Sequential System Synthesis -- Finite State Machine

Sequential System SynthesisSequential System Synthesis-- Finite State Machine-- Finite State Machine

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Outline: Outline: Finite State MachineFinite State Machine

DefinitionsDefinitions FSM RepresentationsFSM Representations

State Transition Graph (STG)State Transition Graph (STG) Flow TableFlow Table Cube TableCube Table

State MinimizationState Minimization Completely Specified FSMCompletely Specified FSM Incompletely Specified Machine (ISM)Incompletely Specified Machine (ISM)

State EncodingState Encoding

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Definition: Definition: Finite State MachineFinite State Machine

A A Finite State Machine (FSM)Finite State Machine (FSM) of of MealyMealy type is a type is a 6 tuple <6 tuple <I,S,I,S,,S,S00,O,,O,> > I: input alphabet, a non-empty set of input values;I: input alphabet, a non-empty set of input values; S: a non-empty, finite set of states;S: a non-empty, finite set of states; : SxI : SxI S, a function defines the next state; S, a function defines the next state; SS00: : S, the set of initial/reset states;S, the set of initial/reset states; O: output alphabet;O: output alphabet; : SxI : SxI O, a function defines the output. O, a function defines the output.

A finite state machine of A finite state machine of MooreMoore type is defined in type is defined in the same way except that the output function the same way except that the output function : S : S O O does not depend on the present inputs. does not depend on the present inputs.

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Example: Example: Finite State MachineFinite State Machine

I = {x,y}I = {x,y} S = {A,B,C}S = {A,B,C} SS00= {A}= {A}

(A,x) = A, (A,x) = A, (B,x) = A, (B,x) = A, (C,x) = C(C,x) = C(A,y) = B, (A,y) = B, (B,y) = C, (B,y) = C, (C,y) = A(C,y) = A

O = {0,1}O = {0,1} (A,x) = 0, (A,x) = 0, (B,x) = 0, (B,x) = 0, (C,x) = 0(C,x) = 0

(A,y) = 1, (A,y) = 1, (B,y) = 0, (B,y) = 0, (C,y) = 1(C,y) = 1

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FSM Representation: FSM Representation: STGSTG

In sum, a STG is a In sum, a STG is a weightedweighted, , directed directed graph where graph where self self loopsloops and and duplicated edgesduplicated edges are allowed. Each node has are allowed. Each node has at most |I| outgoing edges and |I|x|S| incoming edges. at most |I| outgoing edges and |I|x|S| incoming edges. Total number of edges is Total number of edges is |I|x|S|+|S |I|x|S|+|S00|. |.

State Transition Graph:State Transition Graph: Node Node state (S) state (S) Edge Edge transition ( transition ( : SxI : SxI S, S, : SxI : SxI O, S O, S00))

• Direction: from the current state to the next stateDirection: from the current state to the next state• Label: input/output information for the transitionLabel: input/output information for the transition• Special edges: edges without source, their ending Special edges: edges without source, their ending

nodes are initial statesnodes are initial states

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Example: Example: FSM as an STGFSM as an STG

I = {x,y}I = {x,y} S = {A,B,C}S = {A,B,C} SS00= {A}= {A} (A,x) = A, (A,x) = A, (A,y) = B, (A,y) = B,

(B,x) = A, (B,x) = A, (B,y) = C, (B,y) = C, (C,x) = C, (C,x) = C, (C,y) = A(C,y) = A

O = {0,1}O = {0,1} (A,x) = 0, (A,x) = 0, (A,y) = 1, (A,y) = 1,

(B,x) = 0, (B,x) = 0, (B,y) = 0, (B,y) = 0, (C,x) = 0(C,x) = 0, , (C,y) = 1(C,y) = 1

AA

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FSM Representation: FSM Representation: Flow TableFlow Table

The The flow tableflow table of an FSM < of an FSM <I,S,I,S,,S,S00,O,,O,> is a |S|x|I| > is a |S|x|I| table, where the i-th row represents state Stable, where the i-th row represents state Sii, the j-, the j-th column represents input value xth column represents input value xjj. The entry at . The entry at (i,j) is a 2 tuple (i,j) is a 2 tuple <<(S(Sii,x,xjj), ), (S(Sii,x,xjj)>)>. The initial states . The initial states SS00 can be specified separately. can be specified separately.

