Homework. Update on website issue Reading: Chapter 7 Homework: All exercises at end of Chapter 7 Due 9/26

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1 Homework Update on website issue Reading: hapter 7 Homework: All exercises at end of hapter 7 Due 9/26 opyright c UMaine omputer Science Department / 2

2 OS 4: Foundations of omputer Science Karnaugh Maps Fall 28 ircuit Minimization 3 Problem Equivalence Boolean Approach 5 An insight Difficulty Karnaugh Maps 7 Karnaugh Maps Example Another Look at the Map What to ircle Another example Don t cares 8 Including Don t ares Example onclusion 2 Pros/ons More

3 ircuit Minimization 3 / 2 The problem Given a circuit specification, how can we make the best circuit possible? What constitutes better for circuits? Reduce the number of gates Reduce the number of inputs (pins) May also have to use only a particular set of gates Some chips have only one type of gate, and may have that chip NAND and NOR are cheaper to make Must be in a functionally complete set to be able to realize all functions, e.g.: {AND, OR, NOT}, {NAND}, {NOR} opyright c UMaine omputer Science Department 3 / 2 3

4 Equivalence Recall: two circuits are equivalent if they perform the same function, without regard for the gates used, the way the circuit is constructed, etc. Equivalence is also a more general concept Basically, two entities are equivalent if, for all possible inputs, they have the same output Equivalence allows computer scientists to use the right tool for the job by choosing the entity that best suits their needs opyright c UMaine omputer Science Department 4 / 2 Boolean Approach 5 / 2 An insight Given a Boolean circuit specification say, an SOP how would you proceed? Suppose two terms differ only by the sign of a variable one has the variable, the other the complement (negation): an replace via laws of Boolean algebra:...+(a+a)b +... (Distributive Law)...+B +... (Inverse Law) In other words, the value of the variable doesn t matter, and it can be eliminated from that pair The pair is replaced by a new term having one fewer variable Process is repeated until minimal expression found opyright c UMaine omputer Science Department 5 / 2 4

5 Difficulty with Boolean approach Problem: an be difficult to see which terms to combine, in what order D + D + D + D + D + D + D It would be better if there was some way to see which terms can be combined opyright c UMaine omputer Science Department 6 / 2 Karnaugh Maps 7 / 2 Karnaugh Maps A Karnaugh Map is a visual representation of a Boolean SOP expression Each term is represented by a cell in a table (map) Adjacent cells differ in the sign of only one variable E.g., would be adjacent to, also,... So how to draw the map? opyright c UMaine omputer Science Department 7 / 2 5

6 Example: The magic of Karnaugh Maps Suppose you want to create a circuit for the majority function and you want to minimize the circuit, keeping it an SOP. opyright c UMaine omputer Science Department 8 / 2 Example: The magic of Karnaugh Maps reate a Karnaugh Map for the number of variables that you have in the expression opyright c UMaine omputer Science Department 9 / 2 6

7 Example: The magic of Karnaugh Maps Put a in squares that correspond to the terms in the expression. Term square: if variable occurs in the term, if complement occurs E.g.: square For truth tables: Match the input pattern for rows where output is to the square s label E.g.: square opyright c UMaine omputer Science Department / 2 Example: The magic of Karnaugh Maps (Fill in during lecture.) opyright c UMaine omputer Science Department / 2 7

8 Example: The magic of Karnaugh Maps ircle groups of powers of 2 2 (2, 4, 8, etc.) until all ones have been circled. ircle the largest groups possible (ircle the groups.) opyright c UMaine omputer Science Department 2 / 2 Example: The magic of Karnaugh Maps Read the terms from the circled items, leaving out variables that have different values within the group. B (Because B is same, is same, but A = both & ) B + (Because A is same, B is same, but = both & ) (Because A is same, is same, but B = both & ) B + + A opyright c UMaine omputer Science Department 3 / 2 8

9 Another Look at the Map Values for variables are listed so that only one change of value occurs between neighbors. ( Gray code ) With 4 variables 4 rows, 4 columns. opyright c UMaine omputer Science Department 4 / 2 What to ircle ircle: groups of size 2 n, n > Don t have to circle groups of implicit circles must remember to include them in minimized expression, though! ircle largest group possible to cover each Larger groups fewer terms Group of 2 n : n inputs are eliminated A can be in > group: May be needed to increase size of multiple groups Each group: must have at least one not in any other group ircles can wrap around map: side to side, top to bottom all 4 corners opyright c UMaine omputer Science Department 5 / 2 9

10 Example: Another Karnaugh Map Design a minimal circuit for the following expression: D + D + D + D + D + D + D Draw the Karnaugh map and add the values: D opyright c UMaine omputer Science Department 6 / 2 Example: Another Karnaugh Map ircle the groups and read the terms for the minimal circuit. D D BD + D + AD opyright c UMaine omputer Science Department 7 / 2

11 Don t cares 8 / 2 Including Don t ares Put Don t ares in Karnaugh Map as D Include them only in circles if it helps opyright c UMaine omputer Science Department 8 / 2 Example A B D F D D D D D opyright c UMaine omputer Science Department 9 / 2

12 onclusion 2 / 2 Advantages and Limitations of Karnaugh Maps Pros: Easy to work with Handles don t cares no need to manipulate algebraic expression (as some other methods do) ons: Not meant for automation Difficult to use with > 4 variables 5 or 6 variables: map is cube Handle by overlaying tables, but hard to visualize > 6: hypercube opyright c UMaine omputer Science Department 2 / 2 More about Karnaugh Maps and Minimizing ircuits an be used for functions other than SOPs map is read differently Other methods exist that can be automated: Work with more variables E.g., Quine-McKluskey Method But QM is NP-hard (i.e., intractable for many-variable functions) opyright c UMaine omputer Science Department 2 / 2 2

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Minimization CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev Administrative