Combinational Logic (mostly review!)

ombinational Logic (mostly review!)! Logic functions, truth tables, and switches " NOT, N, OR, NN, NOR, OR,... " Minimal set! xioms and theorems of oolean algebra " Proofs by re-writing " Proofs by perfect induction! Gate logic " Networks of oolean functions " Time behavior! anonical forms " Two-level " Incompletely specified functions S 5 - Spring 27 Lec #2: ombinational Logic - Possible Logic Functions of Two Variables! 6 possible functions of 2 input variables: " 2**(2**n) functions of n inputs Y F Y 6 possible functions (F F5) and Y Y xor Y or Y = Y nor Y not ( or Y) not Y not nand Y not ( and Y) S 5 - Spring 27 Lec #2: ombinational Logic - 2

ost of ifferent Logic Functions! Some are easier, others harder, to implement " Each has a cost associated with the number of switches needed " (F) and (F5): require switches, directly connect output to low/high " (F3) and Y (F5): require switches, output is one of inputs " ' (F2) and Y' (F): require 2 switches for "inverter" or NOT-gate " nor Y (F4) and nand Y (F4): require 4 switches " or Y (F7) and and Y (F): require 6 switches " = Y (F9) and " Y (F6): require 6 switches " ecause NOT, NOR, and NN are the cheapest they are the functions we implement the most in practice S 5 - Spring 27 Lec #2: ombinational Logic - 3 Minimal Set of Functions! Implement any logic functions from NOT, NOR, and NN? " For example, implementing and Y is the same as implementing not ( nand Y)! o it with only NOR or only NN " NOT is just a NN or a NOR with both inputs tied together Y nor Y Y nand Y " and NN and NOR are "duals", i.e., easy to implement one using the other nand Y! not ( (not ) nor (not Y) ) nor Y! not ( (not ) nand (not Y) )! ased on the mathematical foundations of logic: oolean lgebra S 5 - Spring 27 Lec #2: ombinational Logic - 4

lgebraic Structure! onsists of " Set of elements " inary operations { +, } " Unary operation { ' } " Following axioms hold:. set contains at least two elements, a, b, such that a # b 2. closure: a + b is in a b is in 3. commutativity: a + b = b + a a b = b a 4. associativity: a + (b + c) = (a + b) + c a (b c) = (a b) c 5. Identity: a + = a a = a 6. distributivity: a + (b c) = (a + b) (a + c) a (b + c) = (a b) + (a c) 7. complementarity: a + a' = a a' = S 5 - Spring 27 Lec #2: ombinational Logic - 5 oolean lgebra! oolean algebra " = {, } " + is logical OR, is logical N " ' is logical NOT! ll algebraic axioms hold S 5 - Spring 27 Lec #2: ombinational Logic - 6

Logic Functions and oolean lgebra! ny logic function that can be expressed as a truth table can be written as an expression in oolean algebra using the operators: ', +, and Y!Y Y ' ' Y Y ' Y'!Y '!Y' (!Y ) + ( '!Y' ) (!Y ) + ( '!Y' )! =!Y, Y are oolean algebra variables S 5 - Spring 27 Lec #2: ombinational Logic - 7 oolean expression that is true when the variables and Y have the same value and false, otherwise xioms and Theorems of oolean lgebra! Identity. + =. =! Null 2. + = 2. =! Idempotency: 3. + = 3. =! Involution: 4. (')' =! omplementarity: 5. + ' = 5. ' =! ommutativity: 6. + Y = Y + 6. Y = Y! ssociativity: 7. ( + Y) + = + (Y + ) 7. ( Y) = (Y ) S 5 - Spring 27 Lec #2: ombinational Logic - 8

