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ENGG3190 Logic Synthesis “Sequential Circuit Synthesis”. Winter 2014 S. Areibi School of Engineering University of Guelph. Outline. Modeling synchronous circuits State-based models. Structural models. State-based optimization methods State minimization .

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### ENGG3190Logic Synthesis“Sequential Circuit Synthesis”

Winter 2014

S. Areibi

School of Engineering

University of Guelph

• Modeling synchronous circuits

• State-based models.

• Structural models.

• State-based optimization methods

• Stateminimization.

• State minimization for completely specified machines

• State minimization for incompletely specified machines.

• State encoding

• State encoding for two-level logic

• State encoding for multiple-level logic

• Structural-based optimization methods

• Retiming

• Combinational logic are very interesting and useful for designing arithmetic circuits (adders, multipliers) or in other words the Data Path of a computer.

• Combinational circuitscannot remember what happened in the past (i.e. outputs are a function of current inputs).

• In certain cases we might need to store some info before we proceed with our computation or take action based on a certain state that happened in the past.

• Sequential circuits are capable of storing information between operations.

• They are useful in designing registers, counters, and CONTROL Circuits, …

Two main types and their classification depends on the times at which their inputs are observed and their internal state changes.

• Synchronous: State changes synchronized by one or more clocks

• Asynchronous: Changes occur independently

Some sequential circuits have memory elements.

• Synchronous circuits have clocked latches.

• Asynchronous circuits may or may not have latches (e.g. C-elements), but these are not clocked.

Feedback (cyclic) is a necessary, but not sufficient condition for a circuit to be sequential.

Synthesis of sequential circuits is not as well developed as combinational. (only small circuits)

Sequential synthesis techniques are not really used in commercial software (except maybe retiming).

Sequential verification is a problem.

Time

Continuous in value & time

Analog

Digital

Discrete in value & continuous in time

Asynchronous

Synchronous

Discrete in value & time

• Synchronous

• Easier to analyze because can factor out gate delays

• Speed of the system is determined by the clock (maybe slowed!)

• Asynchronous

• Potentially faster

• Harder to analyze

We will only look at Synchronous Circuits

0

(1010, 0110)/1

----/1

(--00, 11-0)/0

1

Example

in1

in2

out 1

primary inputs

primary output

in3

in4

Latch

Present State

The above circuit is sequential since primary output depends on the state and primary inputs.

Next State

Registers and Latches (Netlist)

State Transition Graph (STG)

• The behavior of a sequential circuit is determined from:

• Inputs,

• Outputs,

• Present state of the circuit.

• The analysis of a sequential circuit consists of:

• Obtaining a suitable description that demonstrates the time sequence of inputs, outputs and states (STATE DIAGRAM).

Can describe inputs to FF with logic equations

• Similar to truth table with state added

• A sequential circuit with `m’ FFs and `n’ inputs needs 2m+n rows in state table.

State Diagram

Output

• An alternative representation to State Table

Input/Output

“Mealy Model”

State Diagram: Moore

Alternative representation for state table

Inputs

State/Output

Moore model – outputs depend on states only.

Mealy model – outputs depend on inputs & states

• Moore Machine:

• Easy to understand and easy to code.

• Might requires more states (thus more hardware).

• Mealy Machine:

• They are more general than Moore Machines

• More complex since outputs are a function of both the state and input.

• Requires less states in most cases, therefore less components.

• Choice of a model depends on the application and personal preference.

• You can transform a Mealy Machine to a Moore Machine and vice versa.

• Design starts from a specification and results in a logic diagram or a list of Boolean functions.

• The steps to be followed are:

• Derive a state diagram

• Reduce the number of states (Optimization)

• Assign binary values to the states (encoding) (Optimization)

• Obtain the binary coded state table

• Choose the type of flip flops to be used

• Derive the simplified flip flop input equations and output equations

• Draw the logic diagram

100

Present State Next State

C B A C+ B+ A+

0 0 0 x x x

0 0 1 x x x

0 1 0 1 0 1

0 1 1 1 1 0

1 0 0 0 1 0

1 0 1 0 1 1

1 1 0 1 0 0

1 1 1 x x x

101

110

011

Synthesis Example

• Implement simple count sequence: 000, 010, 011, 101, 110

• Derive the state transition table from the state transition diagram

note the don't care conditions that arise from the unused state codes

A+

C+

B+

C

C

C

X 1 0 0

X 0 X 1

X 1 1 0

X 1 X 0

X 0 0 1

X 1 X 1

A

A

A

B

B

B

Don’t cares in FSMs (cont’d)

• Synthesize logic for next state functions derive input equations for flip-flops

C+ = B

B+ = A + B’ C

A+ = A’ C’ + AC

• Some states are not reachable!! Since they are don’t cares.

