Combinational Logic. Review of Combinational Logic 1

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1 Combinational Logic! Switches -> Boolean algebra! Representation of Boolean functions! Logic circuit elements - logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational logic! Incompletely specified functions! Optimization of combinational logic! Arithmetic circuits Review of Combinational Logic 1

2 Switching Algebra! Based on operation of switches x switch conducts if ON switch does not conduct if OFF! Values are either or 1 switch is on or off (A1) X 1 X = X X = 1! Complementation operation which reverses switch state (A2) X = 1 X' = X = X' = 1 Review of Combinational Logic 2

3 Switching Algebra (cont d)! Switch composition is series composition x y x + is parallel composition y (A3) = = 1 (A4) 1 1 = 1 + = (A5) 1 = 1 = + 1 = 1 + = 1 Review of Combinational Logic 3

4 Basic Theorems in Boolean Algebra (T1) + a = a 1 a = a identity (T2) 1 + a = 1 a = null (T3) a + a = a a a = a idempotent (T4) (a ) = a involution (T5) a + a = 1 a a = complement (T6) a + b = b + a a b = b a commutative (T7) a + ( a b ) = a a ( a + b ) = a absorption (T8) (a + b ) b = a b a b + b = a + b (T9) a + ( b + c) = ( a + b ) + c a ( b c) = ( a b ) c associative (T1) a ( b + c ) = a b + a c a + b c = ( a + b ) ( a + c ) distributive duality Review of Combinational Logic 4

5 Applying Boolean Algebra! Proving theorems using the axioms of Boolean algebra: o e.g., prove the theorem: X Y + X Y' = X distributive law (T1) X Y + X Y' = X (Y + Y') complementarity law (T5) X (Y + Y') = X (1) identity (T1) X (1) = X o e.g., prove the theorem: X + X Y = X identity (T1) X + X Y = X 1 + X Y distributive law (T1) X 1 + X Y = X (1 + Y) null (T2) X (1 + Y) = X (1) identity (T1) X (1) = X Review of Combinational Logic 5

6 Proofs in Boolean Algebra! The axioms state what we know about switches! All theorems follow from the axioms! Make use of truth tables! Example: distributive law a ( b + c ) = a b + a c a b c ab ac b+c a(b+c) ab+ac proof by perfect induction Review of Combinational Logic 6

7 Other Theorems! Consensus: ab+a'c+bc = ab+a'c! de Morgan's theorem: (a + b)' = a'b (ab)' = a'+b o in general, to complement an expression: " replace each variable and element with its complement " interchange the + and operations " [f(x 1,x 2,...,x n,,1,+, )]' = f(x' 1,x 2,...,x n,1,,,+)! Expansion theorem (Shannon): f(x 1,x 2,...,x n ) = x 1 f(1,x 2,...,x n ) + x' 1 f(,x 2,...,x n ) = [x 1 +f(,x 2,...,x n )] [x' 1 +f(1,x 2,...,x n )] o How might you use this? Review of Combinational Logic 7

8 Duality - A Meta-Theorem! Any theorem remains true if: o + and are interchanged, and o and 1 are interchanged! Important implication o if you prove one theorem and you have also proven its dual! Example: prove the dual of consensus! Note: not the same as de Morgan's Law o this is a statement about theorems o this is not a way to transform expressions! How would you prove the theorem of Duality? Review of Combinational Logic 8

9 de Morgan s Law! It can be used to convert AND/OR expressions to OR/AND expressions e.g., Z = A' B' C + A' B C + A B' C + A B C' Z' = (A + B + C') (A + B' + C') (A' + B + C') (A' + B' + C) (X + Y)' = X' Y' NOR is equivalent to AND with inputs complemented X Y X' Y' (X + Y)' X' Y' (X Y)' = X' + Y' NAND is equivalent to OR with inputs complemented X Y X' Y' (X Y)' X' + Y' Review of Combinational Logic 9

10 What s the Big Deal with Boolean Algebra? Review of Combinational Logic 1

11 Applying Boolean Algebra! Apply the laws and theorems to simplify Boolean expressions o e.g., full adder's carry-out function Cout = A' B Cin + A B' Cin + A B Cin' + A B Cin Review of Combinational Logic 11

12 Applying Boolean Algebra (cont d)! Apply the laws and theorems to simplify Boolean expressions o e.g., full adder's carry-out function Cout = A' B Cin + A B' Cin + A B Cin' + A B Cin = A' B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin = A' B Cin + A B Cin + A B' Cin + A B Cin' + A B Cin = (A' + A) B Cin + A B' Cin + A B Cin' + A B Cin = (1) B Cin + A B' Cin + A B Cin' + A B Cin = B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin = B Cin + A B' Cin + A B Cin + A B Cin' + A B Cin = B Cin + A (B' + B) Cin + A B Cin' + A B Cin = B Cin + A (1) Cin + A B Cin' + A B Cin = B Cin + A Cin + A B (Cin' + Cin) = B Cin + A Cin + A B (1) = B Cin + A Cin + A B Review of Combinational Logic 12

13 From Switches to logic gates! NOT: X' X ~X X Y X Y 1 1! AND: X Y XY X Y X Y Z X Y Z ! OR: X + Y X Y X Y Z X Y Z ! How do we represent boolean values?! How can we use switches to make gates? Review of Combinational Logic 13

14 Representing Boolean Values! Use voltage 5v 1 Undefined v Review of Combinational Logic 14

15 Only Switches We Need! How do we make an inverter? o input 5v output v o input v output 5v Conducts when control is v Conducts when control is 5v Review of Combinational Logic 15

16 Before we Abstract Voltage away Entirely...! What happens when input isn t 5v or v? o OK for input to be 5v - ε o OK for input to be v + ε Vout 5v IN OUT v 5v Vin Review of Combinational Logic 16

