Digital Logic Design ENEE x. Lecture 12

Transcription

1 Digital Logic Design ENEE x Lecture 12

2 Announcements HW5 due today HW6 up on course webpage, due at the beginning of class on Wednesday, 10/28.

3 Agenda Last time: Quine-McClusky (4.8) Petrick s Method (4.9) Table Reductions (4.10) This time: Multiple Output Simplification Problem (4.12, 4.13)

4 The Multiple-Output Simplification Problem General combinational networks can have several output terminals. The output behavior of the network is described by a set of functions f 1, f 2,, f m, one for each output terminal, each involving the same input variables, x 1, x 2,, x n. The set of functions is represented by a truth table with m + n columns. Objective is to design a multiple-output network of minimal cost. Formally: A set of normal expressions that has associated with it a minimal cost as given by some cost criteria. Cost criteria: number of gates or number of gate inputs in the realization.

5 Pitfalls of Naïve Approach Multiple-output minimization problem is normally more difficult than sharing common terms in independently obtained minimal expressions. Consider: f 1 x, y, z = m 1,3,5 f 2 x, y, z = m(3,6,7) f 1 x, y, z = yz + xz f 2 x, y, z = yz + xy

6 Pitfalls of Naïve Approach Naïve Approach: y z x z f 1 y z x y f 2 Better Approach: y z x y z x y f 1 f 2

7 Multiple Output Prime Implicants A multiple-output prime implicant for a set of Boolean functions f 1, f 2,, f m, is a product term that: Is a prime implicant of one of the individual functions Is a prime implicant of one of the product functions Π i S f i, S {1,.., m}

8 Examples of Multiple Output Prime Implicants x y z f 1 f yz is a prime implicant of f 1

9 Examples of Multiple Output Prime Implicants x y z f 1 f 2 f 1 f xyz is a prime implicant of f 1 f 2

10 Multiple Output Prime Implicants Theorem: Formulas that achieve the multipleoutput minimal sum consist only of sums of multiple-output prime implicants.

11 Tagged Product Terms Term consists of two parts: a kernel and a tag. Kernel: product term involving the variables of the function Tag: Appended to denote which functions are implied by its kernel.

12 Tagged Product Term Example x y z f 1 f Algebraic Form Binary Form x y zf 1 000f 1 x yzf 1 f 2 001f 1 f 2 xyzf 1 f 2 010f 1 f 2 xyz f f 2 xyzf 1 f 2 101f 1 f 2 xyzf 1 f 2 111f 1 f 2

13 Quine-McClusky for tagged multipleoutput prime implicants f f 1 f f 1 f f f 1 f f 1 f 2 (0,1) 0 0 f 1 (0,2) 0 0 f 1 (1,3) 0 1 f 2 (1,5) 0 1 f 1 f 2 (2,3) 0 1 f 2 (3,7) 1 1 f 2 (5,7) 1 1 f 1 f 2 (1,3,5,7) 1 f 2 (1,5,3,7) 1 f 2 Why doesn t (0,1,2,3) appear? The tag of a generated term has f i iff f i appears in both the tags of the generating terms. A generating term is checked only if its tag is identical to the tag of the generated term.

14 Minimal Sums Using Petrick s Method

15 Multiple Outputs Prime Implicant Tables m 0 m 1 m 2 m 5 m 7 m 1 m 2 m 3 m 7 f 1 B x y X X f 1 C x z X X f 2 A z X X X f 2 E xz X X f 1 f 2 D yz X X X f 1 f 2 F xz X X X f 1 f 2 G xyz X X

16 Multiple Outputs Prime Implicant Tables m 0 m 1 m 2 m 5 m 7 m 1 m 2 m 3 m 7 f 1 B x y X X f 1 C x z X X f 2 A z X X X f 2 E xz X X f 1 f 2 D yz X X X f 1 f 2 F xz X X X f 1 f 2 G xyz X X When writing down p-expression, must make a distinction between primes associated with different functions.

17 Multiple Outputs Prime Implicant Tables m 0 m 1 m 2 m 5 m 7 m 1 m 2 m 3 m 7 f 1 B x y X X f 1 C x z X X f 2 A z X X X f 2 E xz X X f 1 f 2 D yz X X X f 1 f 2 F xz X X X f 1 f 2 G xyz X X P-expression: B 1 + C 1 B 1 + D 1 C 1 + G 1 D 1 + F 1 F 1 (A 2 + D 2 )(E 2 + G 2 )(A 2 + E 2 )(A 2 + F 2 )

18 Manipulating P-expression into sum of product form p = B 1 + C 1 B 1 + D 1 C 1 + G 1 D 1 + F 1 F 1 (A 2 + D 2 ) E 2 + G 2 A 2 + E 2 A 2 + F 2 = A 2 B 1 C 1 E 2 F 1 + A 2 B 1 C 1 F 1 G 2 + B 1 C 1 D 2 E 2 F 1 F 2 + A 2 B 1 E 2 F 1 G 1 + A 2 B 1 F 1 G 1 G 2 + B 1 D 2 E 2 F 1 F 2 G 1 + A 2 C 1 D 1 E 2 F 1 + A 2 C 1 D 1 F 1 G 2 + C 1 D 1 D 2 E 2 F 1 F 2 When calculating cost of a product term, we can disregard subscripts. i.e. F 1 F 2 is the same cost as F.