Example: Example: AA

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y/0 y/1

A,1A,1C,0C,0C,0C,0A,0A,0B,1B,1A,0A,0

CCBBAA

yyxx

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FSM Representation: FSM Representation: Cube TableCube Table

The The cube tablecube table of an FSM < of an FSM <I,S,I,S,,S,S00,O,,O,> is a (|S|x|I|)x4 > is a (|S|x|I|)x4 table, where in each row, the first column represents input table, where in each row, the first column represents input value value xxjj, second column is the state , second column is the state SSii, third column is the , third column is the next state next state (S(Sii,x,xjj)), and the last column is the output , and the last column is the output (S(Sii,x,xjj)). The initial states S. The initial states S00 can be can be

specified separately.specified separately. Example: Example:

AA

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11AACCyy00CCCCxx00CCBByy00AABBxx11BBAAyy00AAAAxxOONSNSPSPSII

A,1A,1C,0C,0CC

C,0C,0A,0A,0BB

B,1B,1A,0A,0AA

yyxx

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FSM with Incomplete SpecificationFSM with Incomplete Specification An FSM <An FSM <I,S,I,S,,S,S00,O,,O,> is > is incompletely specifiedincompletely specified

if if and/or and/or are incompletely specified functions. are incompletely specified functions. (I.e., they are not defined on some combinations (I.e., they are not defined on some combinations of inputs and present states.) Otherwise, it is of inputs and present states.) Otherwise, it is completely specified. completely specified. In STG, this means there exist nodes with less than |I| In STG, this means there exist nodes with less than |I|

outgoing edges;outgoing edges; In flow table, this means there exist undefined entries;In flow table, this means there exist undefined entries; In cube table, this means there exist undefined rows.In cube table, this means there exist undefined rows.

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Make Incomplete CompleteMake Incomplete Complete In STG: add a dummy state called In STG: add a dummy state called trap statetrap state.. In flow table: leave the entry empty or fill it by In flow table: leave the entry empty or fill it by

<don’t care, don’t care>.<don’t care, don’t care>. In cube table: delete the undefined row or fill the In cube table: delete the undefined row or fill the

last two columns by don’t cares. last two columns by don’t cares.

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FSM MinimizationFSM Minimization FSMs may contain FSMs may contain redundantredundant states, i.e. states states, i.e. states

whose function can be accomplished by other whose function can be accomplished by other states.states.

Removing the redundant states decreases the Removing the redundant states decreases the number of states in the FSM, and in general number of states in the FSM, and in general results in a simplification in the final results in a simplification in the final implementation.implementation.

State minimization is the transformation of a State minimization is the transformation of a given FSM into an given FSM into an equivalentequivalent FSM with no FSM with no redundant states (I.e. minimal number of states).redundant states (I.e. minimal number of states).

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Binary RelationsBinary Relations Given two sets A and B, a Given two sets A and B, a binary relationbinary relation RR

between A and B is a subset of AxB={(x,y)|between A and B is a subset of AxB={(x,y)|xxA,yA,yB}. We write xB}. We write xRR y if (x,y) y if (x,y)RR . .

Relation Relation RR BxB is BxB is reflexivereflexive iff x iff xRRx for any xx for any xB;B; symmetricsymmetric iff x iff xRR y y y yRR x; x; anti-symmetricanti-symmetric iff x iff xRR y, y y, yRR x x x=y; x=y; transitivetransitive iff x iff xRR y, y y, yRR z z x xRR z. z.

A binary relation A binary relation RR BxB is an BxB is an equivalent equivalent relationrelation if it is reflexive, symmetric, and transitive. if it is reflexive, symmetric, and transitive.

compatibility relationcompatibility relation

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Partition into Equivalent ClassesPartition into Equivalent Classes A A partitionpartition of a set of of a set of BB is a set of subsets is a set of subsets BBiiBB, ,

such that such that BBiiBBii ( (iij) j) iiBBii==BB..

