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 CS231 Boolean Algebra

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. Nex, we’ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits. CS231 Boolean Algebra

Boolean algebra Last time we talked about Boolean functions, Boolean expressions, and truth tables. Today we’ll learn how to how use Boolean algebra to simplify Booleans expressions. Last time, we saw this expression and converted it to a circuit: (x + y’)z + x’ Can we make this circuit “better”? Cheaper: fewer gates Faster: fewer delays from inputs to outputs June 11, 2002 ©2000-2002 Howard Huang

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 CS231 Boolean Algebra

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 CS231 Boolean Algebra

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 0s with 1s The dual of any equation is always true CS231 Boolean Algebra

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’) CS231 Boolean Algebra

Proving the rest of the axioms We can make up truth tables to prove (both parts of) DeMorgan’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 CS231 Boolean Algebra

Simplification with axioms We can now start doing some simplifications x’y’ + xyz + x’y = x’(y’ + y) + xyz [ Distributive; x’y’ + x’y = x’(y’ + y) ] = x’1 + xyz [ Axiom 7; y’ + y = 1 ] = x’ + xyz [ Axiom 2; x’1 = x’ ] = (x’ + x)(x’ + yz) [ Distributive ] = 1  (x’ + yz) [ Axiom 7; x’ + x = 1 ] = x’ + yz [ Axiom 2 ] CS231 Boolean Algebra

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 CS231 Boolean Algebra

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) [ Distributive ] = 1  (x + y) [ x + x’ = 1 ] = x + y [ Axiom 2 ] CS231 Boolean Algebra

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 0s and 1s in the output column(s) f(x,y,z) = x(y’z’ + yz) CS231 Boolean Algebra

Complementing a function algebraically You can use DeMorgan’s law to keep “pushing” the complements inwards 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’) f(x,y,z) = x(y’z’ + yz) f’(x,y,z) = ( x(y’z’ + yz) )’ [ complement both sides ] = x’ + (y’z’ + yz)’ [ because (xy)’ = x’ + y’ ] = x’ + (y’z’)’ (yz)’ [ because (x + y)’ = x’ y’ ] = x’ + (y + z)(y’ + z’) [ because (xy)’ = x’ + y’, twice] CS231 Boolean Algebra

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 The advantage is that any sum of products expression can be implemented using a two-level circuit literals and their complements at the “0th” 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 LogicWorks! f(x,y,z) = y’ + x’yz’ + xz CS231 Boolean Algebra

Minterms A minterm is a special product of literals, in which each input variable appears exactly once. A function with n variables has 2n minterms (since each variable can appear complemented or not) A three-variable function, such as f(x,y,z), has 23 = 8 minterms: Each minterm is true for exactly one combination of inputs: x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz Minterm Is true when… Shorthand x’y’z’ x=0, y=0, z=0 m0 x’y’z x=0, y=0, z=1 m1 x’yz’ x=0, y=1, z=0 m2 x’yz x=0, y=1, z=1 m3 xy’z’ x=1, y=0, z=0 m4 xy’z x=1, y=0, z=1 m5 xyz’ x=1, y=1, z=0 m6 xyz x=1, y=1, z=1 m7 CS231 Boolean Algebra

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’ = m0 + m1 + m2 + m3 + m6 = m(0,1,2,3,6) f’ = xy’z’ + xy’z + xyz = m4 + m5 + m7 = m(4,5,7) f’ contains all the minterms not in f CS231 Boolean Algebra

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 Product of sums expressions can be implemented with two-level circuits literals and their complements at the “0th” level OR gates at the first level a single AND gate at the second level Compare this with sums of products f(x,y,z) = y’ (x’ + y + z’) (x + z) CS231 Boolean Algebra

Maxterms A maxterm is a sum of literals, in which each input variable appears exactly once. A function with n variables has 2n maxterms The maxterms for a three-variable function f(x,y,z): Each maxterm is false for exactly one combination of inputs: x’ + y’ + z’ x’ + y’ + z x’ + y + z’ x’+ y + z x + y’ + z’ x + y’ + z x + y + z’ x + y + z Maxterm Is false when… Shorthand x + y + z x=0, y=0, z=0 M0 x + y + z’ x=0, y=0, z=1 M1 x + y’ + z x=0, y=1, z=0 M2 x + y’ + z’ x=0, y=1, z=1 M3 x’ + y + z x=1, y=0, z=0 M4 x’ + y + z’ x=1, y=0, z=1 M5 x’ + y’ + z x=1, y=1, z=0 M6 x’ + y’ + z’ x=1, y=1, z=1 M7 CS231 Boolean Algebra

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’) = M4 M5 M7 = M(4,5,7) f’ = (x + y + z)(x + y + z’)(x + y’ + z) (x + y’ + z’)(x’ + y’ + z) = M0 M1 M2 M3 M6 = M(0,1,2,3,6) f’ contains all the maxterms not in f CS231 Boolean Algebra

