Circuit analysis summary After finding the circuit inputs

Circuit analysis summary • • After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert between expressions and truth tables. Find the circuit’s inputs and outputs Find a Boolean expression for the circuit Find a truth table for the circuit CS 231 Boolean Algebra 1

Boolean operations summary • • • We can interpret high or low voltage as representing true or false. A variable whose value can be either 1 or 0 is called a Boolean variable. AND, OR, and NOT are the basic Boolean operations. We can express Boolean functions with either an expression or a truth table. Every Boolean expression can be converted to a circuit. Next time, we’ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits. CS 231 Boolean Algebra 2

Expression simplification • • • Normal mathematical expressions can be simplified using the laws of algebra For binary systems, we can use Boolean algebra, which is superficially similar to regular algebra There are many differences, due to – having only two values (0 and 1) to work with – having a complement operation – the OR operation is not the same as addition CS 231 Boolean Algebra 3

Formal definition of Boolean algebra • A Boolean algebra requires – A set of elements B, which needs at least two elements (0 and 1) – Two binary (two-argument) operations OR and AND – A unary (one-argument) operation NOT – The axioms below must always be true (textbook, p. 33) • The magenta axioms deal with the complement operation • Blue axioms (especially 15) are different from regular algebra CS 231 Boolean Algebra 4

Comments on the axioms • The associative laws show that there is no ambiguity about a term such as x + y + z or xyz, so we can introduce multiple-input primitive gates: • The left and right columns of axioms are duals – exchange all ANDs with ORs, and 0 s with 1 s The dual of any equation is always true • CS 231 Boolean Algebra 5

Are these axioms for real? • We can show that these axioms are valid, given the definitions of AND, OR and NOT • The first 11 axioms are easy to see from these truth tables alone. For example, x + x’ = 1 because of the middle two lines below (where y = x’) CS 231 Boolean Algebra 6

Proving the rest of the axioms • • We can make up truth tables to prove (both parts of) De. Morgan’s law For (x + y)’ = x’y’, we can make truth tables for (x + y)’ and for x’y’ • In each table, the columns on the left (x and y) are the inputs. The columns on the right are outputs. In this case, we only care about the columns in blue. The other “outputs” are just to help us find the blue columns. Since both of the columns in blue are the same, this shows that (x + y)’ and x’y’ are equivalent • • CS 231 Boolean Algebra 7

Simplification with axioms • We can now start doing some simplifications x’y’ + xyz + x’y = x’(y’ + y) + xyz = x’ 1 + xyz = x’ + xyz = (x’ + x)(x’ + yz) = 1 (x’ + yz) = x’ + yz [ Distributive; x’y’ + x’y = x’(y’ + y) ] [ Axiom 7; y’ + y = 1 ] [ Axiom 2; x’ 1 = x’ ] [ Distributive ] [ Axiom 7; x’ + x = 1 ] [ Axiom 2 ] CS 231 Boolean Algebra 8

Let’s compare the resulting circuits • • Here are two different but equivalent circuits. In general the one with fewer gates is “better”: – It costs less to build – It requires less power – But we had to do some work to find the second form CS 231 Boolean Algebra 9

Some more laws • Here are some more useful laws (p. 37). Notice the duals again! • We can prove these laws by either – Making truth tables: – Using the axioms: x + x’y = (x + x’)(x + y) = 1 (x + y) =x+y CS 231 Boolean Algebra [ Distributive ] [ x + x’ = 1 ] [ Axiom 2 ] 10

The complement of a function • • The complement of a function always outputs 0 where the original function outputted 1, and 1 where the original produced 0. In a truth table, we can just exchange 0 s and 1 s in the output column(s) f(x, y, z) = x(y’z’ + yz) CS 231 Boolean Algebra 11

Complementing a function algebraically • You can use De. Morgan’s law to keep “pushing” the complements inwards f(x, y, z) = x(y’z’ + yz) f’(x, y, z) = ( x(y’z’ + yz) )’ = x’ + (y’z’ + yz)’ = x’ + (y’z’)’ (yz)’ = x’ + (y + z)(y’ + z’) • [ complement both sides ] [ because (xy)’ = x’ + y’ ] [ because (x + y)’ = x’ y’ ] [ because (xy)’ = x’ + y’, twice] You can also take the dual of the function, and then complement each literal – If f(x, y, z) = x(y’z’ + yz)… – …the dual of f is x + (y’ + z’)(y + z)… – …then complementing each literal gives x’ + (y + z)(y’ + z’)… – …so f’(x, y, z) = x’ + (y + z)(y’ + z’) CS 231 Boolean Algebra 12

