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Lecture 5

Lecture 5. Logistics HW1 due today HW2 available today, due Wed 1/21 Office Hours this week: Me: Friday 10:00-11:00 CSE 668 TA (Josh): Friday 3:30 CSE 002/3 Last lecture Logic gates and truth tables Implementing logic functions Canonical forms Today’s lecture

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Lecture 5

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  1. Lecture 5 • Logistics • HW1 due today • HW2 available today, due Wed 1/21 • Office Hours this week: • Me: Friday 10:00-11:00 CSE 668 • TA (Josh): Friday 3:30 CSE 002/3 • Last lecture • Logic gates and truth tables • Implementing logic functions • Canonical forms • Today’s lecture • Converting to use NAND and NOR • Minimizing functions using Boolean cubes

  2. The “WHY” slide Converting to use NAND and NOR NAND and NOR are more efficient gates than AND or OR (and therefore more common). Your computer is built almost exclusively on NAND and NOR gates. It is good to knowhow to convert any logic circuits to a NAND/NOR circuit. Pushing bubbles It is always good to remember logical/theoretical concepts visually. This is one way to remember the NAND/NOR conversion and De Morgan’s laws easily. Logic Simplification If you are building a computer or a cool gadget, you want to optimize on size and efficiency. Having extra unnecessary operations/gates is not great. We teach nice techniques to allow logic simplifications.

  3. X Z Y X Z Y NAND/NOR more common/efficient • CMOS logic gates are more common and efficient in the inverted forms • NAND, NOR, NOT • Even though Canonical forms discussed so far used AND/OR, NAND/NOR preferred for real hardware implementation X Y Z0 0 10 1 11 0 1 1 1 0

  4. CMOS NAND and NOR Gates NOR NAND

  5. X Y X' Y' (X + Y)' X' • Y'0 0 1 1 1 10 1 1 0 0 01 0 0 1 0 0 1 1 0 0 0 0 NAND and NOR (truth table) (X + Y)' = X' • Y'NOR is equivalent to AND with inputs complemented X Y X' Y' (X • Y)' X' + Y'0 0 1 1 1 10 1 1 0 1 11 0 0 1 1 1 1 1 0 0 0 0 (X • Y)' = X' + Y'NAND is equivalent to OR with inputs complemented

  6. NAND and NOR (logic gates) • de Morgan's • Standard form: A'B' = (A + B)' A' + B' = (AB)' • Inverted: A + B = (A'B')' (AB) = (A' + B')' • AND with complemented inputs  NOR • OR with complemented inputs  NAND • OR  NAND with complemented inputs • AND  NOR with complemented inputs pushing the bubble NOR NOR NAND NAND AND AND OR OR

  7. A B Z C D Converting to use NAND/NOR • Introduce inversions ("bubbles") • Introduce bubbles in pairs • Conserve inversions • Do not alter logic function • Example • AND/OR to NAND/NAND Z = AB + CD = (A'+B')'+(C'+D')' = [(A'+B')(C'+D')]' = [(AB)'(CD)']' A NAND B Z NAND C NAND D

  8. A B Z C D Converting to use NAND/NOR (con’t) • Example: AND/OR network to NOR/NOR Z = AB+CD = (A'+B')'+(C'+D')’ = [(A'+B')'+(C'+D')’]’’ = {[(A'+B')'+(C'+D')']'}' A’ NOR B’ Z NOR C’ NOR D’

  9. Converting to use NAND/NOR (con’t) Example: AND/OR network to NOR/NOR A B Z C D Z = AB+CD = (A'+B')'+(C'+D')’ = [(A'+B')'+(C'+D')’]’’ = {[(A'+B')'+(C'+D')']'}' A NOR B Z NOR C NOR D

  10. A’ A NAND B B’ Z Z NAND C’ C NAND D D’ Converting to use NAND/NOR (con’t) • Example: OR/AND to NAND/NAND

  11. Converting to use NAND/NOR (con’t) Example: OR/AND to NAND/NAND A A NAND B B Z Z NAND C C NAND D D

  12. Converting to use NAND/NOR(con’t) Example: OR/AND to NOR/NOR A NOR A B Z NOR C B Z NOR C D D

  13. Example of bubble pushing: before pushing

  14. Example of bubble pushing: NAND/NAND ‘

  15. Example of bubble pushing: NOR/NOR ‘

  16. Goal: Minimize two-level logic expression Algebraic simplification not an systematic procedure hard to know when we reached the minimum Just program it!! Computer-aided design tools require very long computation times (NP hard) heuristic methods employed – "educated guesses” Visualization methods are useful our brain is good at figuring simple things out many real-world problems are solvable by hand

  17. Key tool: The Uniting Theorem The uniting theorem  A(B’+B) = A The approach: Find some variables don’t change (the A’s above) and others do (the B’s above) Eliminate the changing variables (the B’s) A B F 0 0 1 0 1 1 1 0 0 1 1 0 A has the same value in both “on-set” rows keep A B has a different value in the two rows eliminate B F = A'B'+A'B = A'(B+B') = A'

  18. Boolean cubes Visualization tool for the uniting theorem n input variables = n-dimensional "cube" 01 11 0 1 2-cube 1-cube Y X 00 10 X 1011 1111 0111 0011 011 111 4-cube 0010 3-cube 101 Y Y Z Z W 1100 1000 000 100 X X 0000 0100

  19. Mapping truth tables onto Boolean cubes ON set = solid nodes OFF set = empty nodes Look for on-set adjacent to each other Sub-cube (a line) comprises two nodes A varies within the sub-cube; B does not This sub-cube represents B' A B F 0 0 1 0 1 0 1 0 1 1 1 0 F 01 11 B 00 10 A

  20. Example using Boolean cube Binary full-adder carry-out logic On-set is covered by the OR of three 1-subcubes (A'+A)BCin A(B+B')Cin AB(Cin'+Cin) A B Cin Cout 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 111 B 101 Cin 000 A Cout = BCin+AB+ACin

  21. Another example using Boolean cube Changed one bit from the previous function On-set is covered by the OR of one 2-D subcubes and one 3-D subcubes (A'+A)BCin A(B+B')(Cin+Cin‘) A B Cin D 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 111 101 B Cin 1 000 A D = BCin+A

  22. M-dimensional cubes in n-dimensional space In a 3-cube (three variables): A 0-cube (a single node) yields a term in 3 literals A 1-cube (a line of two nodes) yields a term in 2 literals A 2-cube (a plane of four nodes) yields a term in 1 literal A 3-cube (a cube of eight nodes) yields a constant term "1" 011 111 F(A,B,C) = m(4,5,6,7) On-set forms a square (a 2-D cube) A is asserted (true) and unchanging B and C vary This sub-cube represents the literal A 110 010 B 101 C 000 100 A

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