xioms and Theorems of oolean lgebra (cont d)! istributivity: 8. (Y + ) = ( Y) + ( ) 8. + (Y ) = ( + Y) ( + )! Uniting: 9. Y + Y' = 9. ( + Y) ( + Y') =! bsorption:. + Y =. ( + Y) =. ( + Y') Y = Y. ( Y') + Y = + Y! Factoring: 2. ( + Y) (' + ) = 2. Y + ' = + ' Y ( + ) (' + Y)! onsensus: 3. ( Y) + (Y ) + (' ) = 3. ( + Y) (Y + ) (' + ) = Y + ' ( + Y) (' + ) S 5 - Spring 27 Lec #2: ombinational Logic - 9 xioms and Theorems of oolean lgebra (cont d)! demorgan's: 4. ( + Y +...)' = ' Y'... 4. ( Y...)' = ' + Y' +...! Generalized de Morgan's: 5. f'(,2,...,n,,,+, ) = f(',2',...,n',,,,+)! Establishes relationship between and + S 5 - Spring 27 Lec #2: ombinational Logic -

xioms and Theorems of oolean lgebra (cont d)! uality " ual of a oolean expression is derived by replacing by +, + by, by, and by, and leaving variables unchanged " ny theorem that can be proven is thus also proven for its dual! " Meta-theorem (a theorem about theorems)! uality: 6. + Y +... $ Y...! Generalized duality: 7. f (,2,...,n,,,+, ) $ f(,2,...,n,,,,+)! ifferent than demorgan s Law " This is a statement about theorems " This is not a way to manipulate (re-write) expressions S 5 - Spring 27 Lec #2: ombinational Logic - Proving Theorems (Rewriting Method)! Using the axioms of oolean algebra: " e.g., prove the theorem: Y + Y' = distributivity (8)!Y +!Y' = (Y + Y') complementarity (5) (Y + Y') =!() identity ()!() =! " e.g., prove the theorem: + Y = identity () + Y = + Y distributivity (8) + Y = ( + Y) identity (2) ( + Y) =!() identity () () =! S 5 - Spring 27 Lec #2: ombinational Logic - 2

Proving Theorems (Perfect Induction)! Using perfect induction (complete truth table): " e.g., de Morgan's: ( + Y)' = ' Y' NOR is equivalent to N with inputs complemented ( Y)' = ' + Y' NN is equivalent to OR with inputs complemented Y ' Y' ( + Y)' ' Y' Y ' Y' ( Y)' ' + Y' S 5 - Spring 27 Lec #2: ombinational Logic - 3 Simple Example! -bit binary adder " Inputs:,, arry-in " Outputs: Sum, arry-out in S out in S out S = ' ' in + ' in' + ' in' + in out = ' in + ' in + in' + in S 5 - Spring 27 Lec #2: ombinational Logic - 4

pply the Theorems to Simplify Expressions! Theorems of oolean algebra can simplify oolean expressions " e.g., full adder's carry-out function (same rules apply to any function) out = ' in + ' in + in' + in = ' in + ' in + in' + in + in = ' in + in + ' in + in' + in = (' + ) in + ' in + in' + in = () in + ' in + in' + in = in + ' in + in' + in + in = in + ' in + in + in' + in = in + (' + ) in + in' + in = in + () in + in' + in = in + in + (in' + in) = in + in + () = in + in + S 5 - Spring 27 Lec #2: ombinational Logic - 5 From oolean Expressions to Logic Gates! NOT ' ~ Y Y! N Y Y % Y! OR + Y & Y Y Y Y Y S 5 - Spring 27 Lec #2: ombinational Logic - 6

From oolean Expressions to Logic Gates (cont d)! NN Y Y! NOR! OR " Y! NOR = Y Y Y Y Y Y Y xor Y = Y' + ' Y or Y but not both ("inequality", "difference") xnor Y = Y + ' Y' and Y are the same ("equality", "coincidence") S 5 - Spring 27 Lec #2: ombinational Logic - 7 From oolean Expressions to Logic Gates (cont d)! More than one way to map expressions to gates T2 " e.g., = ' ' ( + ) = (' (' ( + ))) T use of 3-input gate T T2 S 5 - Spring 27 Lec #2: ombinational Logic - 8