Standard forms of expressions • • We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains: – Only OR (sum) operations at the “outermost” level – Each term that is summed must be a product of literals f(x, y, z) = y’ + x’yz’ + xz • • The advantage is that any sum of products expression can be implemented using a two-level circuit – literals and their complements at the “ 0 th” level – AND gates at the first level – a single OR gate at the second level This diagram uses some shorthands… – NOT gates are implicit – literals are reused – this is not okay in Logic. Works! CS 231 Boolean Algebra 13

Minterms • • A minterm is a special product of literals, in which each input variable appears exactly once. A function with n variables has 2 n minterms (since each variable can appear complemented or not) A three-variable function, such as f(x, y, z), has 23 = 8 minterms: x’y’z’ x’y’z x’yz’ x’yz xyz Each minterm is true for exactly one combination of inputs: Minterm x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz Is true when… x=0, y=0, z=0 x=0, y=0, z=1 x=0, y=1, z=0 x=0, y=1, z=1 x=1, y=0, z=0 x=1, y=0, z=1 x=1, y=1, z=0 x=1, y=1, z=1 Shorthand m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 CS 231 Boolean Algebra 14

Sum of minterms form • • • Every function can be written as a sum of minterms, which is a special kind of sum of products form The sum of minterms form for any function is unique If you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. f = x’y’z’ + x’y’z + x’yz’ + x’yz + xyz’ = m 0 + m 1 + m 2 + m 3 + m 6 = m(0, 1, 2, 3, 6) f’ = xy’z’ + xy’z + xyz = m 4 + m 5 + m 7 = m(4, 5, 7) f’ contains all the minterms not in f CS 231 Boolean Algebra 15

The dual idea: products of sums • • Just to keep you on your toes. . . A product of sums (POS) expression contains: – Only AND (product) operations at the “outermost” level – Each term must be a sum of literals f(x, y, z) = y’ (x’ + y + z’) (x + z) • • Product of sums expressions can be implemented with two-level circuits – literals and their complements at the “ 0 th” level – OR gates at the first level – a single AND gate at the second level Compare this with sums of products CS 231 Boolean Algebra 16

Maxterms • • • A maxterm is a sum of literals, in which each input variable appears exactly once. A function with n variables has 2 n maxterms The maxterms for a three-variable function f(x, y, z): x’ + y’ + z’ x + y’ + z’ • x’ + y’ + z x’ + y + z’ x’+ y + z x+y+z Each maxterm is false for exactly one combination of inputs: Maxterm x+y+z x + y + z’ x + y’ + z’ x’ + y’ + z’ Is false when… Shorthand x=0, y=0, z=0 M 0 x=0, y=0, z=1 M 1 x=0, y=1, z=0 M 2 x=0, y=1, z=1 M 3 x=1, y=0, z=0 M 4 x=1, y=0, z=1 M 5 x=1, y=1, z=0 M 6 x=1, y=1, z=1 M 7 CS 231 Boolean Algebra 17

Product of maxterms form • • Every function can be written as a unique product of maxterms If you have a truth table for a function, you can write a product of maxterms expression by picking out the rows of the table where the function output is 0. (Be careful if you’re writing the actual literals!) f = (x’ + y + z)(x’ + y + z’)(x’ + y’ + z’) = M 4 M 5 M 7 = M(4, 5, 7) f’ = (x + y + z)(x + y + z’)(x + y’ + z) (x + y’ + z’)(x’ + y’ + z) = M 0 M 1 M 2 M 3 M 6 = M(0, 1, 2, 3, 6) f’ contains all the maxterms not in f CS 231 Boolean Algebra 18

Minterms and maxterms are related • Any minterm mi is the complement of the corresponding maxterm Mi Minterm x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz • Shorthand m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 Maxterm x+y+z x + y + z’ x + y’ + z’ x’ + y’ + z’ Shorthand M 0 M 1 M 2 M 3 M 4 M 5 M 6 M 7 For example, m 4’ = M 4 because (xy’z’)’ = x’ + y + z CS 231 Boolean Algebra 19