Given an equivalent relation Given an equivalent relation RR BxB, the BxB, the equivalent classequivalent class of x of xB is [x]={yB is [x]={yB|xB|xRRy}. y}. x,yx,yB, [x]=[y] or [x]B, [x]=[y] or [x][y]=[y]=;; If BIf B11,B,B22,…,B,…,Bnn are all the different equivalent classes, are all the different equivalent classes,

then {Bthen {B11,B,B22,…,B,…,Bnn} is a partition of B.} is a partition of B.

An equivalent relation gives a unique partition.An equivalent relation gives a unique partition.

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Refinement of a PartitionRefinement of a Partition Given two partitions PGiven two partitions P11={B={B11

11,B,B2211,…,B,…,Bmm

11} and } and PP22={B={B11

22,B,B2222,…,B,…,Bnn

22} of a set B, P} of a set B, P11 is a is a refinementrefinement of of PP22 if every subset (block) B if every subset (block) Bii

11BBjj22 for some j. for some j.

Let PLet P11={B={B1111,B,B22

11,…,B,…,Bmm11} and P} and P22={B={B11

22,B,B2222,…,B,…,Bnn

22} be } be two sets of subsets of a set B, the two sets of subsets of a set B, the meetmeet of P of P11 and and PP22 is defined as the following set: is defined as the following set: PP11•P•P22={B={Bii

11BBjj22||

i=1,2,…m,j=1,2,…,n}i=1,2,…m,j=1,2,…,n} Theorem: If PTheorem: If P11 and P and P22 are partitions, then P are partitions, then P11•P•P22 is is

also a partition of the same set B, furthermore, it is also a partition of the same set B, furthermore, it is a refinement for both a refinement for both PP11 and P and P22..[Proof:][Proof:]

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Equivalent States of an FSMEquivalent States of an FSM Given two states s and t in an FSM, and a k-string Given two states s and t in an FSM, and a k-string

x=(xx=(x00xx11…x…xk-1k-1), suppose z), suppose zss=(z=(zss00zzss

11…z…zssk-1k-1) and z) and ztt=(z=(ztt

00zztt11…z…ztt

k-k-

11) are the corresponding output strings when states s and ) are the corresponding output strings when states s and t are used as starting state respectively. x is called a t are used as starting state respectively. x is called a length-k distinguishing sequencelength-k distinguishing sequence for states s and t iff z for states s and t iff zss

k-k-

11 z zttk-1k-1. .

xxk-1k-1…x…x11xx00ss

ttzzss

k-1k-1…z…zss11zzss

00

xxk-1k-1…x…x11xx00ss

ttzztt

k-1k-1…z…ztt11zztt

00

==== ??

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Equivalent States of an FSMEquivalent States of an FSM Two states s and t are Two states s and t are k-equivalentk-equivalent, written as s, written as skkt, t,

iff there does not exist a distinguishing sequence iff there does not exist a distinguishing sequence for s and t of length k or less. for s and t of length k or less.

Two states are Two states are equivalentequivalent iff they are |S|- iff they are |S|-equivalent.equivalent.

Define Define kk={(s,t)| s={(s,t)| skkt}, the set of all pairs of k-t}, the set of all pairs of k-equivalent states.equivalent states.

kk is an is an equivalent relationequivalent relation, I.e., it is , I.e., it is Reflexive:Reflexive: s skkss Symmetric:Symmetric: s skkt t t tkkss Transitive:Transitive: r rkks, ss, skkt t r rkktt

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Equivalent States of an FSMEquivalent States of an FSM 11={(A,C),(A,E),(C,E),(B,D),(B,F),(D,F), ={(A,C),(A,E),(C,E),(B,D),(B,F),(D,F),

(C,A),(E,A),(E,C),(D,B),(F,B),(F,D),(C,A),(E,A),(E,C),(D,B),(F,B),(F,D), (A,A),…,(F,F)}(A,A),…,(F,F)}

BB1111={A,C,E}={A,C,E}

BB2211={B,D,F}={B,D,F}

22={(A,C),(A,E),(C,E),(B,D),={(A,C),(A,E),(C,E),(B,D), (C,A),(E,A),(C,A),(E,A),(E,C),(D,B),(E,C),(D,B), (A,A),…,(F,F)}(A,A),…,(F,F)} BB11

22={A,C,E}={A,C,E} BB22

22={B,D}={B,D} BB33

22={F}={F}

33={(A,C),(B,D),={(A,C),(B,D),(C,A),(D,B),(C,A),(D,B),(A,A),…,(F,F)}(A,A),…,(F,F)}

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Equivalent States Checking: Equivalent States Checking: TheoryTheory

Two states are Two states are equivalentequivalent iff they are |S|- iff they are |S|-equivalent.equivalent.