Waveform View of Logic Functions! Just a sideways truth table " ut note how edges don't line up exactly " It takes time for a gate to switch its output! time change in Y takes time to "propagate" through gates S 5 - Spring 27 Lec #2: ombinational Logic - 9 hoosing ifferent Realizations of a Function two-level realization (we don't count NOT gates) multi-level realization (gates with fewer inputs) OR gate (easier to draw but costlier to build) S 5 - Spring 27 Lec #2: ombinational Logic - 2

Which Realization is est?! Reduce number of inputs " literal: input variable (complemented or not) # can approximate cost of logic gate as 2 transistors per literal # why not count inverters? " Fewer literals means less transistors # smaller circuits " Fewer inputs implies faster gates # gates are smaller and thus also faster " Fan-ins (# of gate inputs) are limited in some technologies! Reduce number of gates " Fewer gates (and the packages they come in) means smaller circuits # directly influences manufacturing costs S 5 - Spring 27 Lec #2: ombinational Logic - 2 Which is the est Realization? (cont d)! Reduce number of levels of gates " Fewer level of gates implies reduced signal propagation delays " Minimum delay configuration typically requires more gates # wider, less deep circuits! How do we explore tradeoffs between increased circuit delay and size? " utomated tools to generate different solutions " Logic minimization: reduce number of gates and complexity " Logic optimization: reduction while trading off against delay S 5 - Spring 27 Lec #2: ombinational Logic - 22

re ll Realizations Equivalent?! Under the same inputs, the alternative implementations have almost the same waveform behavior " elays are different " Glitches (hazards) may arise " Variations due to differences in number of gate levels and structure! Three implementations are functionally equivalent S 5 - Spring 27 Lec #2: ombinational Logic - 23 Implementing oolean Functions! Technology independent " anonical forms " Two-level forms " Multi-level forms! Technology choices " Packages of a few gates " Regular logic " Two-level programmable logic " Multi-level programmable logic S 5 - Spring 27 Lec #2: ombinational Logic - 24

anonical Forms! Truth table is the unique signature of a oolean function! Many alternative gate realizations may have the same truth table! anonical forms " Standard forms for a oolean expression " Provides a unique algebraic signature S 5 - Spring 27 Lec #2: ombinational Logic - 25 Sum-of-Products anonical Forms! lso known as disjunctive normal form! lso known as minterm expansion F = F = ''+ ' + ' + ' + F F' F' = ''' + '' + '' S 5 - Spring 27 Lec #2: ombinational Logic - 26

Sum-of-Products anonical Form (cont d)! Product term (or minterm) " Ned product of literals input combination for which output is true " Each variable appears exactly once, in true or inverted form (but not both) minterms ''' m '' m '' m2 ' m3 '' m4 ' m5 ' m6 m7 short-hand notation for minterms of 3 variables F in canonical form: F(,, ) = 'm(,3,5,6,7) = m + m3 + m5 + m6 + m7 = '' + ' + ' + ' + canonical form # minimal form F(,, ) = '' + ' + ' + + ' = ('' + ' + ' + ) + ' = ((' + )(' + )) + ' = + ' = ' + = + S 5 - Spring 27 Lec #2: ombinational Logic - 27 Product-of-Sums anonical Form! lso known as conjunctive normal form! lso known as maxterm expansion F F' F = F = ( + + ) ( + ' + ) (' + + ) F' = ( + + ') ( + ' + ') (' + + ') (' + ' + ) (' + ' + ') S 5 - Spring 27 Lec #2: ombinational Logic - 28