Converting between standard forms • We can convert a sum of minterms to a product of maxterms From before and complementing so • • f f’ = m(0, 1, 2, 3, 6) = m(4, 5, 7) = m 4 + m 5 + m 7 (f’)’ = (m 4 + m 5 + m 7)’ f = m 4’ m 5’ m 7’ [ De. Morgan’s law ] = M 4 M 5 M 7 [ By the previous page ] = M(4, 5, 7) In general, just replace the minterms with maxterms, using maxterm numbers that don’t appear in the sum of minterms: f = m(0, 1, 2, 3, 6) = M(4, 5, 7) The same thing works for converting from a product of maxterms to a sum of minterms CS 231 Boolean Algebra 20

Summary so far • • So far: – A bunch of Boolean algebra trickery for simplifying expressions and circuits – The algebra guarantees us that the simplified circuit is equivalent to the original one – Introducing some standard forms and terminology Next: – An alternative simplification method – We’ll start using all this stuff to build analyze bigger, more useful, circuits CS 231 Boolean Algebra 21

Karnaugh maps • • • Last time we saw applications of Boolean logic to circuit design. – The basic Boolean operations are AND, OR and NOT. – These operations can be combined to form complex expressions, which can also be directly translated into a hardware circuit. – Boolean algebra helps us simplify expressions and circuits. Today we’ll look at a graphical technique for simplifying an expression into a minimal sum of products (MSP) form: – There a minimal number of product terms in the expression. – Each term has a minimal number of literals. Circuit-wise, this leads to a minimal two-level implementation. CS 231 Boolean Algebra 22

Review: Standard forms of expressions • • We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains: – Only OR (sum) operations at the “outermost” level – Each term that is summed must be a product of literals f(x, y, z) = y’ + x’yz’ + xz • • The advantage is that any sum of products expression can be implemented using a two-level circuit – literals and their complements at the “ 0 th” level – AND gates at the first level – a single OR gate at the second level This diagram uses some shorthands… – NOT gates are implicit – literals are reused – this is not okay in Logic. Works! CS 231 Boolean Algebra 23

Terminology: Minterms • • A minterm is a special product of literals, in which each input variable appears exactly once. A function with n variables has 2 n minterms (since each variable can appear complemented or not) A three-variable function, such as f(x, y, z), has 23 = 8 minterms: x’y’z’ x’y’z x’yz’ x’yz xyz Each minterm is true for exactly one combination of inputs: Minterm x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz Is true when… x=0, y=0, z=0 x=0, y=0, z=1 x=0, y=1, z=0 x=0, y=1, z=1 x=1, y=0, z=0 x=1, y=0, z=1 x=1, y=1, z=0 x=1, y=1, z=1 Shorthand m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 CS 231 Boolean Algebra 24

Terminology: Sum of minterms form • • • Every function can be written as a sum of minterms, which is a special kind of sum of products form The sum of minterms form for any function is unique If you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. f = x’y’z’ + x’y’z + x’yz’ + x’yz + xyz’ = m 0 + m 1 + m 2 + m 3 + m 6 = m(0, 1, 2, 3, 6) f’ = xy’z’ + xy’z + xyz = m 4 + m 5 + m 7 = m(4, 5, 7) f’ contains all the minterms not in f CS 231 Boolean Algebra 25

Re-arranging the truth table • A two-variable function has four possible minterms. We can re-arrange these minterms into a Karnaugh map. • Now we can easily see which minterms contain common literals. – Minterms on the left and right sides contain y’ and y respectively. – Minterms in the top and bottom rows contain x’ and x respectively. CS 231 Boolean Algebra 26

Karnaugh map simplifications • Imagine a two-variable sum of minterms: x’y’ + x’y • Both of these minterms appear in the top row of a Karnaugh map, which means that they both contain the literal x’. • What happens if you simplify this expression using Boolean algebra? x’y’ + x’y = x’(y’ + y) = x’ 1 = x’ [ Distributive ] [ y + y’ = 1 ] [x 1=x] CS 231 Boolean Algebra 27

More two-variable examples • Another example expression is x’y + xy. – Both minterms appear in the right side, where y is uncomplemented. – Thus, we can reduce x’y + xy to just y. • How about x’y’ + x’y + xy? – We have x’y’ + x’y in the top row, corresponding to x’. – There’s also x’y + xy in the right side, corresponding to y. – This whole expression can be reduced to x’ + y. CS 231 Boolean Algebra 28

A three-variable Karnaugh map • For a three-variable expression with inputs x, y, z, the arrangement of minterms is more tricky: • Another way to label the K-map (use whichever you like): CS 231 Boolean Algebra 29