Theorem 1. Theorem 1. Let sLet sxx and t and txx be the x-successors of s and t in an be the x-successors of s and t in an FSM, then sFSM, then sk+1k+1t t s skkt and t and xxI, sI, sxxkkttxx..

Theorem 2. Theorem 2. Two states of a given FSM are equivalent iff they Two states of a given FSM are equivalent iff they are are (|S|)-equivalent.(|S|)-equivalent.

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State Equivalence Checking: State Equivalence Checking: PracticePractice Goal: determine Goal: determine |S||S|(S), all pairs of equivalent (S), all pairs of equivalent

states in an FSM S.states in an FSM S. Partition-Refinement procedure:Partition-Refinement procedure:

PPkk={B={B11kk,B,B22

kk,…}: the partition determined by ,…}: the partition determined by kk, the k-, the k-equivalent state pairs. (Pequivalent state pairs. (P00=S={B=S={B11

00}) }) Idea:Idea:

For each block in PFor each block in Pkk • partition it (for all xpartition it (for all xI) if its x-successors are not in I) if its x-successors are not in

the same block;the same block;• Refine the partition by taking the meet of these finer Refine the partition by taking the meet of these finer

partitions;partitions;Stop when PStop when Pk+1k+1=P=Pkk

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PP00={(A,B,C,D,E,F)}={(A,B,C,D,E,F)} (1-block)(1-block)

PP11={(A,C,E),(B,D,F)}={(A,C,E),(B,D,F)}for block Pfor block P11

22=(A,C,E):=(A,C,E): on x=0: next states:on x=0: next states: EECEEC

blk indices:blk indices: 111111PbPb11

00={(A,C,E)}=P={(A,C,E)}=P1122

(no refinement)(no refinement) on x=1:on x=1: next states:next states: DBFDBF


11={(A,C,E)}=P={(A,C,E)}=P1122

(no refinement)(no refinement)PP22={(A,C,E)}={(A,C,E)}

PSPS NS, z NS, zx=0x=0 x=1x=1

AA E, 0E, 0 D, 1D, 1BB D, 0D, 0 F, 0F, 0CC E, 0E, 0 B, 1B, 1DD B, 0B, 0 F, 0F, 0EE C, 0C, 0 F, 1F, 1FF B, 0B, 0 C, 0C, 0

Example: Example: Finding Equivalent States Finding Equivalent States

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inputinput

blk no.blk no.

levellevel

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PP11={(A,C,E),(B,D,F)}={(A,C,E),(B,D,F)}PP22={(A,C,E)}={(A,C,E)}for block Pfor block P22

22=(B,D,F):=(B,D,F): on x=0: next states:on x=0: next states: DBBDBBblk indices:blk indices: 222222PbPb22

00={(B,D,F)}=P={(B,D,F)}=P2222

on x=1:on x=1: next states:next states: FFCFFCblk indices:blk indices: 221221PbPb22

11={(B,D),(F)}={(B,D),(F)} refine:refine: PP22

22=P=P222 2 ••PbPb22

11= Pb= Pb2211

={(B,D),(F)}={(B,D),(F)}PP22={(A,C,E),(B,D),(F)}={(A,C,E),(B,D),(F)}

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PP11={(A,C,E),(B,D,F)}={(A,C,E),(B,D,F)}PP22={(A,C,E),(B,D),(F)}={(A,C,E),(B,D),(F)}

for block Pfor block P1133=(A,C,E):=(A,C,E):

on x=0: next states:on x=0: next states: EECEECblk indices:blk indices: 111111PbPb11

00={(A,C,E)}=P={(A,C,E)}=P1133

on x=1:on x=1: next states:next states: DBFDBFblk indices:blk indices: 223223PbPb11