19 Calculating Cost of Product Terms The term A 2 B 1 F 1 G 1 G 2 yields f 1 x, y, z = x y + xz + xy z f 2 x, y, z = z + xyz The term C 1 D 1 D 2 E 2 F 1 F 2 yields f 1 x, y, z = x z + yz + xz f 2 x, y, z = yz + xy + xz

20 Calculating Cost of multiple output combinational network f 1,, f m m p α i + β j i=1 j=1 Where t 1,, t p is the set of distinct terms, β j is equal to the number of literals in t j, unless the term consists of a single literal, in which case β j = 0. Let α i be the number of terms in f i unless there is only a single term, in which case α i = 0.

21 Calculating Cost of Product Terms The term A 2 B 1 F 1 G 1 G 2 yields f 1 x, y, z = x y + xz + xy z f 2 x, y, z = z + xyz Distinct terms: x y, xz, xy z, z Beta costs: = 7 Alpha costs = = 5 Total cost: 12 The term C 1 D 1 D 2 E 2 F 1 F 2 yields f 1 x, y, z = x z + yz + xz f 2 x, y, z = yz + xy + xz Distinct terms: x z, yz, xz, xy Beta costs: = 8 Alpha costs = 3 +3 = 6 Total cost: 14

22 Minimal Sums using Table Reduction

23 Table Reduction m 0 m 1 m 2 m 5 m 7 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X X 3,4 F xz X X X 3,4 G xyz X X 4,5 Cost in case D is used in f 1 or f 2 but not both, Cost in case D is used in both f 1, f 2

24 Table Reduction m 0 m 1 m 2 m 5 m 7 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X X 3,4 F xz X X X 3,4 G xyz X X 4,5 Essential prime implicant for f 1.

25 Table Reduction m 0 m 1 m 2 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X 3,4 *1 F xz X 1 G xyz X X 4,5 m 7 column cannot be removed from f 2 part since xz is not essential for f 2. f 1 = xz + f 2 =

26 Table Reduction m 0 m 1 m 2 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X 3,4 *1 F xz X 1 G xyz X X 4,5 Dominated Rows f 1 = xz + f 2 =

27 Table Reduction m 0 m 1 m 2 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X 3,4 *1 F xz X 1 G xyz X X 4,5 Dominated Rows Row A dominates Row F Cost for Row A is not greater than cost for Row F. f 1 = xz + f 2 =

28 Table Reduction m 0 m 1 m 2 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X 3,4 G xyz X X 4,5 f 1 = xz + f 2 =

29 Table Reduction m 0 m 1 m 2 m 1 m 2 m 3 m 7 Cost B x y X X 3 C x z X X 3 A z X X X 1 E xy X X 3 D yz X X 3,4 G xyz X X 4,5 Only row that covers m 7 f 1 = xz + f 2 =

30 Table Reduction m 0 m 1 m 2 m 2 Cost B x y X X 3 C x z X X 3 *2 A z 1 E xy X 3 D yz X 3,4 G xyz X X 4,5 Delete m 1, m 3, m 7 f 1 = xz + f 2 = z +

31 Table Reduction m 0 m 1 m 2 m 2 Cost B x y X X 3 C x z X X 3 E xy X 3 D yz X 3 G xyz X X 4,5 Delete row A f 1 = xz + f 2 = z +

32 Table Reduction m 0 m 1 m 2 m 2 Cost B x y X X 3 C x z X X 3 E xy X 3 D yz X 3 G xyz X X 4,5 Row D is dominated by Row B. f 1 = xz + f 2 = z +

33 Table Reduction m 0 m 1 m 2 m 2 Cost B x y X X 3 C x z X X 3 E xy X 3 D yz X 3 G xyz X X 4,5 Row D is dominated by Row B. f 1 = xz + f 2 = z +

34 Table Reduction m 0 m 1 m 2 m 2 Cost B x y X X 3 C x z X X 3 E xy X 3 G xyz X X 4,5 Row D is dominated by Row B. f 1 = xz + f 2 = z +

35 Table Reduction m 0 m 1 m 2 m 2 Cost B x y X X 3 C x z X X 3 E xy X 3 G xyz X X 4,5 Row B is the only row covering m 1 f 1 = xz + f 2 = z +

36 Table Reduction m 0 m 1 m 2 m 2 Cost *1 B x y X X 3 C x z X X 3 E xy X 3 G xyz X X 4,5 Row B is the only row covering m 1 f 1 = xz + x y + f 2 = z +

37 Table Reduction m 2 m 2 Cost *1 B x y 3 C x z X 3 E xy X 3 G xyz X X 4,5 Delete columns m 0, m 1 f 1 = xz + x y + f 2 = z +

38 Table Reduction m 2 m 2 Cost C x z X 3 E xy X 3 G xyz X X 4,5 Delete columns m 0, m 1 f 1 = xz + x y + f 2 = z +

39 Table Reduction m 2 m 2 Cost C x z X 3 E xy X 3 G xyz X X 4,5 Cannot delete dominated rows since their cost is lower. **Table is cyclic** f 1 = xz + x y + f 2 = z +

40 Table Reduction m 2 m 2 Cost C x z X 3 E xy X 3 G xyz X X 4,5 Cannot delete dominated rows since their cost is lower. **Table is cyclic** f 1 = xz + x y + xyz f 2 = z + xyz

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