Sequential System Synthesis -- Finite State Machine

1
Sequential System Synthesis-- Finite State
Machine
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Outline Finite State Machine

• Definitions
• FSM Representations
• State Transition Graph (STG)
• Flow Table
• Cube Table
• State Minimization
• Completely Specified FSM
• Incompletely Specified Machine (ISM)
• State Encoding

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Definition Finite State Machine

• A Finite State Machine (FSM) of Mealy type is a 6
  tuple ltI,S,?,S0,O,?gt
• I input alphabet, a non-empty set of input
  values
• S a non-empty, finite set of states
• ? SxI ? S, a function defines the next state
• S0 ?S, the set of initial/reset states
• O output alphabet
• ? SxI ? O, a function defines the output.
• A finite state machine of Moore type is defined
  in the same way except that the output function
  ? S ? O does not depend on the present inputs.

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Example Finite State Machine

• I x,y
• S A,B,C
• S0 A
• ?(A,x) A, ?(B,x) A, ?(C,x) C
• ?(A,y) B, ?(B,y) C, ?(C,y) A
• O 0,1
• ?(A,x) 0, ?(B,x) 0, ?(C,x) 0
• ?(A,y) 1, ?(B,y) 0, ?(C,y) 1

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FSM Representation STG

• State Transition Graph
• Node ? state (S)
• Edge ? transition (? SxI ? S, ? SxI ? O, S0)
• Direction from the current state to the next
  state
• Label input/output information for the
  transition
• Special edges edges without source, their ending
  nodes are initial states

• In sum, a STG is a weighted, directed graph where
  self loops and duplicated edges are allowed. Each
  node has at most I outgoing edges and IxS
  incoming edges. Total number of edges is ?
  IxSS0.

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Example FSM as an STG

• I x,y
• S A,B,C
• S0 A
• ?(A,x) A, ?(A,y) B,
• ?(B,x) A, ?(B,y) C,
• ?(C,x) C, ?(C,y) A
• O 0,1
• ?(A,x) 0, ?(A,y) 1,
• ?(B,x) 0, ?(B,y) 0,
• ?(C,x) 0, ?(C,y) 1

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FSM Representation Flow Table

• The flow table of an FSM ltI,S,?,S0,O,?gt is a
  SxI table, where the i-th row represents
  state Si, the j-th column represents input value
  xj. The entry at (i,j) is a 2 tuple lt?(Si,xj),
  ?(Si,xj)gt. The initial states S0 can be specified
  separately.
• Example

A,0
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FSM Representation Cube Table

• The cube table of an FSM ltI,S,?,S0,O,?gt is a
  (SxI)x4 table, where in each row, the first
  column represents input value xj, second column
  is the state Si, third column is the next state
  ?(Si,xj), and the last column is the output
  ?(Si,xj). The initial states S0 can be
• specified separately.
• Example

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FSM with Incomplete Specification

• An FSM ltI,S,?,S0,O,?gt is incompletely specified
  if ? and/or ? are incompletely specified
  functions. (I.e., they are not defined on some
  combinations of inputs and present states.)
  Otherwise, it is completely specified.
• In STG, this means there exist nodes with less
  than I outgoing edges
• In flow table, this means there exist undefined
  entries
• In cube table, this means there exist undefined
  rows.

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Make Incomplete Complete

• In STG add a dummy state called trap state.
• In flow table leave the entry empty or fill it
  by ltdont care, dont caregt.
• In cube table delete the undefined row or fill
  the last two columns by dont cares.

y/-
x/0
x/0
A
A
D
D
x/-
y/1
y/1
y/1
y/1
x/0
x/0
?
x/-
y/0
y/0
C
B
C
B
-/-
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FSM Minimization

• FSMs may contain redundant states, i.e. states
  whose function can be accomplished by other
  states.
• Removing the redundant states decreases the
  number of states in the FSM, and in general
  results in a simplification in the final
  implementation.
• State minimization is the transformation of a
  given FSM into an equivalent FSM with no
  redundant states (I.e. minimal number of states).

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Binary Relations

• Given two sets A and B, a binary relation R
  between A and B is a subset of AxB(x,y)x?A,y?B
  . We write xR y if (x,y)?R .
• Relation R ?BxB is
• reflexive iff xRx for any x?B
• symmetric iff xR y ? yR x
• anti-symmetric iff xR y, yR x ? xy
• transitive iff xR y, yR z ? xR z.
• A binary relation R ?BxB is an equivalent
  relation if it is reflexive, symmetric, and
  transitive.