• Can we fix this problem?

A+

C+

B+

C

C

C

1 1 0 0

0 0 1 1

0 1 1 0

0 1 1 0

1 0 0 1

1 1 1 1

Present State Next State

C B A C+ B+ A+

0 0 0 0 1 1

0 0 1 0 1 0

0 1 0 1 0 1

0 1 1 1 1 0

1 0 0 0 1 0

1 0 1 0 1 1

1 1 0 1 0 0

1 1 1 1 1 x

A

A

A

B

B

B

Self-starting FSMs

• Deriving state transition table from don't care assignment

100

101

110

011

Self-starting FSMs

• Start-up states

• at power-up, FSM may be in an used or invalid state

• design must guarantee that it (eventually) enters a valid state

• Self-starting solution

• design FSM so that all the invalid states eventually transition to a valid state may limit exploitation of don't cares

001

111

000

Modeling Synchronous Circuits

• State-based model

• Model circuits as finite-state machines.

• Represented by state tables/diagrams.

• Lacks a direct relation between state manipulation and corresponding area and delay variations.

• You can Applyexact/heuristic algorithms for

• State minimization.

• State encoding.

• Structural-based models

• Represent circuit by synchronouslogic network.

• You can Apply

• Retiming.

• Logic transformations(recall Multi Level Synthesis Transformations!!)

• State transition diagrams can be:

• Transformed into synchronous logic networks by state encoding.

• Recovered from synchronous logic networks by state extraction.

Combinational logic (CL)

Sequential elements

• Combinational optimization

• keep latches/registers at current positions, keep their function

• optimize combinational logic in between

• Sequential optimization

• change latch position/function (retiming)

Specification

State Minimization

Verification/Testing

State Encoding

Logic/Timing Optimization

State-Based Models: Optimization

Initial: FSM description

• provided by the designer as a state table

• extracted from netlist

• derived from HDL description

• obtained as a by-product of high-level synthesis

• translate to netlist, extract from netlist

State minimization: Combine equivalent states to reduce the number of states. For most cases, minimizing the states results in smaller logic, though this is not always true.

State assignment: Assign a unique binary code to each state. The logic structure depends on the assignment, thus this should be done optimally.

Minimization of a node: in an FSM network

Decomposition/factoring: of FSMs, collapsing/elimination

Sequential redundancy removal: using ATPG techniques

• Defined by the quintuple (, , S, , ).

• A set of primary inputspatterns .

• A set of primary outputs patterns .

• A set of states S.

• A state transition function

•  : SS.

• An output function

• : S for Mealy models

•  : S for Moore models.

• Definition: Derive a FSM with similar behavior and minimum number of states.

• Aims at reducing the number of machine states

• reduces the size of transition table.

• State reduction may reduce?

• the number of storage elements.

• the combinational logic due to reduction in transitions

• Types:

• Completely specified finite-state machines

• No don't care conditions.

• Easy to solve.

• Incompletely specified finite-state machines

• Unspecified transitions and/or outputs.

• Intractable problem.

Redundant States: Minimization

0/0

S1

S2

S1

0/0

0/0

0/0

1/1

0/0

1/1

1/1

1/1

1/1

1/0

S3

S4

1/0

S3

S4

0/0

0/0

• Can you distinguish between State S1 and S2?

• States S1, S2 seem to be equivalent!

• Hence we can reduce the machine accordingly

State Minimization for Completely-Specified FSMs

• Def: Equivalent states

• Given any input sequence the corresponding output sequences match.

• Theorem: Two states are equivalent iff

• they lead to identical outputs and

• their next-states are equivalent.

• Since equivalence is symmetric and transitive

• States can be partitioned into equivalence classes.

• Such a partition is unique.

• Two states of an FSM are:

• equivalent (or indistinguishable)

• if for each input they produce the same output and their next states are identical.

Si andSjare equivalent and

merged into a single state.

Si

1/0

Sm

Sm

1/0

0/0

Si,j

1/0

0/0

Sj

Sn

Sn

0/0

• Goal– identify and combine states that have equivalent behavior

• Reduced machine is smaller, faster, consumes less power.

• Algorithm Sketch

1.Place all states in one set

2. Initially partition set based on output behavior

3. Successively partition resulting subsets based on next state transitions

4. Repeat (3) until no further partitioning is required

• states left in the same set are equivalent

Polynomial time procedure

• Example: States A . . . I, Inputs I1, I2, Output, Z

A and D are equivalent

A and E produce same output

Q: Can they be equivalent?

