FSM Decomposition using Partitions on States

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FSM Decomposition using Partitions on States
290N The Unknown Component Problem Lecture 24
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Outline

• Once upon a time
• Motivating example
• Terminology
• Definition of SP-partitions
• Decomposition using SP-partitions
• Computation of SP-partitions
• Extensions for ND FSMs

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Motivating Example
Y1 y1xy1x Y2 y3x Y3 y2y3 Z
y1y2y3

• Y1 y1y2y3xy2y3x y2y3x
• Y2 y2x y1y2y3x y1y3
• Y3 y1xy2y3x y2y3x y1y2x
• Z y2y3

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Terminology

• FSM is I, O, S, ?, ?
• A partition ? on S is a set of disjoint subsets
  of states, whose set-union is S, i.e.
• ? Bk Bi ? Bj ?, i ? j Ui Bi
  S
• Notations
• Subsets Bk are called blocks
• B?(s) denotes the block containing state s
• s ?? t iff s and t are in the same block, i.e.
  B?(s) B?(t)

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Partitions and Equivalence Relations

• Relation R on sets S and T is a subset of pairs
• R (s,t) s R t
• Equivalence relation R
• Reflexive for all s, s R s
• Symmetric if s R t, then t R s
• Transitive if s R t and t R u, then s R u
• Proposition. If R is an equivalence relation on
  S, then the set of equivalence classes is a
  partition ? on S, and conversely, every partition
  ? on S is an equivalence relation R

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Operations on Partitions

• Definition. Product of partitions, ?1 and ?2, is
  a partition ?1??2 on S such that
• s ??1??2 t iff s ? ?1 t and s ? ?2 t
• Definition. Sum of partitions, ?1 and ?2, is a
  partition ?1?2 on S such that
• s ??1?2 t iff there exist a sequence of states
  in S, ss0,s1,,sn, for which si ??1 si1 or si
  ??2 si1
• Definition. Partition ?2 is larger or equal to
  partition ?1, ?1 ? ?2, if and only if ?1 ? ?2
  ?1 (equivalently, ?1 ?2 ?2)

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Example

• S 1,2,3,4,5,6,7,8,9
• ?1 1,2 3,4 5,6 7,8,9
• ?2 1,6 2,3 4,5 7,8 9
• ?1??2 1 2 3 4 5 6 7,8 9
• ?1?2 1,2,3,4,5,6 7,8,9

1 2 3 4 5 6 7 8
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Partitions as a Lattice

• Definition. Let (S, ?) be a partially ordered
  set, and T be a subset of S. Then s ? S is the
  least upper bound (l.u.b.) of T iff
• s ? t for all t in T
• s ? t for all t in T implies that s ? s
• Definition. A lattice is partially ordered set, L
  (S, ?), which has a l.u.b. and a g.l.b.
• Definition. If L is a finite lattice, then it has
  a l.u.b. and g.l.b. for the set of all elements
  in L, denoted by 1 and 0. Element 1 is called
  identity, and 0 is called zero.
• Theorem. Partitions form a lattice.

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Example

• Lattice of subsets of S1,2,3

• Lattice of partitions on S1,2,3

1,2,3
1,2,3
1,2
1,3
2,3
1,2 3
1,32
1 2,3
3
2
1
1 2 3
?
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SP-Partitions

• Definition. A partition ? on the set of states S
  of the machine M (S,I,O,?,?) has the
  substitution property (is SP-partition) iff the
  states in any block, under all inputs, transit
  into another block, i. e. ?x s ??t ?
  ?(s,x) ?? ?(t,x)

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Example

• SP-partition
• ? 1,2 3,4,5 A,B

B
A
B
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?-Image of FSM

• Definition. Let ? be an SP-partition on the set
  of states S of the machine M (S,I,O,?,?). Then,
  the ?-image of M is the machine M? (B?,I,??)
  with ??(B?,x) B? iff ?(B?,x) ? B?.

