1. Prof. Kavita Bala and Prof. Hakim Weatherspoon CS 3410, Spring 2014 Computer Science Cornell University See: P&H Appendix B.2 and B.3 (Also, see B.0 and B.1)

2. From Switches to Logic Gates to Logic Circuits Logic Gates From switches Truth Tables Logic Circuits Identity Laws From Truth Tables to Circuits (Sum of Products) Logic Circuit Minimization Algebraic Manipulations Truth Tables (Karnaugh Maps) Transistors (electronic switch)

3. Acts as a conductor or insulator Can be used to build amazing things… The Bombe used to break the German Enigma machine during World War II

4. Either (OR) Both (AND) + - - ABLight OFF ON OFF ON A B A B ABLight OFF ON OFF ON Truth Table +

5. Either (OR) Both (AND) - - A B A B ABLight OFF ON OFF ON ABLight OFF ON OFF ON Truth Table OR AND

6. Either (OR) Both (AND) - - A B A B ABLight 00 01 10 11 AB 00 01 10 11 Truth Table 0 = OFF 1 = ON OR AND

7. Did you know? George Boole Inventor of the idea of logic gates. He was born in Lincoln, England and he was the son of a shoemaker in a low class family. A B A B George Boole,(1815-1864) OR AND

8. Binary (two symbols: true and false) is the basis of Logic Design

9. NOT: AND: OR: Logic Gates digital circuit that either allows a signal to pass through it or not. Used to build logic functions There are seven basic logic gates: AND, OR, NOT, NAND (not AND), NOR (not OR), XOR, and XNOR (not XOR) [later] ABOut 000 011 101 111 AB 000 010 100 111 A A B A B A

10. NOT: AND: OR: Logic Gates digital circuit that either allows a signal to pass through it or not. Used to build logic functions There are seven basic logic gates: AND, OR, NOT, NAND (not AND), NOR (not OR), XOR, and XNOR (not XOR) [later] ABOut 000 011 101 111 AB 000 010 100 111 A 01 10 A B A B A

11. NOT: AND: OR: Logic Gates digital circuit that either allows a signal to pass through it or not. Used to build logic functions There are seven basic logic gates: AND, OR, NOT, NAND (not AND), NOR (not OR), XOR, and XNOR (not XOR) [later] ABOut 000 011 101 111 AB 000 010 100 111 A 01 10 A B A B A AB 001 010 100 110 AB 001 011 101 110 A B A B NAND: NOR:

12. From Switches to Logic Gates to Logic Circuits Logic Gates From switches Truth Tables Logic Circuits Identity Laws From Truth Tables to Circuits (Sum of Products) Logic Circuit Minimization Algebraic Manipulations Truth Tables (Karnaugh Maps) Transistors (electronic switch)

13. Given a Logic function, create a Logic Circuit that implements the Logic Function… …and, with the minimum number of logic gates Fewer gates: A cheaper ($$$) circuit!

14. NOT: AND: OR: XOR:. ABOut 000 011 101 111 AB 000 010 100 111 A 01 10 A B A B A AB 000 011 101 110 A B

15. NOT: AND: OR: XOR:. ABOut 000 011 101 111 AB 000 010 100 111 A 01 10 A B A B A AB 000 011 101 110 A B AB 001 010 100 110 AB 001 011 101 110 A B A B NAND: NOR: ABOut 001 010 100 111 A B XNOR:

18. Identities useful for manipulating logic equations – For optimization & ease of implementation a + 0 = a + 1 = a + ā = a ∙ 0 = a ∙ 1 = a ∙ ā =

20. functions: gates ↔ truth tables ↔ equations Example: (a+b)(a+c) = a + bc abc 000 001 010 011 100 101 110 111

21. Binary (two symbols: true and false) is the basis of Logic Design More than one Logic Circuit can implement same Logic function. Use Algebra (Identities) or Truth Tables to show equivalence.

23. From Switches to Logic Gates to Logic Circuits Logic Gates From switches Truth Tables Logic Circuits Identity Laws From Truth Tables to Circuits (Sum of Products) Logic Circuit Minimization Algebraic Manipulations Truth Tables (Karnaugh Maps) Transistors (electronic switch)

24. How to standardize minimizing logic circuits?

25. How to implement a desired logic function? abcout 0000 0011 0100 0111 1000 1011 1100 1110

26. How to implement a desired logic function? abcout 0000 0011 0100 0111 1000 1011 1100 1110 1)Write minterms 2)sum of products: OR of all minterms where out=1 minterm a b c

27. How to implement a desired logic function? abcout 0000 0011 0100 0111 1000 1011 1100 1110 1)Write minterms 2)sum of products: OR of all minterms where out=1 minterm a b c

