COMBINATIONAL LOGIC DESIGN PRINCIPLES

1. COMBINATIONAL LOGIC DESIGN PRINCIPLES

2. COMBINATIONAL LOGIC DESIGN PRINCIPLES • SWITCHING ALGEBRA • COMBINATIONAL CIRCUIT ANALYSIS • COMBINATIONAL CIRCUIT SYNTHESIS • TIMING HAZARDS

3. ALGEBRA? • n. 1. A GENERALIZATION OF ARITHMETIC IN WHICH LETTERS REPRESENTING NUMBERS ARE COMBINED ACCORDING TO THE RULES OF ARITHMETIC

4. SWITCHING ALGEBRA • GEORGE BOOLE: 1854, BOOLEAN ALGEBRA • BOOLEAN ALGEBRA: TWO-VALUED ALGEBRAIC SYSTEM • CLAUDE SHANNON: 1938, ADAPTATION OF BOOLEAN ALGEBRA  SWITCHING ALGEBRA

5. SWITCHING ALGEBRA OPERATIONS • NOT OPERATION: X’ • AND OPERATION (LOGICAL MULTIPLICATION): XY, XY • OR OPERATION (LOGICAL ADDITION): X+Y, XY

6. SWITCHING ALGEBRA OPERATIONS

7. AXIOMS • AXIOM = POSTULATE • THE AXIOMS OF A MATHEMATICAL SYSTEM ARE A MINIMAL SET OF BASIC DEFINITIONS THAT WE ASSUME TO BE TRUE, FROM WHICH ALL OTHER INFORMATION ABOUT THE SYSTEM CAN BE DERIVED.

8. AXIOMS (A1) X=0 IF X1 (A1’) X=1 IF X0

9. AXIOMS (A2) IF X=0 THEN X’=1 (A2’) IF X=1 THEN X’=0

10. AXIOMS (A3) 00=0 (A3’) 1+1=1 (A4) 11=1 (A4’) 0+0=0 (A5) 01=10=0 (A5’) 1+0=0+1=1

11. AXIOMS • A1-A5 AND A1’-A5’ COMPLETELY DEFINE SWITCHING ALGEBRA • ALL OTHER FACTS ABOUT SYSTEM CAN BE PROVED USING A1-A5 AND A1’-A5’

12. THEOREM? • n. 1. AN IDEA ACCEPTED OR PROPOSED AS A DEMONSTRABLE TRUTH OFTEN AS PART OF A GENERAL THEORY

13. SINGLE-VARIABLE THEOREMS • PERFECT INDUCTION: • A1: X=0 OR X=1 • PROVE THEOREM FOR BOTH X=0 AND X=1

14. IDENTITIES (T1) X+0=X (T1’) X1=X

15. NULL ELEMENTS (T2) X+1=1 (T2’) X0=0

16. IDEMPOTENCY (T3) X+X=X (T3’) XX=X

17. INVOLUTION (T4) (X’)’=X

18. COMPLEMENTS (T5) X+X’=1 (T5’) XX’=0

19. MORE THEOREMS • TWO- AND THREE-VARIABLE THEOREMS • PROOF: PERFECT INDUCTION OR OTHER THEOREMS AND AXIOMS

20. COMMUTATIVITY (T6) X+Y=Y+X (T6’) XY=YX

21. ASSOCIATIVITY (T7) (X+Y)+Z=X+(Y+Z) (T7’) (XY)Z=X(YZ)

22. DISTRIBUTIVITY (T8) X(Y+Z)=XY+XZ (T8’) (X+Y)(X+Z)=X+YZ

23. EXAMPLE • NOTE: POSSIBLE TO REPLACE VARIABLE WITH EXPRESSION

24. COVERING (T9) X+XY=X (T9’) X(X+Y)=X • X COVERS Y

25. COMBINING (T10) XY+XY’=X (T10’) (X+Y)(X+Y’)=X

26. CONSENSUS (T11) XY+X’Z+YZ=XY+X’Z (T11’) (X+Y)(X’+Z)(Y+Z)=(X+Y)(X’+Z) • T11: YZ IS THE CONSENSUS TERM • T11’: Y+Z IS THE CONSENSUS TERM

27. n-VARIABLE THEOREMS • PROOF FOR MOST: FINITE INDUCTION • FINITE INDUCTION: • STEP 1: PROVE THEOREM FOR n=2 • STEP 2: PROVE THAT IF THEOREM IS TRUE FOR n=i IT IS ALSO TRUE FOR n=i+1

28. GENERALIZED IDEMPOTENCY (T12) X+X+…+X=X (T12’) XX … X=X

29. DeMORGAN’S THEOREMS (T13) (X1X2 … Xn)’=X1’+X2’+…+Xn’ (T13’) (X1+X2+ … +Xn)’=X1’X2’ … Xn’

30. EQUIVALENT CIRCUITS

31. GENERALIZED DeMORGAN’S THEOREM (T14) [F(X1,X2,…Xn,+,)]’=F(X1’,X2’,…,Xn’,,+) • T13 AND T13’ SPECIAL CASES OF T14

32. DUALITY • ANY THEOREM OR IDENTITY IN SWITCHING ALGEBRA REMAINS TRUE IF 0 AND 1 ARE SWAPPED, AND  AND + ARE SWAPPED THROUGHOUT.