Why the funny ordering? • With this ordering, any group of 2, 4 or 8 adjacent squares on the map contains common literals that can be factored out. x’y’z + x’yz = x’z(y’ + y) = x’z 1 = x’z • • “Adjacency” includes wrapping around the left and right sides: = = x’y’z’ + x’yz’ + xyz’ z’(x’y’ + x’y + xy) z’(y’(x’ + x) + y(x’ + x)) z’(y’+y) z’ We’ll use this property of adjacent squares to do our simplifications. CS 231 Boolean Algebra 30

Example K-map simplification • • • Let’s consider simplifying f(x, y, z) = xy + y’z + xz. First, you should convert the expression into a sum of minterms form, if it’s not already. – The easiest way to do this is to make a truth table for the function, and then read off the minterms. – You can either write out the literals or use the minterm shorthand. Here is the truth table and sum of minterms for our example: f(x, y, z) = x’y’z + xyz’ + xyz = m 1 + m 5 + m 6 + m 7 CS 231 Boolean Algebra 31

Unsimplifying expressions • You can also convert the expression to a sum of minterms with Boolean algebra. – Apply the distributive law in reverse to add in missing variables. – Very few people actually do this, but it’s occasionally useful. xy + y’z + xz = (xy 1) + (y’z 1) + (xz 1) = (xy (z’ + z)) + (y’z (x’ + x)) + (xz (y’ + y)) = (xyz’ + xyz) + (x’y’z + xy’z) + (xy’z + xyz) = xyz’ + xyz + x’y’z + xy’z • In both cases, we’re actually “unsimplifying” our example expression. – The resulting expression is larger than the original one! – But having all the individual minterms makes it easy to combine them together with the K-map. CS 231 Boolean Algebra 32

Making the example K-map • • Next up is drawing and filling in the K-map. – Put 1 s in the map for each minterm, and 0 s in the other squares. – You can use either the minterm products or the shorthand to show you where the 1 s and 0 s belong. In our example, we can write f(x, y, z) in two equivalent ways. f(x, y, z) = x’y’z + xyz’ + xyz • f(x, y, z) = m 1 + m 5 + m 6 + m 7 In either case, the resulting K-map is shown below. CS 231 Boolean Algebra 33

K-maps from truth tables • You can also fill in the K-map directly from a truth table. – The output in row i of the table goes into square mi of the K-map. – Remember that the rightmost columns of the K-map are “switched. ” CS 231 Boolean Algebra 34

Grouping the minterms together • The most difficult step is grouping together all the 1 s in the K-map. – Make rectangles around groups of one, two, four or eight 1 s. – All of the 1 s in the map should be included in at least one rectangle. – Do not include any of the 0 s. • Each group corresponds to one product term. For the simplest result: – Make as few rectangles as possible, to minimize the number of products in the final expression. – Make each rectangle as large as possible, to minimize the number of literals in each term. – It’s all right for rectangles to overlap, if that makes them larger. CS 231 Boolean Algebra 35

Reading the MSP from the K-map • Finally, you can find the MSP. – Each rectangle corresponds to one product term. – The product is determined by finding the common literals in that rectangle. • For our example, we find that xy + y’z + xz = y’z + xy. (This is one of the additional algebraic laws from last time. ) CS 231 Boolean Algebra 36

Practice K-map 1 • Simplify the sum of minterms m 1 + m 3 + m 5 + m 6. CS 231 Boolean Algebra 37

Solutions for practice K-map 1 • Here is the filled in K-map, with all groups shown. – The magenta and green groups overlap, which makes each of them as large as possible. – Minterm m 6 is in a group all by its lonesome. • The final MSP here is x’z + y’z + xyz’. CS 231 Boolean Algebra 38

Four-variable K-maps • We can do four-variable expressions too! – The minterms in the third and fourth columns, and in the third and fourth rows, are switched around. – Again, this ensures that adjacent squares have common literals. • Grouping minterms is similar to the three-variable case, but: – You can have rectangular groups of 1, 2, 4, 8 or 16 minterms. – You can wrap around all four sides. CS 231 Boolean Algebra 39

Example: Simplify m 0+m 2+m 5+m 8+m 10+m 13 • The expression is already a sum of minterms, so here’s the K-map: • We can make the following groups, resulting in the MSP x’z’ + xy’z. CS 231 Boolean Algebra 40

K-maps can be tricky! • There may not necessarily be a unique MSP. The K-map below yields two valid and equivalent MSPs, because there are two possible ways to include minterm m 7. y’z + yz’ + xy • y’z + yz’ + xz Remember that overlapping groups is possible, as shown above. CS 231 Boolean Algebra 41