Product-of-Sums anonical Form (cont d)! Sum term (or maxterm) " ORed sum of literals input combination for which output is false " Each variable appears exactly once, in true or inverted form (but not both) maxterms ++ M ++' M +'+ M2 +'+' M3 '++ M4 '++' M5 '+'+ M6 '+'+' M7 F in canonical form: F(,, ) = (M(,2,4) = M M2 M4 = ( + + ) ( + ' + ) (' + + ) canonical form # minimal form F(,, ) = ( + + ) ( + ' + ) (' + + ) = ( + + ) ( + ' + ) ( + + ) (' + + ) = ( + ) ( + ) short-hand notation for maxterms of 3 variables S 5 - Spring 27 Lec #2: ombinational Logic - 29 S-o-P, P-o-S, and demorgan s Theorem! Sum-of-products " F' = ''' + '' + ''! pply de Morgan's " (F')' = (''' + '' + '')' " F = ( + + ) ( + ' + ) (' + + )! Product-of-sums " F' = ( + + ') ( + ' + ') (' + + ') (' + ' + ) (' + ' + ')! pply de Morgan's " (F')' = ( ( + + ')( + ' + ')(' + + ')(' + ' + )(' + ' + ') )' " F = '' + ' + ' + ' + S 5 - Spring 27 Lec #2: ombinational Logic - 3

Four lternative Two-level Implementations of F = + F canonical sum-of-products minimized sum-of-products F2 F3 canonical product-of-sums F4 minimized product-of-sums S 5 - Spring 27 Lec #2: ombinational Logic - 3 Waveforms for the Four lternatives! Waveforms are essentially identical " Except for timing hazards (glitches) " elays almost identical (modeled as a delay per level, not type of gate or number of inputs to gate) S 5 - Spring 27 Lec #2: ombinational Logic - 32

Mapping etween anonical Forms! Minterm to maxterm conversion " Use maxterms whose indices do not appear in minterm expansion " e.g., F(,,) = 'm(,3,5,6,7) = (M(,2,4)! Maxterm to minterm conversion " Use minterms whose indices do not appear in maxterm expansion " e.g., F(,,) = (M(,2,4) = 'm(,3,5,6,7)! Minterm expansion of F to minterm expansion of F' " Use minterms whose indices do not appear " e.g., F(,,) = 'm(,3,5,6,7) F'(,,) = 'm(,2,4)! Maxterm expansion of F to maxterm expansion of F' " Use maxterms whose indices do not appear " e.g., F(,,) = (M(,2,4) F'(,,) = (M(,3,5,6,7) S 5 - Spring 27 Lec #2: ombinational Logic - 33 Incompletely Specified Functions! Example: binary coded decimal increment by " digits encode decimal digits 9 in bit patterns W Y off-set of W on-set of W don't care () set of W these inputs patterns should never be encountered in practice "don't care" about associated output values, can be exploited in minimization S 5 - Spring 27 Lec #2: ombinational Logic - 34

Notation for Incompletely Specified Functions! on't cares and canonical forms " So far, only represented on-set " lso represent don't-care-set " Need two of the three sets (on-set, off-set, dc-set)! anonical representations of the increment by function: " = m + m2 + m4 + m6 + m8 + d + d + d2 + d3 + d4 + d5 " = ' [ m(,2,4,6,8) + d(,,2,3,4,5) ] " = M M3 M5 M7 M9 2 3 4 5 " = ( [ M(,3,5,7,9) (,,2,3,4,5) ] S 5 - Spring 27 Lec #2: ombinational Logic - 35 Simplification of Two-level ombinational Logic! Finding a minimal sum of products or product of sums realization " Exploit don't care information in the process! lgebraic simplification " Not an algorithmic/systematic procedure " How do you know when the minimum realization has been found?! omputer-aided design tools " Precise solutions require very long computation times, especially for functions with many inputs (> ) " Heuristic methods employed "educated guesses" to reduce amount of computation and yield good if not best solutions! Hand methods still relevant " Understand automatic tools and their strengths and weaknesses " bility to check results (on small examples) S 5 - Spring 27 Lec #2: ombinational Logic - 36