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Partition into Equivalent Classes

• A partition of a set of B is a set of subsets
  Bi?B, such that
• Bi?Bi?? (?i?j)
• ?iBiB.
• Given an equivalent relation R ?BxB, the
  equivalent class of x?B is xy?BxRy.
• ?x,y?B, xy or x?y?
• If B1,B2,,Bn are all the different equivalent
  classes, then B1,B2,,Bn is a partition of B.
• An equivalent relation gives a unique partition.

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Refinement of a Partition

• Given two partitions P1B11,B21,,Bm1 and
  P2B12,B22,,Bn2 of a set B, P1 is a refinement
  of P2 if every subset (block) Bi1?Bj2 for some j.
• Let P1B11,B21,,Bm1 and P2B12,B22,,Bn2 be
  two sets of subsets of a set B, the meet of P1
  and P2 is defined as the following set
  P1P2Bi1?Bj2i1,2,m,j1,2,,n
• Theorem If P1 and P2 are partitions, then P1P2
  is also a partition of the same set B,
  furthermore, it is a refinement for both P1 and
  P2.
• Proof

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Equivalent States of an FSM

• Given two states s and t in an FSM, and a
  k-string x(x0x1xk-1), suppose zs(zs0zs1zsk-1)
  and zt(zt0zt1ztk-1) are the corresponding
  output strings when states s and t are used as
  starting state respectively. x is called a
  length-k distinguishing sequence for states s and
  t iff zsk-1? ztk-1.

xk-1x1x0
zsk-1zs1zs0
xk-1x1x0
ztk-1zt1zt0
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Equivalent States of an FSM

• Two states s and t are k-equivalent, written as
  s?kt, iff there does not exist a distinguishing
  sequence for s and t of length k or less.
• Two states are equivalent iff they are
  S-equivalent.
• Define ?k(s,t) s?kt, the set of all pairs of
  k-equivalent states.
• ?k is an equivalent relation, I.e., it is
• Reflexive s?ks
• Symmetric s?kt ? t?ks
• Transitive r?ks, s?kt ? r?kt

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Equivalent States of an FSM

• ?1(A,C),(A,E),(C,E),(B,D),(B,F),(D,F),
  (C,A),(E,A),(E,C),(D,B),(F,B),(F,D),
  (A,A),,(F,F)
• B11A,C,E
• B21B,D,F
• ?2(A,C),(A,E),(C,E),(B,D),
  (C,A),(E,A),(E,C),(D,B), (A,A),,(F,F)
• B12A,C,E
• B22B,D
• B32F
• ?3(A,C),(B,D),(C,A),(D,B),(A,A),,(F,F)

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Equivalent States Checking Theory

• Two states are equivalent iff they are
  S-equivalent.
• Theorem 1.
• Let sx and tx be the x-successors of s and t in
  an FSM, then s?k1t ? s?kt and ?x?I, sx?ktx.
• Theorem 2.
• Two states of a given FSM are equivalent iff
  they are (S)-equivalent.

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State Equivalence Checking Practice

• Goal determine ?S(S), all pairs of equivalent
  states in an FSM S.
• Partition-Refinement procedure
• PkB1k,B2k, the partition determined by ?k,
  the k-equivalent state pairs. (P0SB10)
• Idea
• For each block in Pk
• partition it (for all x?I) if its x-successors
  are not in the same block
• Refine the partition by taking the meet of these
  finer partitions
• Stop when Pk1Pk

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Example Finding Equivalent States

• P0(A,B,C,D,E,F) (1-block)
• P1(A,C,E),(B,D,F)
• for block P12(A,C,E)
• on x0 next states EEC
• blk indices 111
• Pb10(A,C,E)P12
• (no refinement)
• on x1 next states DBF
• blk indices 222
• Pb11(A,C,E)P12
• (no refinement)
• P2(A,C,E)

level
input
blk no.
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Example Finding Equivalent States

• P1(A,C,E),(B,D,F)
• P2(A,C,E)
• for block P22(B,D,F)
• on x0 next states DBB
• blk indices 222
• Pb20(B,D,F)P22
• on x1 next states FFC
• blk indices 221
• Pb21(B,D),(F)
• refine P22P22 Pb21 Pb21
• (B,D),(F)
• P2(A,C,E),(B,D),(F)

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Example Finding Equivalent States

• P1(A,C,E),(B,D,F)
• P2(A,C,E),(B,D),(F)
• for block P13(A,C,E)
• on x0 next states EEC
• blk indices 111
• Pb10(A,C,E)P13
• on x1 next states DBF
• blk indices 223
• Pb11(A,C),(E)
• refine P13P13 Pb11 Pb13
• (A,C),(E)
• P3(A,C),(E)

