Figure 8-1 8-1 Combinational, Single-Output Combinational, Single-Output Circuit Circuit
An equivalent ex
An equivalent example for a map wpld ample for a map wpld be the appearance of be the appearance of an obvious strucan obvious structure as shown in ture as shown in Figure 8-2 forFigure 8-2 for Karnough and Marquand maps of
Karnough and Marquand maps of Y = X
Y = X33XX11 + X + X33XX22 + X + X22XX11
8.2.3 Minimization and Mapping by Observation 8.2.3 Minimization and Mapping by Observation
For a small number of variables, a function to be minimized may be mapped by expanding the expression into a sum For a small number of variables, a function to be minimized may be mapped by expanding the expression into a sum of products form and marking a “one” on the map at
of products form and marking a “one” on the map at all points corresponding to aall points corresponding to a mintermminterm of the of the function. Pointsfunction. Points which are logical distance one apart are connected. (On a Karnough 4-variable map, these would be adjacent points.) which are logical distance one apart are connected. (On a Karnough 4-variable map, these would be adjacent points.) The resulting structures represent reduced terms. The terms of
The resulting structures represent reduced terms. The terms of the largest structures form the prime implicants of thethe largest structures form the prime implicants of the function.
function.
This is a casual approach and as the number f variables and/or the number of minterms of the function increases, the This is a casual approach and as the number f variables and/or the number of minterms of the function increases, the reliability of this method
reliability of this method decreases.decreases.
Figure 8-2
Figure 8-2 Reducing a Reducing a Simple Function Simple Function
8.2.4 Minimization by Algebraic Manipulation 8.2.4 Minimization by Algebraic Manipulation
Algebraic manipulat
Algebraic manipulation of an expressiion of an expression for a function on for a function to change it into to change it into a reduced minimal a reduced minimal form is tedious form is tedious and proneand prone to human error. The
to human error. The same disadvantages may be cited for tabular reduction techniques.same disadvantages may be cited for tabular reduction techniques.
Figure 8-3
Figure 8-3 Algebraic and Algebraic and Map Minimization of Map Minimization of a 5-VariableFunction a 5-VariableFunction
8.3
8.3 Svoboda's Svoboda's Weight Weight Algorithm Algorithm
A convenient alg
A convenient algorithm for manual or orithm for manual or programmed minimizprogrammed minimization of a functation of a function is the Weight ion is the Weight Algorithm developed Algorithm developed byby Svoboda. It is readily applied to
Svoboda. It is readily applied to manual solutions of up to manual solutions of up to eight (8) variables, depending upon eight (8) variables, depending upon the complexity of thethe complexity of the function. To perform the weight algorithm manually, proceed as
function. To perform the weight algorithm manually, proceed as follows:follows:
1.
1. Map all Map all terms wheterms where the re the function Y function Y is true is true as "1" as "1" points on points on a Marquand a Marquand Map. Map. (A Karnough (A Karnough map may map may bebe used but it is
used but it is inconvenient.) Include all "Don't Care" terms as "#" inconvenient.) Include all "Don't Care" terms as "#" points. Unmarked points are those forpoints. Unmarked points are those for which the function Y = 0.
which the function Y = 0.
2.
2. Connect alConnect all pairs l pairs of pointof points of s of logical logical distance distance one, where one, where ppii = 1 or p = 1 or pii = #, where p = #, where piiis the label of point i. Allis the label of point i. All such connections are referred to as
such connections are referred to as "edges"."edges".
3.
3. Using a secUsing a second map (for ond map (for clarity), filclarity), fill in the l in the squares corresponsquares corresponding to the ding to the minterms of minterms of the function the function Y (all pY (all pii = = 1)1) with the number of
with the number of edges connected to that minterm. Include in the edges connected to that minterm. Include in the count edges between minterms wherecount edges between minterms where both p
both pii = 1, and between minterms where one p = 1, and between minterms where one pii = #. = #.
4.
4. Scanning the Scanning the points sequentpoints sequentially from ially from the origin (P0), the origin (P0), find the mfind the minterm with interm with the lowest the lowest edge count oedge count orr weight weight ..
The first search should be for points with weight w = 0.) This is a
The first search should be for points with weight w = 0.) This is a critical point critical point of the function Y. of the function Y.
In his paper, "
In his paper, "Ordering of ImplicantsOrdering of Implicants", Svoboda discusses the natural phenomena that, when coverage ", Svoboda discusses the natural phenomena that, when coverage isis made from the origin
made from the origin forward in sequence, the probability of forward in sequence, the probability of the minimal function expression being obtainedthe minimal function expression being obtained is increased. The ordering concept is
is increased. The ordering concept is the basis of the weight the basis of the weight algorithm, fundamental product coverage, andalgorithm, fundamental product coverage, and the multiple-output minimization.
the multiple-output minimization.
5.
5. Select the Select the term representing term representing the largest the largest structure (edge, structure (edge, face, and face, and cube) which ccube) which covers that overs that minterm. [Notminterm. [Not limited to 3 dimensions.] This is a
limited to 3 dimensions.] This is a prime impl prime implicanicant of the function.t of the function.
6.
6. Record the Record the prime implicprime implicant and ant and mark all mark all minterms minterms of the of the prime impliprime implicant ascant as covered covered by labeling the points by labeling the points as
as Don't CaresDon't Cares..
7.
7. Continue scanContinue scanning from the ning from the last criticlast critical point to al point to find the nexfind the next uncovered t uncovered minterm with minterm with the lowest the lowest weight. Ifweight. If there are none with the last selected weight value, increment the weight value by one and return to the origin there are none with the last selected weight value, increment the weight value by one and return to the origin to begin scanning again.
to begin scanning again.
8.
8. Repeat steps Repeat steps 5 through 7 5 through 7 until all muntil all minterms of interms of the function the function are covered. are covered. The selected The selected minterms fminterms form theorm the critical set of minterms of the function.
critical set of minterms of the function.
The problem with the
The problem with the algorithm is that, where a algorithm is that, where a choice of structures exists, the algorithm fails. In some cases achoice of structures exists, the algorithm fails. In some cases a solution may be obtained by choosing the
solution may be obtained by choosing the structure which covers the most 1s orstructure which covers the most 1s or uncovered uncovered minterms. Where there minterms. Where there are equal choices, there is more than one solution and the algorithm fails to provide a choice of one best solution. In are equal choices, there is more than one solution and the algorithm fails to provide a choice of one best solution. In many cases, there is little practical advantage in pursuing
many cases, there is little practical advantage in pursuing more than one of more than one of the minimal solutions.the minimal solutions.
The algorithm is presented here without proof. Theorems for this and other procedures are presented in Chapter 9.
The algorithm is presented here without proof. Theorems for this and other procedures are presented in Chapter 9.
Figure 8-6 presents a
Figure 8-6 presents a step-by-step solution of a 5-variable function using Marquand Maps. Figure step-by-step solution of a 5-variable function using Marquand Maps. Figure 8-5 presents an8-5 presents an interesting 6-variable, incompletely specifie
interesting 6-variable, incompletely specified function. The latter example appears d function. The latter example appears throughout the text.throughout the text.