dministrative nnouncement! Labs and discussions start in 25 ory next week! on t forget: Lab lecture TOMORROW 2-3 pm! We will try to accommodate all wait-listed students! Lab sections posted on 25 ory and the Web by Monday @ 5 PM S 5 - Spring 27 Lec #2: ombinational Logic - 37 The Uniting Theorem! Key tool for simplification: (' + ) =! Essence of simplification: " Find two element subsets of the ON-set where only one variable changes its value this single varying variable can be eliminated and a single product term used to represent both elements F = ''+' = ('+)' = ' F has the same value in both on-set rows! remains has a different value in the two rows! is eliminated S 5 - Spring 27 Lec #2: ombinational Logic - 38

oolean ubes! Visual technique for indentifying when the uniting theorem can be applied! n input variables = n-dimensional "cube" -cube Y 2-cube 3-cube Y Y W 4-cube S 5 - Spring 27 Lec #2: ombinational Logic - 39 Mapping Truth Tables onto oolean ubes! Uniting theorem combines two "faces" of a cube into a larger "face"! Example: F F two faces of size (nodes) combine into a face of size (line) ON-set = solid nodes OFF-set = empty nodes -set = )'d nodes varies within face, does not this face represents the literal ' S 5 - Spring 27 Lec #2: ombinational Logic - 4

Three Variable Example! inary full-adder carry-out logic in out ('+)in (in'+in) (+')in the on-set is completely covered by the combination (OR) of the subcubes of lower dimensionality - note that is covered three times out = in++in S 5 - Spring 27 Lec #2: ombinational Logic - 4 Higher imensional ubes! Sub-cubes of higher dimension than 2 F(,,) = 'm(4,5,6,7) on-set forms a square i.e., a cube of dimension 2 represents an expression in one variable i.e., 3 dimensions 2 dimensions is asserted (true) and unchanged and vary This subcube represents the literal S 5 - Spring 27 Lec #2: ombinational Logic - 42

m-imensional ubes in an n-imensional oolean Space! In a 3-cube (three variables): " -cube, i.e., a single node, yields a term in 3 literals " -cube, i.e., a line of two nodes, yields a term in 2 literals " 2-cube, i.e., a plane of four nodes, yields a term in literal " 3-cube, i.e., a cube of eight nodes, yields a constant term ""! In general, " m-subcube within an n-cube (m < n) yields a term with n m literals S 5 - Spring 27 Lec #2: ombinational Logic - 43 Karnaugh Maps! Flat map of oolean cube " Wrap around at edges " Hard to draw and visualize for more than 4 dimensions " Virtually impossible for more than 6 dimensions! lternative to truth-tables to help visualize adjacencies " Guide to applying the uniting theorem " On-set elements with only one variable changing value are adjacent unlike the situation in a linear truth-table 2 3 F S 5 - Spring 27 Lec #2: ombinational Logic - 44

Karnaugh Maps (cont d)! Numbering scheme based on Gray code " e.g.,,,, " Only a single bit changes in code for adjacent map cells 2 3 6 4 7 5 4 5 2 8 3 9 2 3 6 4 7 5 3 7 2 6 5 4 3 = = S 5 - Spring 27 Lec #2: ombinational Logic - 45 djacencies in Karnaugh Maps! Wrap from first to last column! Wrap top row to bottom row S 5 - Spring 27 Lec #2: ombinational Logic - 46

Karnaugh Map Examples! F =! out =! f(,,) = 'm(,4,6,7) + in + in in + + obtain the complement of the function by covering s with subcubes S 5 - Spring 27 Lec #2: ombinational Logic - 47 More Karnaugh Map Examples G(,,) = F(,,) = 'm(,4,5,7) = + F' simply replace 's with 's and vice versa F'(,,) = ' m(,2,3,6) = + S 5 - Spring 27 Lec #2: ombinational Logic - 48

Karnaugh Map: 4-Variable Example! F(,,,) = 'm(,2,3,5,6,7,8,,,4,5) F = + + find the smallest number of the largest possible subcubes to cover the ON-set (fewer terms with fewer inputs per term) S 5 - Spring 27 Lec #2: ombinational Logic - 49 Karnaugh Maps: on t ares! f(,,,) = ' m(,3,5,7,9) + d(6,2,3) " without don't cares # f = + S 5 - Spring 27 Lec #2: ombinational Logic - 5