11={(A,C),(E)}={(A,C),(E)} refine:refine: PP11

33=P=P113 3 ••PbPb11

11= Pb= Pb1133

={(A,C),(E)}={(A,C),(E)}PP33={(A,C),(E)}={(A,C),(E)}

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PP33={(A,C),(E)}={(A,C),(E)}for block Pfor block P22

33=(B,D):=(B,D): on x=0: next states:on x=0: next states: DBDB


00={(B,D)}=P={(B,D)}=P2233

on x=1:on x=1: next states:next states: FFFFblk indices:blk indices: 3333

PbPb111 1 ={(B,D)}=P={(B,D)}=P22

33

PP33={(A,C),(E),(B,D)}={(A,C),(E),(B,D)}

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PP33={(A,C),(E),(B,D)}={(A,C),(E),(B,D)}for block Pfor block P33

33=(F):=(F):contains single state, contains single state, cannot be partitioned.cannot be partitioned.

PP33={(A,C),(E),(B,D),(F)}={(A,C),(E),(B,D),(F)}PP33={(A,C),(E),(B,D),(F)}={(A,C),(E),(B,D),(F)}

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PP11={(A,C,E),(B,D,F)}={(A,C,E),(B,D,F)}PP22={(A,C,E),(B,D),(F)}={(A,C,E),(B,D),(F)}PP33={(A,C),(E),(B,D),(F)}={(A,C),(E),(B,D),(F)}

One can compute POne can compute P44 in the in the same way, which givessame way, which gives

PP44={(A,C),(E),(B,D),(F)}={(A,C),(E),(B,D),(F)}

PP44=P=P33 so we stopso we stopConclusion: Conclusion:

A and C are equivalentA and C are equivalentB and D are equivalentB and D are equivalent

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FSM Minimization with Equivalent StatesFSM Minimization with Equivalent States

STG: collapse states in the same equivalent STG: collapse states in the same equivalent class to one state; update edges.class to one state; update edges.

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Equivalent classes: (A,C), (B,D), (E), (F)

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Definition: Definition: Finite State MachineFinite State Machine

RecallRecall: A Finite State Machine (FSM) of Mealy type : A Finite State Machine (FSM) of Mealy type is a 6 tuple <is a 6 tuple <I,S,I,S,,S,S00,O,,O,> > I: input alphabet, a non-empty set of input values;I: input alphabet, a non-empty set of input values; S: a non-empty, finite set of states;S: a non-empty, finite set of states; : SxI : SxI S, a function defines the next state; S, a function defines the next state; SS00: : S, the set of initial/reset states;S, the set of initial/reset states; O: output alphabet;O: output alphabet; : SxI : SxI O, a function defines the output. O, a function defines the output.

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FSM Equivalence CheckingFSM Equivalence Checking

MM11=<=<II11,S,S11,,11,S,S0011,O,O11,,11> > MM22=<=<II22,S,S22,,22,S,S00

22,O,O22,,22>> What do we mean by What do we mean by MM11 and and MM22 are equivalent? are equivalent?

For any input, they should produce the same output.For any input, they should produce the same output. II11 = = II22

OO1 1 = = OO22

How to verify that How to verify that MM11 and and MM22 are equivalent? are equivalent? Assuming that Assuming that SS00

11={s={s0011} } and and SS00

22={s={s0022}}, then if there is , then if there is

no input string can distinguish no input string can distinguish ss001 1 and and ss00

22, we claim , we claim that that MM11 and and MM2 2 are equivalent.are equivalent.

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The Product MachineThe Product Machine The The product machineproduct machine of two FSMs, of two FSMs,

MM11=<=<I,SI,S11,,11,S,S0011,O,,O,11> and > and MM22=<=<I,SI,S22,,22,S,S00

22,O,,O,22>, >, is defined as is defined as MM1212=<=<I,SI,S1212,,1212,S,S00

1212,{0,1},,{0,1},1212>> SS1212=S=S11xSxS22={(={(ss11,s,s22)| )| ss11SS11,s,s22SS22}} 1212: S: S1212xI xI SS1212 1212(s(s1212,x)=t,x)=t1212=(t=(t11,t,t22))

1212(s(s1212,x)=(,x)=(11(s(s11,x), ,x), 22(s(s22,x)) =(t,x)) =(t11,t,t22)) SS00

12 12 =S=S0011xSxS00

22={(={(ss0011,s,s00

22)| )| ss0011SS00

11,s,s0022SS00

22}}

1212: S: S1212xI xI {0,1}{0,1} 1212(s(s1212,x)=,x)=11 iff iff 11(s(s11,x)=,x)=22(s(s22,x),x)

MM1 1 and and MM22 are equivalent are equivalentMM1212 always outputs 1.always outputs 1.