28. How does one find the most efficient equation? – Manipulate algebraically until…? – Use Karnaugh maps (optimize visually) – Use a software optimizer For large circuits – Decomposition & reuse of building blocks

29. abcout 0000 0011 0100 0111 1001 1011 1100 1110

30. abc 0000 0011 0100 0111 1001 1011 1100 1110 Sum of minterms yields out = Karnaugh maps identify which inputs are (ir)relevant to the output 0001 1101 00 01 11 10 0 1 c ab

31. abcout 0000 0011 0100 0111 1001 1011 1100 1110 0001 1101 00 01 11 10 0 1 c ab

32. 0111 0010 00 01 11 10 0 1 c ab 1111 0010 00 01 11 10 0 1 c ab

33. 1001 0000 0000 1001 00 01 11 10 00 01 ab cd 11 10 0000 1001 1001 0000 00 01 11 10 00 01 ab cd 11 10

34. “Don’t care” values can be interpreted individually in whatever way is convenient assume all x’s = 1 out = d assume middle x’s = 0 assume 4 th column x = 1 out = 100x 0xx0 0xx0 1001 00 01 11 10 00 01 ab cd 11 10 0000 1xxx 1xx1 0000 00 01 11 10 00 01 ab cd 11 10

35. A multiplexer selects between multiple inputs out = a, if d = 0 out = b, if d = 1 Build truth table Minimize diagram Derive logic diagram a b d abdout 000 001 010 011 100 101 110 111

36. Binary (two symbols: true and false) is the basis of Logic Design More than one Logic Circuit can implement same Logic function. Use Algebra (Identities) or Truth Tables to show equivalence. Any logic function can be implemented as “sum of products”. Karnaugh Maps minimize number of gates.

37. From Transistors to Gates to Logic Circuits Logic Gates From transistors Truth Tables Logic Circuits Identity Laws From Truth Tables to Circuits (Sum of Products) Logic Circuit Minimization Algebraic Manipulations Truth Tables (Karnaugh Maps) Transistors (electronic switch)

38. Transistors: 6:10 minutes (watch from from 41s to 7:00) http://www.youtube.com/watch?v=QO5FgM7MLGg Fill our Transistor Worksheet with info from Video

39. NMOS Transistor Connect source to drain when gate = 1 N-channel VDVD V S = 0 V VGVG V G = V S V G = 0 V PMOS Transistor Connect source to drain when gate = 0 P-channel VSVS V D = 0 V VGVG V G = V S V G = 0 V

40. NMOS Transistor Connect source to drain when gate = 1 N-channel VDVD V S = 0 V VGVG V G = 1 V G = 0 PMOS Transistor Connect source to drain when gate = 0 P-channel VSVS V D = 0 V VGVG V G = 1 V G = 0

41. InOut 01 10 Function: NOT Called an inverter Symbol: Useful for taking the inverse of an input CMOS: complementary-symmetry metal–oxide– semiconductor in out Truth table inout V supply (aka logic 1) (ground is logic 0)

42. ABout 001 101 011 110 Function: NAND Symbol: b a out A V supply B B A

43. ABout 001 100 010 110 Function: NOR Symbol: b a out A V supply B B A

44. NOT: AND: OR: NAND and NOR are universal Can implement any function with NAND or just NOR gates useful for manufacturing

45. NOT: AND: OR: NAND and NOR are universal Can implement any function with NAND or just NOR gates useful for manufacturing b a b a a

46. One can buy gates separately ex. 74xxx series of integrated circuits cost ~$1 per chip, mostly for packaging and testing Cumbersome, but possible to build devices using gates put together manually

47. The first transistor on a workbench at AT&T Bell Labs in 1947 Bardeen, Brattain, and Shockley An Intel Haswell –1.4 billion transistors –177 square millimeters –Four processing cores http://techguru3d.com/4th-gen-intel-haswell-processors-architecture-and-lineup/

48. Hide complexity through simple abstractions Simplicity –Box diagram represents inputs and outputs Complexity –Hides underlying NMOS- and PMOS-transistors and atomic interactions in out Vdd Vss inout a d b a b d

49. Most modern devices are made from billions of on /off switches called transistors We will build a processor in this course! Transistors made from semiconductor materials: –MOSFET – Metal Oxide Semiconductor Field Effect Transistor –NMOS, PMOS – Negative MOS and Positive MOS –CMOS – complementary MOS made from PMOS and NMOS transistors Transistors used to make logic gates and logic circuits We can now implement any logic circuit Can do it efficiently, using Karnaugh maps to find the minimal terms required Can use either NAND or NOR gates to implement the logic circuit Can use P- and N-transistors to implement NAND or NOR gates