33. DUAL OF A LOGIC EXPRESSION FD(X1,X2,…,Xn,+,,’)= F(X1,X2,…,Xn,,+,’)

34. GENERALIZED DeMORGAN’S THEOREM [F(X1,X2,…Xn,+,)]’=F(X1’,X2’,…,Xn’,,+) [F(X1,X2,…,Xn)]’= FD(X1’,X2’,…,Xn’,)

35. SHANNON’S EXPANSION THEOREMS (T15) F(X1,X2,…,Xn)=X1F(1,X2,…,Xn)+ +X1’F(0,X2,…,Xn) (T15’) F(X1,X2,…,Xn)=[X1+F(0,X2,…,Xn)] [X1’+F(1,X2,…,Xn)]

36. LOGIC FUNCTION REPRESENTATION • TRUTH TABLE ROW X Y F 0 0 0 F(0,0) 1 0 1 F(0,1) 2 1 0 F(1,0) 3 1 1 F(1,1)

37. DEFINITIONS • LITERAL: A VARIABLE OR ITS COMPLEMENT. EXAMPLES: Y, Y’ • PRODUCT TERM: LITERAL, OR LOGICAL PRODUCT OF LITERALS. EXAMPLES: X, ZY’ • SUM TERM: LITERAL OR LOGICAL SUM OF LITERALS. EXAMPLES: Z, X+Y+W’

38. DEFINITIONS • PRODUCT-OF-SUMS: LOGICAL PRODUCT OF SUM TERMS. EXAMPLE: (X+Y)(Z+W’) • SUM-OF-PRODUCTS: LOGICAL SUM OF PRODUCT TERMS. EXAMPLE: XY+ZW’ • NORMAL TERM: PRODUCT OR SUM TERM IN WHICH NO VARIABLE APPEARS MORE THAN ONCE. EXAMPLE: WXY’. EXAMPLE OF NONNORMAL TERM: WW’X

39. MINTERM, MAXTERM • n-VARIABLE MINTERM: NORMAL PRODUCT TERM WITH n LITERALS. EXAMPLE OF 2-VARIABLE MINTERM: WX • n-VARIABLE MAXTERM: NORMAL SUM TERM WITH n LITERALS. EXAMPLE OF 2-VARIABLE MAXTERM: W+X

40. MINTERM, MAXTERM • MINTERM: PRODUCT TERM THAT IS 1 FOR EXACTLY ONE ROW OF THE TRUTH TABLE • MAXTERM: SUM TERM THAT IS 0 FOR EXACTLY ONE ROW OF THE TRUTH TABLE

41. MINTERM, MAXTERM ROW X Y F MINTERM MAXTERM 0 0 0 F(0,0) X’Y’ X+Y 1 0 1 F(0,1) X’Y X+Y’ 2 1 0 F(1,0) XY’ X’+Y 3 1 1 F(1,1) XY X’+Y’ • MINTERM i, MAXTERM i

42. MINTERM, MAXTERM • CANONICAL SUM = MINTERM LIST = ON-SET • CANONICAL PRODUCT = MAXTERM LIST = OFF-SET • CONVERSION IS EASY: A,B,C(0,1,2,3)=A,B,C(4,5,6,7) X,Y,Z,W(0,1,2,3,5,7,11,13)= =X,Y,Z,W(4,6,8,9,10,12,14,15)

43. COMBINATIONAL LOGIC REPRESENTATIONS • TRUTH TABLE • CANONICAL SUM • MINTERM LIST (-NOTATION) • CANONICAL PRODUCT • MAXTERM LIST (-NOTATION)

44. COMBINATIONAL LOGIC DESIGN PRINCIPLES • SWITCHING ALGEBRA • COMBINATIONAL CIRCUIT ANALYSIS • COMBINATIONAL CIRCUIT SYNTHESIS • TIMING HAZARDS

45. CIRCUIT ANALYSIS • WHY? • DETERMINE BEHAVIOR FOR VARIOUS INPUTS • MANIPULATE ALGEBRAIC DESCRIPTION • TRANSFORM ALGEBRAIC DESCRIPTION INTO STANDARD FORM • USE ALGEBRAIC DESCRIPTION IN ANALYSIS OF LARGER CIRCUIT

46. EXAMPLES

47. COMBINATIONAL LOGIC DESIGN PRINCIPLES • SWITCHING ALGEBRA • COMBINATIONAL CIRCUIT ANALYSIS • COMBINATIONAL CIRCUIT SYNTHESIS • TIMING HAZARDS

48. WHERE DO WE START? • WORD DESCRIPTION • “AND”, “OR”, “NOT” • TRUTH TABLE • CANONICAL SUM OR PRODUCT

49. EXAMPLES

50. COMBINATIONAL CIRCUIT MINIMIZATION • WHY? • COST • HOW? • MINIMIZE NUMBER OF FIRST LEVEL GATES • MINIMIZE NUMBER OF INPUTS TO FIRST LEVEL GATES • MINIMIZE NUMBER OF INPUTS TO SECOND LEVEL GATES