Minterms and maxterms are related Any minterm mi is the complement of the corresponding maxterm Mi For example, m4’ = M4 because (xy’z’)’ = x’ + y + z Minterm Shorthand x’y’z’ m0 x’y’z m1 x’yz’ m2 x’yz m3 xy’z’ m4 xy’z m5 xyz’ m6 xyz m7 Maxterm Shorthand x + y + z M0 x + y + z’ M1 x + y’ + z M2 x + y’ + z’ M3 x’ + y + z M4 x’ + y + z’ M5 x’ + y’ + z M6 x’ + y’ + z’ M7 CS231 Boolean Algebra

Converting between standard forms We can convert a sum of minterms to a product of maxterms In general, just replace the minterms with maxterms, using maxterm numbers that don’t appear in the sum of minterms: The same thing works for converting from a product of maxterms to a sum of minterms From before f = m(0,1,2,3,6) and f’ = m(4,5,7) = m4 + m5 + m7 complementing (f’)’ = (m4 + m5 + m7)’ so f = m4’ m5’ m7’ [ DeMorgan’s law ] = M4 M5 M7 [ By the previous page ] = M(4,5,7) f = m(0,1,2,3,6) = M(4,5,7) CS231 Boolean Algebra

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 and analyze bigger, more useful, circuits CS231 Boolean Algebra

Karnaugh maps So far, 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. Next, we’ll look at a graphical technique for simplifying an expression into a minimal sum of products (MSP) form: There are 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. CS231 Boolean Algebra

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 The advantage is that any sum of products expression can be implemented using a two-level circuit literals and their complements at the “0th” 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 LogicWorks! f(x,y,z) = y’ + x’yz’ + xz CS231 Boolean Algebra

Terminology: Minterms A minterm is a special product of literals, in which each input variable appears exactly once. A function with n variables has 2n minterms (since each variable can appear complemented or not) A three-variable function, such as f(x,y,z), has 23 = 8 minterms: Each minterm is true for exactly one combination of inputs: x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz Minterm Is true when… Shorthand x’y’z’ x=0, y=0, z=0 m0 x’y’z x=0, y=0, z=1 m1 x’yz’ x=0, y=1, z=0 m2 x’yz x=0, y=1, z=1 m3 xy’z’ x=1, y=0, z=0 m4 xy’z x=1, y=0, z=1 m5 xyz’ x=1, y=1, z=0 m6 xyz x=1, y=1, z=1 m7 CS231 Boolean Algebra

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’ = m0 + m1 + m2 + m3 + m6 = m(0,1,2,3,6) f’ = xy’z’ + xy’z + xyz = m4 + m5 + m7 = m(4,5,7) f’ contains all the minterms not in f CS231 Boolean Algebra

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. CS231 Boolean Algebra

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) [ Distributive ] = x’  1 [ y + y’ = 1 ] = x’ [ x  1 = x ] CS231 Boolean Algebra

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. CS231 Boolean Algebra

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): CS231 Boolean Algebra

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. “Adjacency” includes wrapping around the left and right sides: We’ll use this property of adjacent squares to do our simplifications. x’y’z + x’yz = x’z(y’ + y) = x’z  1 = x’z x’y’z’ + xy’z’ + x’yz’ + xyz’ = z’(x’y’ + xy’ + x’y + xy) = z’(y’(x’ + x) + y(x’ + x)) = z’(y’+y) = z’ CS231 Boolean Algebra

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 + xy’z + xyz’ + xyz = m1 + m5 + m6 + m7 CS231 Boolean Algebra

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. 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. 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 CS231 Boolean Algebra

Making the example K-map Next up is drawing and filling in the K-map. Put 1s in the map for each minterm, and 0s in the other squares. You can use either the minterm products or the shorthand to show you where the 1s and 0s belong. In our example, we can write f(x,y,z) in two equivalent ways. In either case, the resulting K-map is shown below. f(x,y,z) = x’y’z + xy’z + xyz’ + xyz f(x,y,z) = m1 + m5 + m6 + m7 CS231 Boolean Algebra

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.” CS231 Boolean Algebra

Grouping the minterms together The most difficult step is grouping together all the 1s in the K-map. Make rectangles around groups of one, two, four or eight 1s. All of the 1s in the map should be included in at least one rectangle. Do not include any of the 0s. 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. CS231 Boolean Algebra

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.) CS231 Boolean Algebra

Practice K-map 1 Simplify the sum of minterms m1 + m3 + m5 + m6. CS231 Boolean Algebra

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 m6 is in a group all by its lonesome. The final MSP here is x’z + y’z + xyz’. CS231 Boolean Algebra

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. CS231 Boolean Algebra

Example: Simplify m0+m2+m5+m8+m10+m13 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. CS231 Boolean Algebra

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 m7. Remember that overlapping groups is possible, as shown above. y’z + yz’ + xy y’z + yz’ + xz CS231 Boolean Algebra