Exact State Minimization of Non-Deterministic FSMs

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Exact State Minimization of Non-Deterministic FSMs
290N The Unknown Component Problem Lecture 17
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Outline

• Introduction
• The algorithm for exact state minimization
• Binate covering problem (weighted SAT)
• The implicit implementation of the algorithm
• Conclusions

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Exact State Minimization

• Given a non-deterministic FSM, find a contained
  FSM, which has the minimum number of states
• An implicit implementation of the algorithm is
  presented in
• T. Kam, T. Villa, R. Brayton, A.
  Sangiovanni-Vincentelli. A Fully Implicit
  Algorithm for Exact State Minimization. Proc. DAC
  1994, pp. 684-690

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The Algorithm

• Generate the compatibles
• Compute the constraints
• Covering conditions
• Closure conditions
• Determine prime compatibles
• Solve the resulting binate covering problem
• Derive the minimized FSM from the solution

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Compatibles

• Definition A compatible is a subset of states
  that can be combined together
• Compatibles can overlap
• Unlike classes of equivalent state of the CS-FSM
• The number of compatibles can be exponential
• Computation of the complete set of compatibles
• Find all the pairs of incompatible states
• Start with one-step incompatible pairs
• Traverse backwards to find all pairs of
  incompatible states
• Find all the subsets of states, which contain at
  least one incompatible state pair
• Find all the non-empty subsets of states, which
  do not belong to the above set (these are
  compatibles)
• Correction Pair-wise compatibility does not
  guarantee group-wise compatibility. Additional
  checking is needed to show that the pair-wise
  compatible states belonging to a compatible, as
  computed above, are group-wise compatible.

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Variables and Constraints

• Introduce a Boolean variable for each compatible
• This variable is 1 iff the compatible is selected
  to become a state after of the FSM after state
  minimization
• Constraints are of two types
• Covering constraints
• Express the property that each state of the
  original machine is contained in at least one
  selected compatible
• Example c1 c3 c6
• Closure constraints
• Express the property that, if a compatible is
  selected, then for every input there is a
  transition into another selected compatible
• Example c1 gt c5 c7 c8

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Class Sets

• Definition. For compatible C and input i, an
  implied set is a subset of states reachable from
  states in C under input i.
• Definition. A class set of compatible C is its
  implied set D satisfying the conditions
• D has more than one state
• D is not contained in C
• D is not contained in any other implied set D of
  C

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Prime Compatibles

• To solve exactly the state minimization problem,
  it is sufficient to consider a subset of
  compatibles called prime compatibles.
• It was shown that at least one minimum cover of
  the FSM consists entirely of prime compatibles
• A. Grasselli and F. Luccio. A method for
  minimizing the number of internal states in
  incompletely specified sequential networks. IRE
  Trans. Electr. Computers, EC-14(3) 350-359, June
  1965.
• Definition. A compatible C dominates a
  compatible C if
• C contains C
• The class set of C is contained in the class set
  of C
• Definition. A compatible C is a prime compatible
  if it is not dominated by any other compatible

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Binate Covering Problem

• BCP is a problem of finding an assignment of
  Boolean variables such that
• This assignment satisfies the covering and
  closure constraints expressed in the CNF form
• The number of positive polarity variables in the
  satisfying assignment is minimum
• BCP is a special case of weighted SAT problem
• The weights are 1 if the variable is positive and
  0 if the variable is negative

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Setting Up the BCP Problem

• Given is a set of prime compatibles with class
  sets computed
• Covering constraints
• For each state S, which belongs to the compatible
  C, create the sum of prime compatible Ci, which
  contains this state
• Example C1 C2 C5.
• Correction This constraint is needed only for
  the initial state.
• Closure constraints
• For each compatible C and for each class set D of
  C, create the sum of compatibles Ci containing D
• Example C gt C2 C7 C12.
• The number of constraints
• There are as many covering constraints as there
  are states
• There are as many closure constrains as there are
  class sets in all compatibles

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Implicit Implementation

• Given are the transition relation T(i,cn,ns) and
  the output relation O(i,cs,o) of the FSM
• Use positional notation to represent sets of
  states
• Introduce as many Boolean variables as there are
  states in the FSM
• A subset of states can be represented as one
  minterm
• For example, n 6, subset is s1,s2, minterm is
  011000.
• A set of subsets of states (a set of compatibles)
  can be represented by a characteristic function
• Each on-set minterm of the characteristic
  function corresponds to one compatible belonging
  to the set of compatibles

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Operations on Positional Sets

• Equality (XY) ?i (xi?yi)
• Containment (X?Y) ?i (xi ? yi)
• Strict containment (X?Y) (X?Y) !(XY)
• Complementation, union, intersection and sharp
  are Boolean operations NOT, OR, AND, and AND-NOT

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Set Containment and Set Union

• Set-containment is true iff positional-set A
  contains positional-set B
• SetContainx(A,B) ?x?A(x) ? ?B(x)
• Set-union of a number of positional sets is a
  positional set containing all the elements from
  the original positional sets
• SetUnionx(A,Y) ?i yi ? ?x(?A(x) xi)

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Maximal and Tuplen,k(x)

• Maximalx(?A) ?A(x) !?y?A(y) (y ?x)
• Tuplen,k(x) contains all positional sets of size
  n, containing exactly k ones
• Tuple(i,k)
• if (k lt 0 or i lt k ) return 0
• if ( i 0 and i k ) return 1
• return ITE( xi, Tuple(i-1,k-1), Tuple(i-1,k) )

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Output Incompatible Pairs

• Definition. A pair is output incompatible there
  exists an input, for which the outputs are not
  compatible
• OICP(y,z) Tuple1(y) Tuple1(z)
• ?i !?o O(i,y,o) O(i,z,o)

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Incompatible Pairs

• ICP(y,z) OICP(y,z)
• ICPk1(y,z) ICPk(y,z) ?i,u,vT(i,y,u)
  T(i,z,v)ICPk(u,v)
• ICPk1(y,z) ICPk(y,z) ICP(y,z)

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Incompatibles and Compatibles

• IC(c) ?y,z ICP(y,z) ?i (yizi ?ci)
• C(c) !Tuple0(c) !IC(c)

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Implied Set and Class Set

• F(c,i,n) ?p C(c) (c ?p) T(i,p,n)
• CI(c,d) ?i ?n F(c,i,n) SetUnionn(F(c,i,n),d)
  
• CCS(c,d) !Tuple0(c) !(c ?d) Maximald
  CI(c,d)

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Prime Compatibles

• Dominance(c,c) (c ?p) SetContaindCCS(c,d)
  , CCS(c,d)
• PC(c) C(c) !?cC(c) Dominance(c,c)