Karnaugh Maps: on t ares (cont d)! f(,,,) = ' m(,3,5,7,9) + d(6,2,3) " f = ' + '' without don't cares " f = ' + ' with don't cares by using don't care as a "" a 2-cube can be formed rather than a -cube to cover this node don't cares can be treated as s or s depending on which is more advantageous S 5 - Spring 27 Lec #2: ombinational Logic - 5 esign Example: Two-bit omparator N N2 LT EQ GT block diagram and truth table < = > LT EQ GT we'll need a 4-variable Karnaugh map for each of the 3 output functions S 5 - Spring 27 Lec #2: ombinational Logic - 52

esign Example: Two-bit omparator (cont d) K-map for LT K-map for EQ K-map for GT LT = EQ = GT = ' ' + ' + ' '''' + '' + + ' ' ' + ' + ' = ( xnor ) ( xnor ) LT and GT are similar (flip / and /) S 5 - Spring 27 Lec #2: ombinational Logic - 53 esign Example: Two-bit omparator (cont d) two alternative implementations of EQ with and without OR EQ EQ NOR is implemented with at least 3 simple gates S 5 - Spring 27 Lec #2: ombinational Logic - 54

esign Example: 2x2-bit Multiplier 2 2 block diagram and truth table P P2 P4 P8 2 2 P8 P4 P2 P 4-variable K-map for each of the 4 output functions S 5 - Spring 27 Lec #2: ombinational Logic - 55 esign Example: 2x2-bit Multiplier (cont d) 2 K-map for P8 K-map for P4 2 P4 = 22' + 2'2 2 P8 = 22 2 2 K-map for P2 K-map for P P = 2 2 P2 = 2'2 + 2' + 22' + 2' 2 S 5 - Spring 27 Lec #2: ombinational Logic - 56

esign Example: Increment by I I2 I4 I8 block diagram and truth table O O2 O4 O8 I8 I4 I2 I O8 O4 O2 O 4-variable K-map for each of the 4 output functions S 5 - Spring 27 Lec #2: ombinational Logic - 57 esign Example: Increment by (cont d) I8 O8 O4 I8 I2 I4 I O8 = I4 I2 I + I8 I' O4 = I4 I2' + I4 I' + I4 I2 I O2 = I8 I2 I + I2 I' I2 I4 I I8 O2 O = I' O I8 I I I2 I2 I4 I4 S 5 - Spring 27 Lec #2: ombinational Logic - 58

efinition of Terms for Two-level Simplification! Implicant " Single element of ON-set or -set or any group of these elements that can be combined to form a subcube! Prime implicant " Implicant that can't be combined with another to form a larger subcube! Essential prime implicant " Prime implicant is essential if it alone covers an element of ON-set " Will participate in LL possible covers of the ON-set " -set used to form prime implicants but not to make implicant essential! Objective: " Grow implicant into prime implicants (minimize literals per term) " over the ON-set with as few prime implicants as possible (minimize number of product terms) S 5 - Spring 27 Lec #2: ombinational Logic - 59 Examples to Illustrate Terms 6 prime implicants: '', ',, '',, ' essential minimum cover: + ' + '' 5 prime implicants:, ',, ', '' essential minimum cover: 4 essential implicants S 5 - Spring 27 Lec #2: ombinational Logic - 6

lgorithm for Two-level Simplification! lgorithm: minimum sum-of-products expression from a K-map " Step : choose an element of the ON-set " Step 2: find "maximal" groupings of s and s adjacent to that element # consider top/bottom row, left/right column, and corner adjacencies # this forms prime implicants (number of elements always a power of 2) " Repeat Steps and 2 to find all prime implicants " Step 3: revisit the s in the K-map # if covered by single prime implicant, it is essential, participates in final cover # s covered by essential prime implicant do not need to be revisited " Step 4: if there remain s not covered by essential prime implicants # select the smallest number of prime implicants that cover the remaining s S 5 - Spring 27 Lec #2: ombinational Logic - 6 lgorithm for Two-level Simplification (example) 2 primes around ''' 2 primes around ' 3 primes around ''' 2 essential primes minimum cover (3 primes) S 5 - Spring 27 Lec #2: ombinational Logic - 62

