state merging

8/3/2019 State Merging

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Advanced Logic Circuits

Design of asingle state

variables

system

NALUPAs Group


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Contents

State Assignment

State Merging

Equivalent State


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3

EQUIVALENT STATE Definition

2 states are equivalent if for each possible input:

Both states give exactly same output and go to samenext state, or

Both states give exactly same output and go to sameequivalent next states


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4

Example

Present Next State Output

State x = 0 x = 1 x = 0 x = 1

a c b 0 1

b d a 0 1

c a d 1 0

d b d 1 0

States a & b have same outputfor the same input

Next states for a & b: c & d for x = 0

b & a for x = 1

If c & d are equivalent, then a &b are equivalent because sameor equivalent next states

(a, b) imply(c, d)

States c & d have same outputfor the same input

Next states for c & d:

a & b for x = 0

d & d for x = 1 If a & b are equivalent, then c &

d are equivalent because sameor equivalent next states

(c, d) imply(a, b)


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Example

Since (a, b) imply(c, d) and (c, d) imply(a, b), then

both pairs are equivalent a equivalent to b

c equivalent to d


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Next State OutputX = 0 1 X = 0 1

a b c 0 1b d b 0 0c e a 0 1d d e 0 0e e e 0 0

Example


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STATE MERGING - (Merging Diagram/Graph)

Merger graph is state reducing tool used toreduce states in the incompletely specifiedmachine. It is prepared for a primitive flow tableto determine all the possible compatible states

(maximal compatible states) and from this aminimal collection of compatibles covering allthe states.


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The merger graph is defined as follows:

1.It contains the same number of vertices as thestate table contains states.

2.Each compatible state pair is indicated by linedrawn between the two state vertices.

3.Every potentially compatible state pair, withoutputs not in conflict but whose next states aredifferent, is connected by a broken line.

4. If two states are incompatible no connecting lineis drawn.


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The flow table is required to be examined forall the possible pairs of states. All the pairs arechecked and the merger graph is obtained. Thuswe see that the merger graph displays all

possible pairs of compatible states and theirimplied pairs. Next it is necessary to checkwhether the incompatible pair(s) does notinvalidate any other implied pair. If any implied

pair is invalidated it is neglected. All theremaining valid compatible pairs form a group ofmaximal compatibles.


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The maximal compatible set can be used toconstruct the reduced flow table by assigningone row to each member of the group. However,the maximal compatibles do not necessarily

constitute the set of minimal compatibles. Theset of minimal compatibles is a smaller collectionof compatibles that will satisfy the row merging.


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The conditions that must be satisfied for rowmerging are:

the set of chosen compatibles must cover all thestates, and

the set of chosen compatibles must be closed.


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The condition of covering requires inclusion ofall the states of the primitive flow graph in the setof chosen compatibles. This condition onlydefines a lower bound on the number of states

in the minimal set. However, if none of theirimplied pairs are contained in the set, the set isnot sufficient and this is referred to as closedcondition not being satisfied. Therefore,

condition of closed covering is essentiallyrequired for row merging.


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EXAMPLE


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The diagram indicates that row A can be mergedwith H, but only if H is not merged with G,because there is no line joining A and G. Row Bcan be merged with rows F and G. Since it is

also possible to merge F and G, it follows that B,F, and G are pairwise compatible. Any set ofrows that are pairwise compatible for all pairs inthe set can be merged into a single state. Thus

states B, F, and G can be merged into a singlestate, but only if states Gand H are not merged.State C cannot be merged with any other state.States D and E canbe merged.


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Complete Merger Diagram


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State Assignment

A set of heuristic rules that attempts to reducethe cost of thee combinational logic in a finite statemachine.


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State Assignment

In order to reduce the amount of logic required, we willmake a state assignment using the following guidelines;

I. States that have the same next state (NS) for a giveninput should be given adjacent assignments (look at thecolumns of the state table).

II. States that are the next states of the same state shouldbe given adjacent

assignments (look at the rows).

III. States that have the same output for a given inputshould be given adjacent assignments.


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State Assignment


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State Assignment


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Advanced Logic Circuits

Prepared by:

Nalupa,Mark Lister V.Caindo, Bethany Rose Z.

Macapayag, Brichiruss

state merging

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