Sequentinal INtro and Partition Refinement

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Synthesis of Sequential Machines

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Introduction

• Study of Two Level logic Synthesis

• Finite State Machine

• Optimization – Reducing the number of states and assign binary codes.

• Restrict on study of networks of machines.

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General Introduction

• Digital electronics Circuits• Combinational Logic Ciruits• Sequential Logic Circuits

• Combinational Logic • Basic element are Gates• Circuit Behavior: Output depends on the present input• Examples: Comparator, Arithmetic Circuits

• Sequential Logic• Basic elements are Flip Flops and Latches• Further Classified into Synchronous and Asynchronous based on

the clock.• Circuit Behavior: Output depends on the past history.• Examples: Counters, Shift Registers.

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Classification of FSM

• FSM Classification

• Class A Machines / Mealy Machine• Class B Machine / Moore Machine• Class C Machine • Class D Machine• Class E Machine.

We restrict to Class A and Class B Machines.

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Class A / Mealy machine

• Mealy Machine: M(S, I, O, δ, λ) but– λ: S x I → O (i.e. output depends on both

present state and present input)– for digital circuits, typically I = {0,1}m and O = {0,1}n

• In addition, (for both Moore and Mealy machines) certain states are classified as reset or initial states

• Finite automata are similar to FSM’s, but– they do not produce any outputs,– they just accept input sequences (an accepting

set of states is given).

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Class A/Mealy Machine

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Class B /Moore Machine

• Moore Machine: is a quintuple: M(S, I, O, δ, λ)

– S: finite non-empty set of states

– I: finite non-empty set of inputs

– O: finite non-empty set of outputs

– δ: S x I → S transition (or next state) function

– λ: S → O output function (note: output only a function of present state)

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Class B /Moore Machine

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Other Class Machine

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Simplified FSM Design Flow

Specification

State Minimization

State Encoding

Verification /Testing

Logic/Timing Optimization

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Steps Involved In Designing The FSM

1. Problem Analysis2. Defining the Black Box3. Draw the State Diagram (STG)/ASM4. Draw the state Table/Flow table5. Draw the State Transition Table6. Draw the excitation Table7. Using K-map/QM method find the output and

next state decoder function.8. Implement the design.

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Example

• Design an sequential circuit which detects the sequence 0101 continuously in the data sequence using D-FF.

• Solution:

Black Box

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Step 2 State Diagram

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State Table

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State Transition Table

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Reduction Using K-map

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Implementation

CLK

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Moore Machine

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Moore Machine-State Table

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Moore Machine

• Repeat the Steps as done for Mealy Machine.

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State Reduction

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State Reduction

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Incompletely Specified FSM

• An FSM is incompletely specified if the next state function and/or the output functions are specified only for some combinations of inputs and present states.

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Finite Automata

• Similar to FSM whose task is only to declare inputs accepted or rejected.

• Classification– Deterministic– Non-deterministic

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State Equivalence and State Minimization

• K-equivalence– Two State Si and Sj, of Machine M are distinguishable

if and only if there exists at least one finite input sequence which can be applied to M,causes different output sequences, depending on whether Si and Sj is the initial state.

– If there exists for pair (Si,Sj) a distinguishing sequence of length k,the states in (Si,Sj) are said to be k-distinguishtable.

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Flow Table

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 B,0

C E,0 B,1

D F,0 B,0

E C,0 F,1

F B,0 C,0

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Contd..

• states A and B are 1-distinguishable, since a 1 input applied to A yields an output 1, versus an output 0 from B.

• states A and E are 3-distinguishable, since input sequence 111 applied to A yields output 100, versus an output 101 from E.

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Definition

• States Si and Sj of Machine M are said to be equivalent if and only if ,for every possible input sequence, the same output sequence will be produced regardless of whether Si and Sj is the initial state.

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Minimization Procedure

• Objective: – To describe a procedure for determining the

sets of equivalent states of as specified machine M.

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Step One

• Partition the states of M into subsets such that all states in the same subset are 1-equivalent.

• Consider the example shown

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Example

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 D,0

C E,0 B,1

D F,0 B,0

E C,0 F,1

F B,0 C,0

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P0=(ABCDEF)

0-distinguistablility

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 D,0

C E,0 B,1

D F,0 B,0

E C,0 F,1

F B,0 C,0

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P0=(ABCDEF) P1=(ACE)

(BDF)

• 1-equivalent• Obtained by

inspecting table and placing those states having same outputs under all inputs, in the same block.