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Run and Reachable StateRun and Reachable State For an FSM For an FSM MM11=<=<II11,S,S11,,11,S,S00

11,O,O11,,11>, an input string >, an input string xx00xx11…x…xk-1k-1 produces a sequence of states produces a sequence of states ss00ss11…s…skk (called a (called a runrun, where , where ss0 0 is the starting state)is the starting state) and an and an output string output string zz00zz11…z…zk-1k-1..

A state A state t is reachable from state st is reachable from state s if there exists an if there exists an input string that produces a run with s as the input string that produces a run with s as the starting state and t as the ending state. starting state and t as the ending state.

The The reachable statesreachable states of an FSM < of an FSM <I,S,I,S,,S,S00,O,,O,> is > is defined as:defined as: {t{tSS| t is reachable from s, s| t is reachable from s, sSS00}}

We only need to check the reachable states for FSM We only need to check the reachable states for FSM equivalence.equivalence.

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FSM Equivalence CheckingFSM Equivalence Checking

MM11=<=<II11,S,S11,,11,S,S0011,O,O11,,11> > MM22=<=<II22,S,S22,,22,S,S00

22,O,O22,,22>> If If II11 II22 or or OO1 1 OO22 return not equivalentreturn not equivalent;; Build the product machine Build the product machine MM1212;; Start with the initial state sStart with the initial state s00

1212, traverse the STG of the , traverse the STG of the FSM FSM MM1212;; For each reachable state in For each reachable state in MM1212, , if it can output 0, if it can output 0, return not return not

equivalentequivalent;; Return equivalentReturn equivalent;;

To get a distinguishing sequence in the case of not To get a distinguishing sequence in the case of not equivalent, we need to store the predecessor information equivalent, we need to store the predecessor information and do backtracking.and do backtracking.

ENEE 644 32

Example: Example: FSM Equivalence CheckingFSM Equivalence Checking

MM11=<=<{x,y},{A,B,C},{x,y},{A,B,C},11,{A},{0,1},,{A},{0,1},11>> MM2 2 =<=<{x,y},{D,E,F},{x,y},{D,E,F},22,{D},{0,1},,{D},{0,1},22> > MM1212=<{x,y},=<{x,y},SS1212,,1212,{(A,D)},{0,1},,{(A,D)},{0,1},1212>>

|S|S1212| = 9, however, only 3 states are reachable: (A,D),(B,E),(C,F)| = 9, however, only 3 states are reachable: (A,D),(B,E),(C,F) Every reachable state outputs 1 on all inputs.Every reachable state outputs 1 on all inputs. So MSo M11 and M and M22 are equivalent. are equivalent.

AA

CCBB

x/0

x/0

y/1x/0

y/0 y/1

DD

FFEE

x/0

x/0

y/1x/0

y/0 y/1

A,DA,D

C,FC,FB,EB,E

x/1

x/1

y/1x/1

y/1 y/1

A,EA,Ex/1

B,FB,Fy/0

••••••

ENEE 644 33

Example: Example: FSM Equivalence CheckingFSM Equivalence Checking

Now, MNow, M11 and M and M22 are not equivalent. are not equivalent. Consequently, one of the reachable state (C,F) Consequently, one of the reachable state (C,F)

outputs 0 on input x.outputs 0 on input x. Backtracking to find the distinguishing sequence.Backtracking to find the distinguishing sequence.

AA

CCBB

x/0

x/1

y/1x/0

y/0 y/1

DD

FFEE

x/0

x/0

y/1x/0

y/0 y/1

A,DA,D

C,FC,FB,EB,E

x/1

x/0

y/1x/1

y/1 y/1

A,EA,Ex/1

B,FB,Fy/0

••••••

Documents

Sequential System Synthesis -- Finite State Machine