A: Yes, if B and D were equivalent

and C and G were equivalent.

• Stepwise partition refinement.

• Let i , i= 1, 2, …., n denote the partitions.

• Initially

• 1 = States belong to the same block when outputs are the same for any input.

• Refine partition blocks:

While further splitting is possible

• k+1 = States belong to the same block if they were previously in the same block and their next-states are in the same block of k for any input.

• At convergence  i+1 =i

• Blocks identify equivalent states.

• 1 = {(s1, s2), (s3, s4), (s5)}.

• Split s3, s4

• 2 = {(s1, s2), (s3), (s4), (s5)}.

• 2 = is a partition into equivalence classes

• States (s1, s2) are equivalent.

Cont .. Example …

Original FSM

Minimal FSM

Original FSM

{OUT_0} = IN_0 LatchOut_v1' + IN_0 LatchOut_v3' + IN_0' LatchOut_v2'

v4.0 = IN_0 LatchOut_v1' + LatchOut_v1' LatchOut_v2'

v4.1 = IN_0' LatchOut_v2 LatchOut_v3 + IN_0' LatchOut_v2'

v4.2 = IN_0 LatchOut_v1' + IN_0' LatchOut_v1 + IN_0' LatchOut_v2 LatchOut_v3

sis> print_stats

pi= 1 po= 1 nodes= 4 latches= 3

lits(sop)= 22 #states(STG)= 5

Minimal FSM

{OUT_0} = IN_0 LatchOut_v1' + IN_0 LatchOut_v2 + IN_0' LatchOut_v2'

v3.0 = IN_0 LatchOut_v1' + LatchOut_v1' LatchOut_v2‘

v3.1 = IN_0' LatchOut_v1' + IN_0' LatchOut_v2'

sis> print_stats

pi= 1 po= 1 nodes= 3 latches= 2

lits(sop)= 14 #states(STG)= 4

Sequence Present State X=0 X=1 X=0 X=1

Reset S0 S1 S2 0 0

0 S1 S3 S4 0 0

1 S2 S5 S6 0 0

00 S3 S0 S0 0 0

01 S4 S0 S0 1 0

10 S5 S0 S0 0 0

11 S6 S0 S0 1 0

S0

0/0

1/0

S1

S2

0/0

1/0

0/0

1/0

S3

S4

S5

S6

1/0

1/0

1/0

1/0

0/0

0/1

0/0

0/1

State Minimization Example

• Sequence Detector for 010 or 110

Sequence Present State X=0 X=1 X=0 X=1

Reset S0 S1 S2 0 0

0 S1 S3 S4 0 0

1 S2 S5 S6 0 0

00 S3 S0 S0 0 0

01 S4 S0 S0 1 0

10 S5 S0 S0 0 0

11 S6 S0 S0 1 0

Method of Successive Partitions

( S0 S1 S2 S3 S4 S5 S6 )

( S0 S1 S2 S3 S5 ) ( S4 S6 )

( S0 S1 S2 ) ( S3 S5 ) ( S4 S6 )

( S0 ) ( S1 S2 ) ( S3 S5 ) ( S4 S6 )

S1 is equivalent to S2

S3 is equivalent to S5

S4 is equivalent to S6

Sequence Present State X=0 X=1 X=0 X=1

Reset S0 S1' S1' 0 0

0 + 1 S1' S3' S4' 0 0

X0 S3' S0 S0 0 0

X1 S4' S0 S0 1 0

S0

X/0

S1’

0/0

1/0

S4’

S3’

X/0

0/1

1/0

Minimized FSM

State minimized sequence detector for 010 or 110

• Polynomially-bound algorithm.

• There can be at most |S| partition refinements.

• Each refinement requires considering each state

• Complexity O(|S|2).

• Actual time may depend upon

• Data-structures.

• Implementation details.

Implication Table Method

B

C

D

E

F

G

H

I

EH

BD

CG

BD

CG

EH

AB

FG

CF

CF

CD

AC

CD

AC

BC

AG

AC

AF

EG

AH

GHDH

GH

DH

A B C D E F G H

B

C

D

E

F

G

H

I

Equivalent states:

S1: A, D, G

S2: B, C, F

S3: E, H

S4: I

EH

BD

CG

BD

CG

EH

CF

AB

FG

CF

BC

AG

CD

AC

CD

AC

AC

AF

EG

AH

GHDH

GH

DH

A B C D E F G H

Original

Minimized

Number of flip-flops is reduced

from 4 to 2.

