CMSC 2833 Lecture 26. Step Expression Justification

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1 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture Minimiation with Theorems onsider the Boolean function: FF(xx, yy, ) = xxʹyyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx Step Expression Justification. xxʹyyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx Original expression. xxʹyyʹ + xyʹʹ + xyʹ + xyʹ + xxxxʹ + xxxxxx Idempotetent Law (OR Form). xxʹyyʹ + xyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx ommutative Law (OR Form). (xxʹ + xx)yyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx Distributive Law(OR Form). ()yyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx Inverse Law(OR Form). yyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx Identity Law (AND Form). yyʹ + xx(yyʹʹ + yyʹ + yyʹ + yyyy) Distributive Law (AND Form). yyʹ + xx(yyʹ(ʹ + ) + yy(ʹ + )) Distributive Law (AND Form) times. yyʹ + xx(yyʹ() + yy()) Inverse Law (OR Form) times. yyʹ + xx(yyʹ + yy) Identity Law (AND Form) times. yyʹ + xx() Inverse Law (OR Form). yyʹ + xx Identity Law(AND Form), xx + yyʹ ommutative Law(OR Form) Example.- (xx, yy, ) = xxʹyyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx Minimiation with a Karnaugh Map Simplification using K-Maps (Karnaugh Maps) Plot all the terms in the expression for FF(xx, yy, ) = xxʹyyʹ + xyʹʹ + xyʹ + xxxxʹ + xxxxxx. Plot xxʹyyʹ = =. Plot xyʹʹ = =

2 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture. Plot xyʹ = =. Plot xxxxʹ = =. Plot xxxx = =. Now, minimie the number of groupings required to cover all entries in the map and at the same time maximie the number of entries covered in each grouping.

3 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture. Find the entries that are most isolated first the entries that form a group by themselves or the entries that can be grouped with only one other entry.. ontinue to find groups containing more entries until all entries have been selected until all entries have been covered.. Read the map. In a -variable map, a group containing four () entries is represented by a single variable. The four entries that are grouped in the darkly shaded area have variable xx in common. Therefore, our minimied representation for includes xx.. Read the map. In a -variable map, a group containing two () entries is represented by the product of two variables. The lightly shaded group is formed by the product yy.

4 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture. We can now say that our minimied representation for FF is FF = xx + yy. FF = xx + yy The Karnaugh Map B A D.- (a) Two-variable map.- (b) Three-variable map.- (c) Four-variable map A Karnaugh map (K-Map) is a truth table organied so that a minimied representation of the output function can be easily recognied. A two-variable K-Map is drawn so that: MINTERM is adjacent to,,,, A three-variable K-Map is drawn so that: MINTERM is adjacent to,,,,,,,,,,,,,,,,

5 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture A four-variable K-Map is drawn so that: MINTERM is adjacent to,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Plotting A Karnaugh Map Minterm number x mm ii xx yy FF mm mm mm mm y Figure.- Plotting a two-variable K-Map FF(xx, yy) = (,) = (,) mm ii xx yy FF mm mm mm mm mm mm mm mm Figure.- Plotting a three-variable K-Map FF(xx, yy, ) = (,,,,) = (,,)

6 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture mm ii ww xx yy FF mm mm wx mm mm mm mm mm mm mm mm mm mm mm mm mm mm Figure.- Plotting a four-variable K-Map FF(ww, xx, yy, ) = (,,,,,,,,,,,) = (,,,) y Labeling a K-Map y x x y x y y x y x have x in common have x in common. y x y x have y in common. have y in common

7 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture x y x y x y x y have x in common have x in common have y in common have y in common have in common have in common

8 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture wx y w x y w w wx y w x y w w wx y w x y w w wx y w x y w w wx y wx have w in common y wx have w in common y wx have x in common y wx have x in common y wx have y in common y wx have y in common y

