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- Industrial Automation And Control - IAC
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**KIET Group of Institutions - KIET**- 1 Topics
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Lexical Analysis - Part 3 Converting an NFA to a DFA Given: A non-deterministic finite state machine (NFA) Goal: Convert to an equivalent deterministic finite state machine (DFA) Why? Faster recognizer! Approach: Consider simulating a NFA. Work with sets of states. IDEA: Each state in the DFA will correspond to a set of NFA states. Worst-case: There can be an exponential number O(2N) of sets of states. The DFA can have exponentially many more states than the NFA ... but this is rare. © Harry H. Porter, 2005 1 Lexical Analysis - Part 3 NFA to DFA Input: A NFA S = States = { s0, s1, ..., sN} = SNFA ! = Move function = MoveNFA Move’(S, a) " Set of states Output: A DFA S = States = {?, ?, ..., ?} = SDFA ! = Move function = MoveDFA Move(s, a) " Single state from SDFA Main Idea: Each state in SDFA will be a set of states from the NFA SDFA = { {...}, {...} , ..., {...} } 3 5 7 a a {3} NFA a {5,7} DFA (The names of the states is arbitrary and can be changed later, if desired.) 1 © Harry H. Porter, 2005 a 2 2

Lexical Analysis - Part 3 Algorithm: Convert NFA to DFA We’ll use... MoveNFA(S,a) #-Closure(s) #-Closure(S) the transition function from NFA where s is a single state from NFA where S is a set of states from NFA We’ll construct... SDFA the set of states in the DFA Initially, we’ll set SDFA to {} Add X to SDFA where X is some set of NFA states Example: “Add {3,5,7} to SDFA” We’ll “mark” some of the states in the DFA. Marked = “We’ve done this one” ($) Unmarked = “Still need to do this one” The transition function from DFA To add an edge to the growing DFA... MoveDFA(T,b) Set MoveDFA(T,b) to S b T S ...where S and T are sets of NFA states © Harry H. Porter, 2005 3 Lexical Analysis - Part 3 Example 0 # # 1 # Start state: #-Closure (0) = © Harry H. Porter, 2005 2 4 a b # 3 5 %= {a,b} # 6 # # 7 a 8 b 9 b 10 # 4

Lexical Analysis - Part 3 Example 0 # # 1 # Start state: #-Closure (0) = {0, 1, 2, 4, 7} 2 4 a b # 3 5 %= {a,b} # 6 # 7 # a 8 b 9 b 10 # © Harry H. Porter, 2005 5 Lexical Analysis - Part 3 Example 0 # # 1 # Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A © Harry H. Porter, 2005 2 4 a b # 3 5 %= {a,b} # 6 # # 7 a 8 b 9 b 10 # 6

Lexical Analysis - Part 3 Example 0 # # 1 # Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A 2 4 a b # 3 5 %= {a,b} # 6 # 7 # a 8 b 9 b 10 # MoveDFA(A,a) = MoveDFA(A,b) = © Harry H. Porter, 2005 7 Lexical Analysis - Part 3 Example 0 # # 1 # Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A 2 4 a b # 3 5 %= {a,b} # 6 # # 7 a 8 b 9 b 10 # MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = MoveDFA(A,b) = © Harry H. Porter, 2005 8

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