• Next state and output functions have don’t cares.

• However, for an implementation,  and  are functions,

• thus they are uniquely defined for each input and state combination.

• Don’t cares arise when some combinations are of no interest:

• they will not occur or

• their outputs will not be observed

• For these, the next state or output may not be specified.

• (In this case,  and  are relations, but of special type. We should make sure we want these as don’t cares.)

• Such machines are called incompletely specified.

… State Minimizationfor Incompletely Specified FSMs

• Minimum finite-state machine is not unique.

• Implication relations make problem intractable.

• Example

• Replace * by 1.

• {(s1, s2), (s3), (s4), (s5)}.

Minimized to 4 states

… State Minimizationfor Incompletely Specified FSMs

• Minimum finite-state machine is not unique.

• Example

• Replace * by 0.

• {(s1, s5), (s2, s3, s4)}.

0

0

It is now completely specified

Unfortunately, there is an exponential

number of completely specified FSMs

in correspondence to the choice of the

don’t care values!!

1/1

1/1

s1

s2

0/0

By adding a dummy state this can be converted to a machine with only the output incompletely specified.

Could also specify “error” as the output when transitioning to the dummy state.

Alternatively (better for optimization), can interpret undefined next state as allowing any next state.

s1

s2

0/0

1/-

0/-

1/-

non-accepting

state

1/1

to all states and

output any value

d

-/-

s1

s2

0/0

0/-

0/-

1/-

• Binary and Gray encoding use the minimum number of bits for state register

• Gray and Johnson code:

• Two adjacent codes differ by only one bit

• Reduce simultaneous switching

• Reduce crosstalk

• Reduce glitch

• The cost & delay of FSM implementation depends on encoding of symbolic states.

• e.g., 4 states can be encoded in 4! = 24 different ways

• There are more than n! different encodings for n states.

• exploration of all encodings is impossible, therefore heuristics are used

• Heuristics Used:

• One-hot encoding

• minimum-bit change

• Uses redundant encoding in which one flip-flop is assigned to each state.

• Each state is distinguishable by its own flip-flop having a value of 1 while all others have a value of 0.

A

00

S = 0

Z = 1

0

1

S

A

B

C

B

01

S = 1

10

C

Z = 0

0

1

Z

• Assigns codes to states so that the total number of bit changes for all state transitions is minimized.

• In other words, if every arc in the state diagram has a weight that is equal to the number of bits by which the source and destination encoding differ, this strategy would select the one that minimizes the sum of all these weights.

Encoding with 6 bit

changes

Encoding with 4 bit

changes

• Inputs are A and B

• State variables are Y1 and Y2

• An output is F(A, B, Y1, Y2)

• A next state function is G(A, B, Y1, Y2)

A

Karnaugh map of

output function or

next state function

• Larger clusters

produce smaller

logic function.

• Clustered minterms

differ in one variable.

Y2

Y1

B

• Number of product terms determines number of gates.

• Number of literals in a product term determines number of gate inputs, which is proportional to number of transistors.

• Hardware α (total number of literals)

• Examples of four minterm functions:

• F1 = ABCD +ABCD +ABCD +ABCD has 16 literals

• F2 = ABC +ACD has 6 literals

Rule 1 (Priority #1), Common Dest

States that have the same next state for some fixed input should be assigned logically adjacent codes.

Fixed

Inputs

Combinational logic

Outputs

Si

Sj

Sk

Next

state

Present

state

Flip-flops

0/0

Clock

Clear

S0

S1

Rule #1: (S1, S2)

The input value of 0 will move both states into the same state S3

1/0

0/0

0/1, 1/1

S3

S2

0/1

Rule 2 (Priority #2), A Common Source

States that are the next states of the same state under logically adjacent inputs, should be assigned logically adjacent codes.

I1

I2

Inputs

Combinational logic

Outputs

Si

Sk

Sm

Fixed

present

state

Next

state

Flip-flops

0/0

Clock

Clear

S0

S1

Rule #2: (S1, S2)

They are both next states of the state S0

1/0

0/0

0/1, 1/1

S3

S2

0/1

Rule 3 (Priority #3), A Common Output

States that have the same output value for the same input value, should be assigned logically adjacent codes.

