CA Lecture 5a: Digital Logic

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CA Lecture 5a Digital Logic

• Combinational Logic,
• Truth table,
• Logic gates
• Read Appendix A of textbook p441 p447

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Digital logic level

• The digital logic level the computers real
  hardware are constructed from logic gates.
• Logic gates are actually made from transistors,
  capacitors and wires etc. see diagrams in next
  slide.
• A circuit is a combination of logic gates.
• Different combinations of logic gates can make up
  circuits for doing arithmetic, CPUs and memory
  units.
• This subject is on the boundary of computer
  science and electrical engineering.

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Inside logic gates
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Definitions

• Combinational logic a digital logic circuit in
  which logical decisions are based only on
  combinations of the current inputs.
• Sequential logic a digital logic circuit in
  which logical decisions are based on combinations
  of the current inputs as well as the past history
  of inputs, e.g. a memory unit.
• Finite state machine a circuit which has an
  internal state, and whose outputs are functions
  of both current inputs and its internal state.
  e.g. a vending machine controller.

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Combinational Logic Unit

• A combinational logic unit (CLU) translates a set
  of inputs into a set of outputs according to one
  or more mapping functions.
• Inputs and outputs for a CLU normally have two
  distinct (binary) values high and low, 1 and 0,
  0 and 1, or 5 V and 0 V for example.
• The outputs of a CLU are strictly functions of
  the inputs, and the outputs are updated
  immediately after the inputs change.

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Combinational Logic Unit Cont.

• A set of inputs i0 - in are presented to the CLU,
  which produces a set of outputs according to
  mapping functions f0 - fm.

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Truth Table

• Developed in 1854 by George Boole and further
  developed by Claude Shannon at Bell laboratory.
• A truth table captures logical relationships in a
  tabular form. Outputs are computed for all
  possible input combinations.
• In a truth table, all possible input combinations
  of binary variables are enumerated and a
  corresponding output value of 0 and 1 is assigned
  for each input combination.

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Truth Table Cont.

• Consider a room with two light switches.
• Truth table of the switches A, B, and the light
  Z.

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Truth Table Cont.

• An alternate assignment of the outputs and the
  truth table.

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Truth Table Cont.

• We can make the assignment of output values to
  input combinations any way that we want to
  achieve the desired input-output behaviour.
• In general, for n inputs, there are 2n input
  combinations, and 22n possible assignments of
  output values to input combinations.

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Logic Gates

• If we enumerate all possible assignments of
  switch settings for two input variables, we will
  obtain 16 assignments, shown in next slide.
• The assignments are referred to as Boolean logic
  functions.
• In a Boolean logic function, 1 means true and 0
  means false.

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Truth Tables Showing All Possible Functions of
Two Binary Variables
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Logic Gates Cont.

• The AND function is true only when A and B are
  true. Denoted as AB.
• The OR function is true when A or B is true or
  when both A and B are true. Denoted as A B.
• The False function is always false.
• The True function is always true.
• A variables complement is true if the variables
  value is false and vice versa. Denoted as

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Logic Gates Cont.

• The NAND and NOR are the complements of AND and
  OR.
• The XOR (exclusive-OR) function is true when
  either of its inputs, but not both, is true. In
  general, XOR produces a 1 at its output whenever
  the number of 1s at its inputs is odd.
• The XNOR is the complement XOR.

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Logic Gates Cont.

• A logic gate is a physical device (see slide 3)
  that implements a simple Boolean function.
• Each logic gate has a symbol. For each functions,
  A and B are binary inputs and F is the output.
• The buffer simply copies its input to its output.
• A NOT gate is also called an inverter. The circle
  at the output of the NOT gate denotes the
  complement operation.

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Logic Gates Cont.
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Logic Gates Cont.
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Logic Gates Cont.
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Common Notations Used at Circuit Intersections
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Tutorial four questions

• There exist 4 boolean functions of a single
  binary variable and 16 functions of 2 binary
  variables. How many functions of 3 binary
  variables?
• Use a truth table to show that
• P (P and Q) OR (P and NOT Q)
• Textbook appendix A A.4, A.6, A.7