### Interactive circuit
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<iframe src="https://circuitverse.org/simulator/embed/xor-gate-a240131e-a9c3-4240-a52b-e14412fdd654?theme=default&display_title=false&clock_time=true&fullscreen=true&zoom_in_out=true" style="border-width:; border-style: solid; border-color:;" name="myiframe" id="projectPreview" scrolling="no" frameborder="1" marginheight="0px" marginwidth="0px" height="250" width="500" allowFullScreen></iframe>
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## NOR gate
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> The NOR gate inverts the function of an OR gate
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### Symbol
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### Truth conditions
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| $P$ | $Q$ | $P \lor Q$ |
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| --- | --- | ---------- |
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| 1 | 1 | 0 |
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| 1 | 0 | 0 |
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| 0 | 1 | 0 |
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| 0 | 0 | 1 |
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### Interactive circuit
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<iframe src="https://circuitverse.org/simulator/embed/nor-gate-ac7946b4-f5d0-4c87-afd1-a2e92326d006?theme=default&display_title=false&clock_time=true&fullscreen=true&zoom_in_out=true" style="border-width:; border-style: solid; border-color:;" name="myiframe" id="projectPreview" scrolling="no" frameborder="1" marginheight="0px" marginwidth="0px" height="250" width="500" allowFullScreen></iframe>
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