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Example

• SP-partition
• ? 1,2 3,4,5 A,B

B
A
B
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Observations

• Observation 1. Machine M? performs only part of
  the computation of machine M, because it only
  keeps track of which block of ? contains the
  given state.
• Observation 2. If ? is an SP-partition, and we
  know the block of ?, which contains the given
  state of M, then we can compute the block of ?,
  to which this state of M is transformed by any
  input sequence. Machine M? performs this
  computation.
• On the other hand, if ? is not an
  SP-partition, then it is not possible to predict
  where the given state will go under some input
  sequences.
• In other words, an SP-partition defines an
  uncertainty about the states of M, which does not
  spread as the machine operates.

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SP-Partitions as a Sub-Lattice

• Lemma. SP-partitions are closed under product and
  sum operations.
• Theorem. SP-partitions form a sub-lattice of the
  lattice of all partitions

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Example

• ?1 1,2 3,4 5,6 7,8
• ?2 1,2,3,4 5,6,7,8
• ?3 1 2 3 4,5 6 7 8
• ?4 1,2 3,4,5,6 7,8
• ?5 1 2 3,6 4 5 7 8
• ?6 1 2 3,6 4,5 7 8

1
?4
?2
?5
?3
?1
?6
0
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Observations

• Lattice of SP-partitions shows all non-trivial
  parallel-serial decompositions of the FSM
• The lattice is a picture of FSM structure
• Algebraic properties of the lattice are reflected
  in the machine properties, and vice versa
• FSMs can be classified according to their
  lattices

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Decomposition of FSMs

• Definition. Machine M is decomposable into two
  machines, M1 and M2, if the set of i/o strings
  produced by M is equal to the set of i/o strings
  produced by the composition of M1 and M2.
• Definition. Decomposition of M into two machines,
  M1 and M2, is non-trivial if M1 and M2 have fewer
  states than M.

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Parallel Composition

• Definition. Given the state machines
• M1 (S1, I, ?1) and M2 (S2, I, ?2),
• and the output function ? S1? S2 ? I ? O,
• the parallel composition of M1 and M2 is the
  machine M (S1? S2, I, O, ?, ?), where
  ?((s1,s2), (x1, x2)) (?1(s1, x1), ?2(s2, x2)).

M
I1
?
S1
M1
O
S2
M2
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Serial Composition

• Definition. Given the state machines
• M1 (S1, I1, ?1) and M2 (S2, I2, ?2)
• with I2 S1 ? I1, a set of output symbols O,
  and an output function ? S1? S2 ? I1 ? O, then
  the serial composition of M1 and M2 with the
  output function ? is the machine M (S1? S2, I1,
  O, ?, ?), where ?((s,t),x) (?1(s, x),
  ?2(t,(s,x))).

M
I1
S1
S2
O
?
M1
M2
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Theory

• Theorem 1. Machine M has a non-trivial parallel
  decomposition iff there exist two non-trivial
  SP-partitions ?1 and ?2 on the states of M, such
  that ?1 ? ?2 0.
• Theorem 2. Machine M has a non-trivial serial
  decomposition iff there exist a non-trivial
  SP-partition ? on the states of M.

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Parallel Decomposition (part 1)

• SP-partitions
• ?1 0,1,2 3,4,5 A,B
• ?2 0,5 1,4 2,3 I,II,III

A
B
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Parallel Decomposition (part 2)
M
M
S1
M1
?
I
S2
M2
M1
M2
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Serial Decomposition (part 1)

• SP-partition
• ? 1,2 3,4,5 A,B

B
A
B
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Serial Decomposition (part 2)

• Another partition
• ? 1,3 2,4 5 a,b,c

a
c
b
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Serial Decomposition (part 3)
M
M
M1
M2
M2
M1
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Computation of All SP-partitions

• For every pair of states, s and t, compute the
  SP-partition containing these states in one
  block.
• The resulting non-trivial partitions are the
  smallest partitions of the lattice of
  SP-partitions
• Find all possible sums of the above partitions
• The resulting non-trivial partitions are the
  remaining partitions of the lattice of
  SP-partitions