Sohail Aslam Compiler Construction Notes
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NFA ? DFA Construction
The algorithm is called subset construction. In the transition table of an NFA, each entry
is a set of states. In DFA, each entry is a single state. The general idea behind NFA-to-
DFA construction is that each DFA state corresponds to a set of NFA states. The DFA
uses its state to keep track of all possible states the NFA can be in after reading each
input symbol.
We will use the following operations.
• ε-closure(T):
set of NFA states reachable from some NFA state s in T on ?-transitions alone.
• move(T,a):
set of NFA states to which there is a transition on input a from some NFA state s
in set of states T.
Before it sees the first input symbol, NFA can be in any of the state in the set
ε-closure(s0), where s0 is the start state of the NFA. Suppose that exactly the states in set
T are reachable from s0 on a given sequence of input symbols. Let a be the next input
symbol. On seeing a, the NFA can move to any of the states in the set move(T,a). Let a
be the next input symbol. On seeing a, the NFA can move to any of the states in the set
move(T,a). When we allow for ε-transitions, NFA can be in any of the states in
ε-closure(move(T,a)) after seeing a.
Subset Construction
Algorithm:
Input: NFA N with state set S, alphabet Σ, start state s0, final states F
Output: DFA D with state set S’, alphabet Σ, start states,
s0’ = ε-closure(s0), final states F’ and transition table: S’ x Σ ? S’
// initially, e-closure(s0) is the only state in D states S’ and it is unmarked
s0’ = ε-closure(s0)
S’ = {s0’ } (unmarked)
while (there is some unmarked state T in S’)
mark state T
for all a in Σ do
U = ?εclosure( move(T,a) );
if U not already in S’
add U as an unmarked state to S’
Dtran(T,a) = U;
end for
end while