9 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture wx have in common y wx have in common y Mapping Examples: FF(AA, BB, ) = AB + AA. Plot the first term AB = =. Plot the second term AA. The second term designates the intersection of the AA columns with the row. In the figure below, entries shaded blue identify the AA columns and the entries shaded yellow mark the row. The entries shaded green are the intersection of the AA columns with the row. ircle the shaded entries. FF(AA, BB, ) = AB + AA

10 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture FF(AA, BB, ) = AA + AA + BB. Plot the first term AA. Mark the intersection of the AA columns with the row.. Plot the second term AA. Mark the intersection of the AA columns with the row.. Plot the third term BB. Mark the BB columns.. ircle terms.

11 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture Example: Simplify FF(xx, yy) = yy + (xxxx). Reduce FF(xx, yy) to sum-of-products form. In particular the term (xxxx) is not in SOP form. Step Expression Justification. yy + (xxxx) Original expression. yy + (xx + yy ) DeMorgan s Law (AND Form). yy + (yy + xx ) ommutative Law (OR Form). (yy + yy ) + xx Associative Law (OR Form). + xx Inverse Law (OR Form). Dominance Law (OR Form) Example FF(xx, yy) = yy + (xxxx) = Example: Simplify FF(xx, yy, ) = xx yyyy + xx yyyy + xxxx. Since the expression for F is in SOP form, we plot and combine terms on a K-Map. First, plot terms. Plot xx yyyy.. Plot xx yyyy.. Plot xxxx the intersection of the xx-columns with the -row.. ircle adjacent cells so as to minimie the number of terms.

12 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture. Read the minimied expression for FF(xx, yy, ) = xy + xxxx. Products identified in a four-variable map Looping one cell represents one minterm, giving a product of four literals Looping two cells represents a product of three literals Looping four cells represents product of two literals Looping eight cells represents a product of one literals Looping all sixteen cells produces a function that is always equal to. Example: Simplify FF(ww, xx, yy, ) = (,,,,,,,,,, )using a K-Map wx y FF(ww, xx, yy, ) = yy + ww + xx Figure. FF(ww, xx, yy, ) = (,,,,,,,,,, ) Example: Simplify FF(AA, BB,, DD) = AA BB + BB DD + AA BB DD + AA BB D Figure. FF(AA, BB,, DD) = AA BB. Identify the AA BB column. Find the intersection of AA BB column and the rows.

13 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture. Mark the two cells D Figure. FF(AA, BB,, DD) = AA BB + BB DD. Identify the DD row.. Find the intersection of a DD row and the BB columns.. Mark the two cells D Figure. FF(AA, BB,, DD) = AA BB + BB DD + AA BB DD. onvert the minterm AA BB DD to its corresponding subscript as shown below. AA BB DD

14 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture Mark cell as shown. D Figure. FF(AA, BB,, DD) = AA BB + BB DD + AA BB DD + AA BB. Identify the AB column.. Find the intersection of the AB column and the rows.. Mark the two cells D Figure. FF(AA, BB,, DD) = AADD. Find the smallest group of cells that has at least one cell that is not included in the group.. Draw a loop that includes the group.. Add the product-term to the sum for function FF.

15 omputer Organiation I. Karnaugh Maps and Minimiation MS Lecture D Figure. FF(AA, BB,, DD) = AADD + BB DD. Find the smallest group of cells that has at least one cell that is not included in the group. Please note that finding the largest possible group is the goal.. Draw a loop that includes the group.. Add the product-term to the sum for function FF. D Figure. FF(AA, BB,, DD) = AADD + BB DD + BB. Find the smallest group of cells that has at least one cell that is not included in the group. Please note that finding the largest possible group is the goal.. Draw a loop that includes the group.. Add the product-term to the sum for function FF. Prime Implicants. All the minterms of the function are covered (looped) when cells are combined.. The number of product-terms in the expression is minimied. There are no redundant terms (i.e., minterms already covered by other terms).