I1

I2

Inputs

Combinational logic

Outputs

Si

Sk

Sm

Fixed

present

state

Next

state

Flip-flops

0/0

0/0

Clock

Clear

S0

01

11

S1

1/0

1/0

Rule #3: (S0, S1), and (S2, S3), states S0 and S1 have the same output value 0 for the same input value 0

0/0

0/0

0/1, 1/1

0/1, 1/1

00

S3

10

S2

0/1

0/1

(Rule 1)

(Rule 1)

A

0/1

1/0

1/0

0/0

D

B

1/1

0/0

1/0

0/0

(Rule 2)

C

0 1

(Rule 2)

0

1

Verify that BC and

A = 00, B = 01, C = 10, D = 11

X

X

Y1*

Z

Y2

Y2

Y1

Y1

X

Y2*

Result: 5 products, 10 literals.

Y2

Y1

Z

Combinational logic

X

Y1*

Y2*

Y1

CLEAR

Y1

Y2

CK

Y2

32 transistors

Logic Minimization for Arbitrary State Assignment B = 01, C = 11, D = 10

X

X

Y1*

Z

Y2

Y2

Y1

Y1

X

Y2*

Current Result: 6 products, 14 literals.

Y2

Y1

Previous Result: 5 products, 10 literals.

Circuit for Arbitrary State Assignment B = 01, C = 11, D = 10

Comb.

logic

Z

X

Y1*

Y2*

Y1

CLEAR

Y1

Y2

42 transistors

CK

Y2

Best Encoding Strategy B = 01, C = 11, D = 10

• To determine the encoding with the minimum cost and delay, we need to:

• Generate K-Maps for the next-state and output functions

• Derive excitation equations from the next-state map.

• Derive output equations from the output function map.

• Implement above equations using two-level NAND gates.

• Calculate cost and delay

Optimizing Sequential Circuits by B = 01, C = 11, D = 10RetimingNetlist of Gates

Netlist of gates and registers:

Various Goals:

• Reduce clock cycle time

• Reduce area

• Reduce number of latches

Inputs

Outputs

Retiming B = 01, C = 11, D = 10

Problem

• Pure combinational optimization can be myopic since relations across register boundaries are disregarded

Solutions

• Retiming: Move register(s) so that

• clock cycle decreases, or number of registers decreases and

• input-output behavior is preserved

• RnR: Combine retiming with combinational optimization techniques

• Move latches out of the way temporarily

• optimize larger blocks of combinational

Synchronous Logic Network … B = 01, C = 11, D = 10

• Synchronous Logic Network

• Variables.

• Boolean equations.

• Synchronous delay annotation.

• Synchronous network graph

• Vertices  equations  I/O , gates.

• Edges  dependencies  nets.

• Weights  synch. delays  registers.

Circuit Representation B = 01, C = 11, D = 10

Circuit representation: G(V,E,d,w)

• V  set of gates

• E  set of wires

• d(v) = delay of gate/vertex v, (d(v)0)

• w(e) = number of registers on edge e, (w(e)0)

… Synchronous Logic Network B = 01, C = 11, D = 10

+ B = 01, C = 11, D = 10

7

Operation delay

 3

+ 7

Circuit Representation

Example: Correlator (from Leiserson and Saxe) (simplified)

0

Host

0

0

0

2

3

3

0

(x, y) = 1 if x=y

0 otherwise

Retiming Graph (Directed)

a

b

Circuit

Every cycle in Graph has at least one register i.e. no combinational loops.

7 B = 01, C = 11, D = 10

Preliminaries

For a path p :

Clock cycle

Path Delay

Path weight

Path with

w(p)=0

0

0

0

0

2

3

3

0

For correlator c = 13

Basic Operation B = 01, C = 11, D = 10

• Movement of registers from input to output of a gate or vice versa

• A positive value corresponds to shifting registers from the outputs to inputs

• This should not affect gate functionality's

• Mathematical formulation:

• A retiming of a network G(V,E,W) is:

• r: V  Z, an integer vertexlabeling, that transforms G(V,E,W) into G’(V,E,W’), where for each edge e=(u,v), the weight after retiming wr(e) is given by:

• wr(e) = w(e) + r(v) - r(u) for edge e = (u,v)

Retime by -1

Retime by +1

Summary B = 01, C = 11, D = 10

• Sequential Logic Synthesis is an important phase of the Front End Tool for VLSI Circuits.

• Optimization of Sequential Circuits involves:

• State Minimization

• State Encoding

• Retiming

• State Minimization may be applied to completely specified Machines or Incompletely specified Machines.

• State Encoding utilizes different heuristics to further minimize the logic

• Retiming plays an important role in reducing the latency of the circuit.

End Slides B = 01, C = 11, D = 10