17 All Possible Functions of One Variable a a a' 1 Review of Combinational Logic 17

18 All Possible Functions of Two Variables a b A a b X O N X b b a a N 1 N O R O N A D R R O a b N R D AND = ab OR = a+b XOR = NEQ = ab +a b = a b a b = a +b b a = a+b NAND = (ab) NOR = (a+b) XNOR = EQ = ab+a b = a b Review of Combinational Logic 18

19 How Many Functions of N Variables? Review of Combinational Logic 19

20 Minimal Set of Gates! How many different gates do we need? Review of Combinational Logic 2

21 Representing of Boolean Functions! Boolean expression: a'b + ab' + a'c! truth table:! n-dimensional cube: a b c a'b ab' a'c' a'b+ab'+a'c' a c b! circuit with logic gates Review of Combinational Logic 21

22 Canonical forms: Sum Of Products! Truth table is the unique signature of a Boolean function! Many alternative expressions may have the same truth table! Canonical form o standard form for a Boolean expression o Sum-of-products form a.k.a. disjunctive normal form or minterm expansion A B C F F' F = A'BC + AB'C' + AB'C + ABC' + ABC F' = A'B'C' + A'B'C + A'BC' Review of Combinational Logic 22

23 Canonical Forms: SOP (cont d)! Product term (or minterm) o ANDed product of literals in which each variable appears exactly once, in true or complemented form (but not both) A B C minterms A'B'C' m 1 A'B'C m1 1 A'BC' m2 1 1 A'BC m3 1 AB'C' m4 1 1 AB'C m5 1 1 ABC' m ABC m7 short-hand notation for minterms of 3 variables F in canonical form: F(A, B, C) = Σm(3,4,5,6,7) = m3 + m4 + m5 + m6 + m7 = A'BC + AB'C' + AB'C + ABC' + ABC canonical form minimal form F(A, B, C) = AB'(C + C') + A'BC + AB(C' + C) = AB' + A'BC + AB = A(B' + B) + A'BC = A + A'BC = A + BC Review of Combinational Logic 23

24 Product-of-sums canonical form! Also known as conjunctive normal form! Also known as maxterm expansion A B C F F' F = 1 1 F = (A + B + C) (A + B' + C) (A' + B + C) F' = (A + B + C') (A + B' + C') (A' + B + C') (A' + B' + C) (A' + B' + C') Review of Combinational Logic 24

25 Canonical Forms: POS! Product-of-sums form a.k.a. conjunctive normal form or maxterm expansion! Sum term (or maxterm) o ORed sum of literals in which each variable appears exactly once, in true or complemented form (but not both) A B C maxterms A+B+C M 1 A+B+C' M1 1 A+B'+C M2 1 1 A+B'+C' M3 1 A'+B+C M4 1 1 A'+B+C' M5 1 1 A'+B'+C M A'+B'+C' M7 F in canonical form: F(A, B, C) = ΠM(,1,2) = M M1 M2 = (A + B + C) (A + B + C') (A + B' + C) canonical form minimal form F(A, B, C) = (A + B + (C C')) (A + (B B') + C) = (A + B) (A + C) = A + BC short-hand notation for maxterms of 3 variables Review of Combinational Logic 25

26 Implementations of two-level logic! Sum-of-products o AND gates to form product terms (minterms) o OR gate to form sum! Product-of-sums o OR gates to form sum terms (maxterms) o AND gates to form product Review of Combinational Logic 26

27 Two-level logic using NAND and NOR gates! NAND-NAND and NOR-NOR networks o de Morgan's law: (A + B) = A' B' (A B)' = A' + B' o written differently: A + B = (A' B')' (A B) = (A' + B')'! In other words o OR is the same as NAND with complemented inputs o AND is the same as NOR with complemented inputs o NAND is the same as OR with complemented inputs o NOR is the same as AND with complemented inputs OR OR AND AND NAND NAND NOR NOR Review of Combinational Logic 27

28 Conversion between forms! Convert from networks of ANDs and ORs to networks of NANDs and NORs o introduce appropriate inversions ("bubbles")! Each introduced "bubble" must be matched by a corresponding "bubble" o conservation of inversions o do not alter logic function! Example: AND/OR to NAND/NAND A B C D Z A B C D NAND NAND NAND Z Review of Combinational Logic 28

29 Conversion between forms (cont d)! Example: map AND/OR network to NOR/NOR network A B C Z D A B C D NOR NOR Z \A \B \C \D NOR NOR NOR Z conserve "bubbles" Step 1 Step 2 conserve "bubbles" Review of Combinational Logic 29

30 Incompleteley specified functions! Example: binary coded decimal increment by 1 o BCD digits encode the decimal digits 9 in the bit patterns 11 A B C D W X Y Z X X X X X X X X 1 1 X X X X X X X X X X X X X X X X off-set of W on-set of W don't care (DC) set of W these inputs patterns should never be encountered in practice we "don't care" about associated output values, and this can be exploited in minimization Review of Combinational Logic 3

31 Notation for incompletely specified functions! Don't cares and canonical forms o so far, only represented on-set o must also represent don't-care-set o need two of the three sets (on-set, off-set, dc-set)! Canonical representations of the BCD increment by 1 function: o Z = m + m2 + m4 + m6 + m8 + d1 + d11 + d12 + d13 + d14 + d15 o Z = Σ [ m(,2,4,6,8) + d(1,11,12,13,14,15) ] o Z = M1 M3 M5 M7 M9 D1 D11 D12 D13 D14 D15 o Z = Π [ M(1,3,5,7,9) D(1,11,12,13,14,15) ] Review of Combinational Logic 31

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