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 D,0

C E,0 B,1

D F,0 B,0

E C,0 F,1

F B,0 C,0

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P0=(ABCDEF) P1=(ACE)

(BDF)

P2=(ACE)(BD)(F)

• 2-equivalent• Observed iff states

are 1-equivalent.

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 D,0

C E,0 B,1

D F,0 B,0

E C,0 F,1

F B,0 C,0

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P0=(ABCDEF)

P1=(ACE)(BDF)

P2=(ACE)(BD)(F)

P3=(AC)(E)(BD)(F)

P4=(AC)(E)(BD)(F)

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 D,0

C E,0 B,1

D F,0 B,0

E C,0 F,1

F B,0 C,0

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In General

• PK+1 partition is obtained from PK by placing in the same block of PK and whose Ii –successors for every possible Ii are also common block of PK .

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Theorem

• The equivalence partition is unique

• If two states Si and Sj of machine M are distinguishable then they are distinguishable by a sequence of length

n-1 or less ,where n is the number of states in M.

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Machine equivalence

• Two Machines M1 and M2 are said to be equivalent iff for every state in M1,there is a corresponding equivalent state in M2 and vice versa.

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Machine equivalence

P3=(AC)(E)(BD)(F)

(AC) s1

(E) s2

(BD) s3

(F) s4

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Reduced Flow Table

PS NS,Z

X=0 X=1

s1 s2,0 s3,1

s2 s1,0 s4,1

s3 s4,0 s3,0

s4 s3,0 s1,0

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Example

PS NS,Z

X=0 X=1

A E,0 D,1

B F,0 D,0

C E,0 B,1

D B,0 F,0

E C,0 F,1

F B,0 C,0

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State Equivalence

P1=P0={B1}={A,B,C,D,E,F}

X=0=> O0={0,0,0,0,0,0}

X=1=>O1={1,0,1,0,1,0}

P11={{A,C,E}{B,D,F}}

B1={B,D,F}

t0= {D,B,B} when x=0 => b0= {1,1,1}

t1={F,F,C} when x=1 => b1={1,1,2}

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State Equivalence

P2={{B,D},{F}}B2={A,C,E} yield no refinement.P2={{B,D},{F},{A,C,E}}For k=3 first two blocks yield no refinement

B3={A,C,E} t0={E,E,C} when x=0; b0={3,3,3}t1={D,B,F} when x=1; b1={1,1,2}P3={{B,D},{F},{A,C},{E}}

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State Equivalence

• P1=P0={{A,B,C,D,E,F,G}}

• O0=(0,0,0,0,1,0,0)

• O1=(0,1,1,0,0,0,0)

• P1={{A,B,C,D,F,G},{E}} when x=0 --- 1

• P1={{A,D,E,F,G}{BC}} when x=1 ---- 2

• Refine=> 1 Intersection 2

• P1={{A,D,F,G}{B,C}{E}}

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State Equivalence

• Consider B1

• t0={E,G,E,D} b0=(3,1,3,1) when x=0

• t1={C,A,D,G} b1=(2,1,1,1) when x=1

• Pb0={{A,F}{D,G}} Pb1={{A},{D,F,G}}

• Refine P2={{A}{D,G}{F}}.

• So P2={{A}{B,C}{D,G}{E}{F}}

• IIIlarly P3={{A}{B}{C}{D}{E}{F}{G}}

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Moore Machine

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Partition for a Moore Machine

• Similar as done with Mealy Machine.• Imagine for the different inputs same output.

• P0= (3,4,5,6)(1,2,7,8)

• P1= (3,4,5,6)(1,2)(7,8)

a=(1,2) b=(3,4,5,6) c=(7,8)

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Iterative Collapsing Approach

• Collapsed Flow table is constructed, in which each occurrence of any member of a given class as a next is replaced by its representative.

• One Checks if the partition P1 is consistent with the modified table ie all members of each equivalence class must have identical entries in the new flow table.

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Example

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Iterative Collapsed Method

• P1={{A,D,F,G},{B,C},{E}}

• We select representives A,B,E and construct the collapsed flow table.

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This is not consistent, since rows A,D,F,G are not identical, although rows D and G are and B and C are.

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• The partition is refined into

P2 = {{A},{B,C}{D,G}{E},{F}}

as before.

The next step is to collapse the original table by replacing G by block 3 representive D and C by block 2 representive B.

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This flow table is still not consistent ,since rows D,G are not identical,neither B,C.

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• At this point the partition is refined to the minimum partition ,which of course is consistent ,so

P3 = {{A},{B},{C},{D},{E},{F},{G}}

Sequentinal INtro and Partition Refinement

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