Finite Automata. Lecture 4 Sections Robb T. Koether. Hampden-Sydney College. Wed, Jan 21, 2015
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1 Finite Automt Lecture 4 Sections Ro T. Koether Hmpden-Sydney College Wed, Jn 21, 2015 Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
2 1 Nondeterministic Finite Automt 2 Deterministic Finite Automt 3 Converting n NFA to DFA 4 Assignment Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
3 Outline 1 Nondeterministic Finite Automt 2 Deterministic Finite Automt 3 Converting n NFA to DFA 4 Assignment Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
4 Nondeterministic Finite Automt Definition (Nondeterministic Finite Automton) A nondeterministic finite utomton, or NFA, consists of A finite set S of sttes. An input lphet Σ. A trnsition function δ tht mps to ech stte-symol pir set of next sttes. The symol my e ε. A set F of finl sttes (or ccepting sttes), where F S. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
5 The Trnsition Function Exmple (The Trnsition Function) The trnsition function cn e represented s trnsition digrm. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
6 The Trnsition Function Exmple (The Trnsition Function) Stte Next Stte 1 {1, 2} 2 {3} 3 {4} 4 The trnsition function cn lso e represented s trnsition tle. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
7 Acceptnce of Strings A string w is ccepted y n NFA if there exists pth through the NFA tht w cn follow tht leds to cceptnce. The string is rejected if ll possile pths led to rejection. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
8 Acceptnce of Strings A string w is ccepted y n NFA if there exists pth through the NFA tht w cn follow tht leds to cceptnce. The string is rejected if ll possile pths led to rejection. In the previous exmple, is there pth leding to cceptnce for the string? Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
9 Acceptnce of Strings A string w is ccepted y n NFA if there exists pth through the NFA tht w cn follow tht leds to cceptnce. The string is rejected if ll possile pths led to rejection. In the previous exmple, is there pth leding to cceptnce for the string? In the previous exmple, is there pth leding to rejection for the string? Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
10 Acceptnce of Strings A string w is ccepted y n NFA if there exists pth through the NFA tht w cn follow tht leds to cceptnce. The string is rejected if ll possile pths led to rejection. In the previous exmple, is there pth leding to cceptnce for the string? In the previous exmple, is there pth leding to rejection for the string? How out the string? Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
11 Outline 1 Nondeterministic Finite Automt 2 Deterministic Finite Automt 3 Converting n NFA to DFA 4 Assignment Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
12 Deterministic Finite Automt Definition (Deterministic Finite Automton) A determimistic finite utomton, or DFA, is the sme s n NFA except The empty string ε is not included in the set of symols. (No ε-moves.) For ech stte nd for ech symol, there is exctly one next stte. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
13 Deterministic Finite Automt Exmple (Deterministic Finite Automt) Design trnsition digrm for DFA tht ccepts L(( ) ). Write the trnsition tle for this DFA. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
14 Outline 1 Nondeterministic Finite Automt 2 Deterministic Finite Automt 3 Converting n NFA to DFA 4 Assignment Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
15 NFAs vs. DFAs In generl, it is esier to design n NFA thn it is to design DFA. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
16 NFAs vs. DFAs In generl, it is esier to design n NFA thn it is to design DFA. Therefore, common strtegy is to design n NFA for lnguge. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
17 NFAs vs. DFAs In generl, it is esier to design n NFA thn it is to design DFA. Therefore, common strtegy is to design n NFA for lnguge. But how cn computer simulte n NFA, given tht NFAs re inherently nondeterministic? Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
18 NFAs vs. DFAs In generl, it is esier to design n NFA thn it is to design DFA. Therefore, common strtegy is to design n NFA for lnguge. But how cn computer simulte n NFA, given tht NFAs re inherently nondeterministic? There is n lgorithm tht will convert n NFA into n equivlent DFA. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
19 Converting n NFA to DFA Definition (ε-closure) Let S e the sttes of NFA nd let s S e stte. The ε-closure of s is the set of ll sttes tht re rechle from s (including s itself) through ny sequences of ε-moves. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
20 Converting n NFA to DFA Given n NFA, define the sttes of DFA to e P(S), i.e., sets of sttes in the NFA. For every stte s P(S) nd every symol Σ, the trnsition function δ(s, ) is the ε-closure of ll sttes in the NFA tht re reched from sttes in s y reding. Tht is, First find ll sttes reched from s y following -moves. Then find the ε-closure of tht set of sttes. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
21 Exmple Exmple (Converting n NFA to DFA) Drw trnsition digrm for n NFA tht ccepts the lnguge of ll strings over {, } tht either contin n even numer of s or contin the sustring. It is esy to drw if we use ε-moves. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
22 Exmple Exmple (Converting n NFA to DFA) 1 ε 2 3 ε Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
23 Exmple Exmple (Converting n NFA to DFA) The digrm hs 7 sttes. Tht is, S = 7. Therefore, P(S) = 2 7 = 128. Fortuntely, we will use very few of these sttes. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
24 Exmple Exmple (Converting n NFA to DFA) Stte ε-closure 1 {1, 2, 4} 2 {2} 3 {3} 4 {4} 5 {5} 6 {6} 7 {7} Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
25 Exmple Exmple (Converting n NFA to DFA) The strt stte is ε-closure(1) = {1, 2, 4}. The trnsition function, for exmple, includes δ({1, 2, 4}, ) = ε-closure({3, 4}) = {3, 4} δ({1, 2, 4}, ) = ε-closure({2, 4, 5}) = {2, 4, 5} Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
26 Exmple Exmple (Converting n NFA to DFA) The strt stte is ε-closure(1) = {1, 2, 4}. The trnsition function, for exmple, includes δ({1, 2, 4}, ) = ε-closure({3, 4}) = {3, 4} δ({1, 2, 4}, ) = ε-closure({2, 4, 5}) = {2, 4, 5} Work out the remining trnsitions. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
27 Exmple Exmple (Converting n NFA to DFA) Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
28 Exmple Exmple (Converting n NFA to DFA) A reduced DFA Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
29 Exmple Exmple (Converting n NFA to DFA) Relel the sttes Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
30 Outline 1 Nondeterministic Finite Automt 2 Deterministic Finite Automt 3 Converting n NFA to DFA 4 Assignment Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
31 Assignment Assignment Red Sections p. 151: 2(i), 3, 4, 5(). p. 166: 1. Ro T. Koether (Hmpden-Sydney College) Finite Automt Wed, Jn 21, / 23
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