Implementations of Two-level Logic! Sum-of-products " N gates to form product terms (minterms) " OR gate to form sum! Product-of-sums " OR gates to form sum terms (maxterms) " N gates to form product S 5 - Spring 27 Lec #2: ombinational Logic - 63 Two-level Logic Using NN Gates! Replace minterm N gates with NN gates! Place compensating inversion at inputs of OR gate S 5 - Spring 27 Lec #2: ombinational Logic - 64

Two-level Logic Using NN Gates (cont d)! OR gate with inverted inputs is a NN gate " de Morgan's: ' + ' = ( )'! Two-level NN-NN network " Inverted inputs are not counted " In a typical circuit, inversion is done once and signal distributed S 5 - Spring 27 Lec #2: ombinational Logic - 65 Two-level Logic Using NOR Gates! Replace maxterm OR gates with NOR gates! Place compensating inversion at inputs of N gate S 5 - Spring 27 Lec #2: ombinational Logic - 66

Two-level Logic Using NOR Gates (cont d)! N gate with inverted inputs is a NOR gate " de Morgan's: ' ' = ( + )'! Two-level NOR-NOR network " Inverted inputs are not counted " In a typical circuit, inversion is done once and signal distributed S 5 - Spring 27 Lec #2: ombinational Logic - 67 Two-level Logic Using NN and NOR Gates! NN-NN and NOR-NOR networks " de Morgan's law: ( + )' = ' ' ( )' = ' + ' " written differently: + = (' ') ( ) = (' + ')'! In other words " OR is the same as NN with complemented inputs " N is the same as NOR with complemented inputs " NN is the same as OR with complemented inputs " NOR is the same as N with complemented inputs OR OR N N NN NN NOR NOR S 5 - Spring 27 Lec #2: ombinational Logic - 68

onversion etween Forms! onvert from networks of Ns and ORs to networks of NNs and NORs " Introduce appropriate inversions ("bubbles")! Each introduced "bubble" must be matched by a corresponding "bubble" " onservation of inversions " o not alter logic function! Example: N/OR to NN/NN NN NN S 5 - Spring 27 Lec #2: ombinational Logic - 69 NN onversion etween Forms (cont d)! Example: verify equivalence of two forms NN NN NN = [ ( )' ( )' ]' = [ (' + ') (' + ') ]' = [ (' + ')' + (' + ')' ] = ( ) + ( )! S 5 - Spring 27 Lec #2: ombinational Logic - 7

onversion etween Forms (cont d)! Example: map N/OR network to NOR/NOR network NOR NOR \ \ \ \ NOR NOR NOR conserve "bubbles" Step Step 2 S 5 - Spring 27 Lec #2: ombinational Logic - 7 conserve "bubbles" onversion etween Forms (cont d)! Example: verify equivalence of two forms \ \ NOR NOR \ \ NOR = { [ (' + ')' + (' + ')' ]' }' = { (' + ') (' + ') }' = (' + ')' + (' + ')' = ( ) + ( )! S 5 - Spring 27 Lec #2: ombinational Logic - 72

ombinational Logic Summary! Logic functions, truth tables, and switches " NOT, N, OR, NN, NOR, OR,..., minimal set! xioms and theorems of oolean algebra " Proofs by re-writing and perfect induction! Gate logic " Networks of oolean functions and their time behavior! anonical forms " Two-level and incompletely specified functions! Simplification " Two-level simplification S 5 - Spring 27 Lec #2: ombinational Logic - 73