290N: The Unknown Component
Problem
Lecture 13
Subset Construction
Outline
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Subset construction
 General flow of the algorithm
 Examples
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When the number of states is exponential (2^n-1)
When the number of states is reduced
Computing reachable subsets of states
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Explicit
• Enumerating minterms
• Partitioning Boolean space
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Implicit
• Cofactoring monolithic transition relation
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Hybrid
• Using the transition relations for each state
Determinization by Subset
Construction
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Assume that ND transitions in the ND automaton happen
at the same time
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It means that, at any moment, the ND automaton is in a subset
of its states
The subset may contain more than one state
The point of determinization is to enumerate through all
the subsets of states reachable from the initial state
under any possible inputs
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Each subset of states of the ND automaton becomes a single
state of the new deterministic automaton
The languages accepted by the ND automaton and its
determinized version are the same
Determinization Algorithm
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The automaton
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Additional data structures
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The linked lists of states {s}, with the accepting states marked
Q: The FIFO queue of reached subsets of states Sk
H: The hash table mapping each reached subsets of states Sk into the
corresponding state of the determinized automaton
Initialization
Create the initial state of the determinized automaton by creating the
subset of states {s0} composed of the initial state of the ND automaton
insert {s0} into Q and H
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Computation
while Q is not empty, extract one subset of states Si from Q
for all subsets of states Sj reachable in one transition from Si
if Sj is not in H (that is, Sj has not been visited)
create the new state of the determinized automaton
make the new state accepting if some state of Sj is accepting
insert Sj into Q and into H
else find the new state corresponding to Sj using the hash table H
add the transition from Si into Sj
Example when Subset Construction
Leads to Exponential Number of States
Example when Subset Construction
Reduces Number of States
Computing Reachable Subsets
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Given a subset of states, what are other subsets of states that can be
reached in one transition from the given subset?
Naïve explicit approach (using STG)
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Enumerate the minterms of the Boolean space of conditions
For each minterm, find the subset of states reachable from the given
subset in one iteration
Collect unique subsets
State subset {2,3}
Minterm 00: 2  {1}
3  {1}
{2,3}  {1}
Minterm 01: 2  {3}
3  {1}
{2,3}  {1,3}
Minterm 10: 2  {3}
3  {1}
{2,3}  {1,3}
Minterm 11: 2  {3}
3  {1,3}
{2,3}  {1,3}
Computing Reachable Subsets
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Improved explicit approach (using STG)
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Compute partitioning on the condition space defined by states in the subset
Compute the product of partitions for all states in the subset
Each partition corresponds to one subset of next states
Collect unique subsets
This approach does not require enumerating through the minterms
State subset {2,3}
State 2 partition: (00)  {1} (01,10,11)  {3}
State 3 partition: (00,01,10)  {1} (11)  {1,3}
Product of partitions:
(00)  {1} (01,10)  {1,3}
(11)  {1,3}
Unique next state subsets: {1} and {1,3}
(00)  {1} (01,10,11)  {1,3}
Transition Relation of the Subset
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Given the subset si, compute Rsi(x,s)
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This relation for each input x, gives the set of next states {sj}
Example: State subset {2,3}
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Input variables {x1,x2}
State variables {s1,s2}
Transition relation of state 2:
R2(x1,x2,s1,s2) = x1’x2’s1’s2 + (x1+x2)s1s2
Transition relation of state 3:
R3(x1,x2,s1,s2) = x1x2s1s2 + s1’s2
Transition relation of state subset {2,3}:
R(x1,x2,s1,s2) = R2 + R3 =
x1’x2’s1’s2+ (x1 + x2)s2
Code 01
Code 10
Code 11
Computing Transition Relation of
the Subset
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Implicit approach
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The monolithic transition relation R(x,cs,ns) is available
Restrict the monolithic transition relation R(x,cs,ns) to the given
subset of states Si(cs): R(x,s) = cs [R(x,cs,ns) & Si(cs)]ns s
Hybrid approach
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The individual state transition relations Ri(x,s) are available
Add the transition relations for all states in the subset
R(x,s) = iSi Ri(x,s)
Computing Reachable Subsets
using Transition Relation of the Subset
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Compute orthonormal expansion of the transition
relation of the subset R(x,s) w.r.t. variables in {x}
R(x,s) = i [ fi(x) & gi(s) ], where
(1) fi(x) & fj(x) = 0, i  j
(2) gi(s) = gj(s)  i = j
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In the orthonormal expansion, functions gi(s) are sets
of next states reachable under conditions fi(x) from the
given subset
Computing Orthonormal Expansion
using BDD Variable Ordering
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Orthonormal expansion is
R(x,s) = i [ fi(x) & gi(s) ], where
Code 01
(1) fi(x) & fj(x) = 0, i  j
(2) gi(s) = gj(s)  i = j
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BDD represents the function as a set of disjoint Code 10
paths - condition (1)
BDD reduction guarantees merging identical
cofactors - condition (2)
Building BDD with variables {x1,x2} on top lead to
the orthonormal expansion
x1
Example: R(x1,x2,s1,s2) = x1’x2’s1’s2 + (x1+x2)s2
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s1’s2 encodes state {1} (condition x1’x2’)
s2 encodes state subset {1,3} (condition x1+x2)
x2
s1
s2
Code 11
Computing Orthonormal Expansion
using General Method
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Given a state subset Si and its transition relation R(x,s)
while R(x,s) is not empty, enumerate through the elements
of the expansion (reachable subsets):
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Extract one minterm m(x,s) from R(x,s)
Restrict m(x,s) to only input variables x (call it m(x))
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Find Sj reachable from Si under m(x): Sj(s) = x[R(x,s) & m(x)]
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Find Cij(x) labeling transition Si  Sj: Cij(x)=s[R(x,s)  Sj(s)]
Subtract this transition from R(x,s): R(x,s) = R(x,s) & NOT(Cij(x))
Example
R(x1,x2,s1,s2) = x1’x2’s1’s2 + (x1+x2)s2
Extract minterm: m(x,s) = x1x2s1s2
Restrict to variables x: m(x) = x1x2
Find the related set of states: Sj(s) = s2
(subset {1,3})
Find the related condition: Cij(x) = x1+x2
Subtract this transition from the relation: R(x1,x2,s1,s2) = x1’x2’s1’s2
Extract minterm: m(x,s) = x1’x2’s1’s2
Restrict to variables x: m(x) = x1’x2’
Find the related set of states: Sj(s) = s1’s2 (subset {1})
Find the related condition: Cij(x) = x1’x2’
Subtract this transition from the relation: R(x1,x2,s1,s2) = 0
Quit the while-loop