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Example Finding Equivalent States

• P1(A,C,E),(B,D,F)
• P2(A,C,E),(B,D),(F)
• P3(A,C),(E)
• for block P23(B,D)
• on x0 next states DB
• blk indices 22
• Pb10(B,D)P23
• on x1 next states FF
• blk indices 33
• Pb11 (B,D)P23
• P3(A,C),(E),(B,D)

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Example Finding Equivalent States

• P1(A,C,E),(B,D,F)
• P2(A,C,E),(B,D),(F)
• P3(A,C),(E),(B,D)
• for block P33(F)
• contains single state, cannot be partitioned.
• P3(A,C),(E),(B,D),(F)
• P3(A,C),(E),(B,D),(F)

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Example Finding Equivalent States

• P1(A,C,E),(B,D,F)
• P2(A,C,E),(B,D),(F)
• P3(A,C),(E),(B,D),(F)
• One can compute P4 in the same way, which gives
• P4(A,C),(E),(B,D),(F)
• P4P3 so we stop
• Conclusion
• A and C are equivalent
• B and D are equivalent

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FSM Minimization with Equivalent States

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

Equivalent classes (A,C), (B,D), (E), (F)
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Definition Finite State Machine

• Recall A Finite State Machine (FSM) of Mealy
  type is a 6 tuple ltI,S,?,S0,O,?gt
• I input alphabet, a non-empty set of input
  values
• S a non-empty, finite set of states
• ? SxI ? S, a function defines the next state
• S0 ?S, the set of initial/reset states
• O output alphabet
• ? SxI ? O, a function defines the output.

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FSM Equivalence Checking

• M1ltI1,S1,?1,S01,O1,?1gt M2ltI2,S2,?2,S02,O2,?2
  gt
• What do we mean by M1 and M2 are equivalent?
• For any input, they should produce the same
  output.
• I1 I2
• O1 O2
• How to verify that M1 and M2 are equivalent?
• Assuming that S01s01 and S02s02, then if
  there is no input string can distinguish s01 and
  s02, we claim that M1 and M2 are equivalent.

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The Product Machine

• The product machine of two FSMs,
  M1ltI,S1,?1,S01,O,?1gt and M2ltI,S2,?2,S02,O,?2gt,
  is defined as M12ltI,S12,?12,S012,0,1,?12gt
• S12S1xS2(s1,s2) s1?S1,s2?S2
• ?12 S12xI ?S12 ?12(s12,x)t12(t1,t2)
• ?12(s12,x)(?1(s1,x), ?2(s2,x)) (t1,t2)
• S012 S01xS02(s01,s02) s01?S01,s02?S02
• ?12 S12xI ?0,1 ?12(s12,x)1 iff
  ?1(s1,x)?2(s2,x)
• M1 and M2 are equivalent?M12 always outputs 1.

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Run and Reachable State

• For an FSM M1ltI1,S1,?1,S01,O1,?1gt, an input
  string x0x1xk-1 produces a sequence of states
  s0s1sk (called a run, where s0 is the starting
  state) and an output string z0z1zk-1.
• A state t is reachable from state s if there
  exists an input string that produces a run with s
  as the starting state and t as the ending state.
• The reachable states of an FSM ltI,S,?,S0,O,?gt is
  defined as t?S t is reachable from s, s?S0
• We only need to check the reachable states for
  FSM equivalence.

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FSM Equivalence Checking

• M1ltI1,S1,?1,S01,O1,?1gt M2ltI2,S2,?2,S02,O2,?2
  gt
• If I1 ? I2 or O1 ? O2 return not equivalent
• Build the product machine M12
• Start with the initial state s012, traverse the
  STG of the FSM M12
• For each reachable state in M12, if it can output
  0, return not equivalent
• Return equivalent
• To get a distinguishing sequence in the case of
  not equivalent, we need to store the predecessor
  information and do backtracking.

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Example FSM Equivalence Checking

• M1ltx,y,A,B,C,?1,A,0,1,?1gt
• M2 ltx,y,D,E,F,?2,D,0,1,?2gt
• M12ltx,y,S12,?12,(A,D),0,1,?12gt
• S12 9, however, only 3 states are reachable
  (A,D),(B,E),(C,F)
• Every reachable state outputs 1 on all inputs.
• So M1 and M2 are equivalent.

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Example FSM Equivalence Checking

• Now, M1 and M2 are not equivalent.
• Consequently, one of the reachable state (C,F)
  outputs 0 on input x.
• Backtracking to find the distinguishing sequence.