Computer logic Data and programs in digital computers are represented and processed by electronic circuit networks called digital logic circuits or logic circuits, for short. Logic circuits are the heart of computer hardware. Logic circuits operate on two logical values, usually called bit 0 and bit 1, and the operations are based on the principles of Boolean Algebra.

Computer logic Logical circuits are made from two basic devices Logic gate Flip-flop Flip-flops provide memory for storing data while logic gates provide operations on, or functions of, the values stored in these memory devices. Logic gate is a combinational circuit that performs an elementary logic operation.

Logic function and truth table The values of input x1 and x2 and the corresponding value y of each basic logic gate is represented in it truth table. The truth table lists all possible switch setting (input values) along with value of the result (output value) for each setting.

Logic Function and Truth Table In general logic terms, this truth table represents the function values y of two variables x1 and x2 values as y = f(x1, x2) and may be extended to n variable as the logic function y = f(x1,x2,...,xn) Basic logic gates can be used to construct logic network (network of logic gates) or logic circuit that implements more complex logic function (Boolean function).

Logic Function and Truth Table At any moment, the output signal of a gate is a function of the input signals at that moment. The AND gate can have two or more inputs and one output. The output is 1 if and only if all the inputs are 1s. Otherwise it is 0. The OR gate can have two or more inputs and one output. The output is 1 if any of the inputs is 1. Not gate or “inverter” has one input and one output which is always the opposite of the input.

Logic Function and Truth Table Truth table is important way for describing logic function values. Any logic function with n inputs and one output, the corresponding truth table will have one column for each input, and one column for the output. The truth table will have one row for every possible combination of inputs; 2n rows in all. The output column in each row simply specifies the output for that combination of inputs.

Logic Function and Truth Table The logic function : Y = A·B + A·B The corresponding truth table : 1 Y= A· B + A· B A . B B A

Logic Function and Truth Table The corresponding logic circuit Y = A·B + A·B A B

Boolean Function and Boolean Expression There are two important problems of Boolean functions and Boolean expressions : Given the values of a Boolean function, how can a Boolean expression that represent this function be found ? Is there a smaller set of operators that can be used to represent all Boolean functions ?

Boolean Function and Boolean Expression Both of these problems have practical importance in circuit design. Methods of representing of Boolean functions Sum-of-products expansions (minterm expansions ) Product-of-sums expansions (maxterm expansions )

Boolean Function and Boolean Expression Find Boolean expressions that represent the Boolean functions F(x,y,z) and G(x,y,z) below : F(x,y,z) = x  y  z = xyz + xy z G(x,y,z)

exclusive-OR ( x  y ) Sum-of-products expansions f(x, y) = 1 when x = 1 and y = 0 x = 0 and y = 1 x y = x•y + x•y

exclusive-OR ( x  y ) Product-of-sums expansions f(x, y) = 0 when x = 1 and y = 1 x = 0 and y = 0 x y = (x+y) • (x+y)

Boolean Function The Boolean functions F and G of n variables are equal if and only if F( b1 , b2 , … , bn) = G( b1 , b2 , … , bn ) whenever b1, b2 , … , bn belong to B = { 0 , 1 } Two different Boolean expressions that represent the same Boolean function are called equivalent Boolean expressions.

Boolean Function Example To show F( x , y, z ) = G( x , y, z ) where F( x , y, z ) = x + y•z G( x , y,z ) = ( x + y ) •( x + z )

F( x , y, z ) = x + y•z G( x , y,z ) = ( x + y ) •( x + z )

Aj Bj C Circuit C2 S1 S2 C1 A1 B1 A2 B2 Cm-1 Sm Am Bm Cm Cj-1 Sj = Aj + Bj Cj Aj Bj Cj-1 C Circuit Output C2 S1 S2 C1 A1 B1 A2 B2 Cm-1 Sm Am Bm C Circuit Cm Input

Elements of computer hardware Transistors Group of transistors ICs Circuit boards Computer

Elements of computer hardware Transistors are the smallest computational elements. Boolean functions and memories are formed by groups of transistors. Elementary addition and multiplication is done by boolean functions and memory elements called flip-flops. Integrated circuits (ICs) contain a large number of such functions.

Elements of computer hardware Printed circuits boards contain severeral integrated circuits to form a either a full computer or specific I/O (input/output) functions. Massive storage such as disks and tapes interfaces the computer through its main bus.

Functional organization CPU Main memory I/O interface Bus translator Bus

Integrated circuits The logic gates AND, OR, NOT are the basic components of digital logic circuits of computers. However, these three gates are in principle sufficient to construct a logic circuit of any kind. The number of gates that are combined in a single integrated circuit (IC) is often used to distinguish different levels of IC manufacturing.

Integrated circuits Four levels of IC chips are recognized : Small-scale integration ( SSI ) 1 to 9 gates per chip Medium-scale integration ( MSI ) 10 to 99 gates per chip Large-scale integration ( LSI ) 100 to 100,000 gates per chip Very large-scale integration ( VLSI ) over 100,000 gates per chip

Integrated circuits An integrated circuit (IC) is a microelectronic device consisting of many interconnected transistors and other components. ICs are constructed (‘fabricated’) on a small rectangle, called a ‘die’, cut from a silicon (or for special applications, sapphire) wafer.

Integrated circuits ICs can be classified into analog, digital, or hybrid (both digital and analog on the same chip). Digital ICs can contain anything from one to millions of logic gates, flip-flops, multiplexers, etc. in a few square millimeters. The small size of ICs allows high speed, low power dissipation, and reduced manufacturing cost compare with board-level integration.