Non-deterministic Finite Automata
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1 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Non-deterministic Finite Automt Informtics 2A: Lecture 4 Alex Simpson School of Informtics University of Edinburgh ls@inf.ed.c.uk 24 September, / 23
2 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs 1 Non-deterministic finite utomt (NFAs) 2 Equivlence of DFAs nd NFAs The gol: converting NFAs to DFAs Worked exmple The generl construction 2 / 23
3 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Vrition on theme: Non-deterministic finite utomt In n NFA, for ny current stte nd ny symbol, there my be zero, one or mny new sttes we cn jump to. 0,1 1 0,1 0,1 0,1 0,1 q0 q1 q2 q3 q4 q5 Here there re two trnsitions for 1 from q0, nd none from q5. NFAs re useful becuse... We often wish to ignore certin detils of system, nd model just the rnge of possible behviours. Some lnguges cn be specified much more concisely by NFAs thn by DFAs. Certin useful fcts bout regulr lnguges re most conveniently proved using NFAs. 3 / 23
4 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The lnguge ccepted by n NFA The lnguge ssocited with n NFA is defined to consist of ll strings tht re ccepted under some possible execution run. Exmple: 0,1 1 0,1 0,1 0,1 0,1 q0 q1 q2 q3 q4 q5 The ssocited lnguge is {x Σ the fifth symbol from the end of x is 1} To ponder: Could you design DFA for the sme lnguge? 4 / 23
5 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Forml definition of NFAs Formlly, n NFA N with lphbet Σ consists of: A set Q of sttes, A trnsition reltion Q Σ Q, A set S Q of possible strting sttes. A set F Q of ccepting sttes. Note: ny DFA is n NFA! 5 / 23
6 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Exmple forml definition 0,1 1 0,1 0,1 0,1 0,1 q0 q1 q2 q3 q4 q5 Q = {q0, q1, q2, q3, q4, q5} = { (q0, 0, q0), (q0, 1, q0), (q0, 1, q1), (q1, 0, q2), (q1, 1, q2), (q2, 0, q3), (q2, 1, q3), (q3, 0, q4), (q3, 1, q4), (q4, 0, q5), (q4, 1, q5) } S = {q0} F = {q5} 6 / 23
7 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Forml definition of cceptnce From the forml definition of n NFA, we cn define mny-step trnsition reltion Q Σ Q: (q, ɛ, q ) iff q = q (q, xu, q ) iff q. (q, x, q ) & (q, u, q ) The lnguge ccepted by the NFA is then L(N) = {x Σ s, q. s S & (s, x, q) & q F } 7 / 23
8 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs DFAs nd NFAs The gol: converting NFAs to DFAs Worked exmple The generl construction By definition, regulr lnguge is one tht is recognized by some DFA. Every DFA is n NFA, but not vice vers. So you might wonder whether NFAs re more powerful thn DFAs. Are there lnguges tht cn be recognized by n NFA but not by ny DFA? The min gol of the lecture is to show tht the nswer is No. In fct, ny NFA cn be converted into DFA with exctly the sme ssocited lnguge. So regulr lnguges cn eqully well be defined s those tht re recognized by some NFA. This mkes it esy to prove some further useful fcts bout regulr lnguges. 8 / 23
9 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Clicker question The gol: converting NFAs to DFAs Worked exmple The generl construction Consider the following NFA over {, b, c}: b c Wht is the minimum number of sttes of n equivlent DFA? / 23
10 Solution Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The gol: converting NFAs to DFAs Worked exmple The generl construction An equivlent DFA must hve t lest 5 sttes! b c b c (grbge stte),b,c 10 / 23
11 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Clicker question The gol: converting NFAs to DFAs Worked exmple The generl construction Consider our first exmple NFA over {0, 1}: 0,1 1 0,1 0,1 0,1 0,1 q0 q1 q2 q3 q4 q5 In wht rnge is the number of sttes of the smllest equivlent DFA? / 23
12 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs NFAs to DFAs: the ide The gol: converting NFAs to DFAs Worked exmple The generl construction Given n NFA N over Σ nd string x Σ, how would you in prctice decide whether x L(N)? q0,b q2,b q1 String to process: b Ide: At ech stge in processing the string, keep trck of ll the sttes the mchine might possibly be in. 12 / 23
13 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Stge 0: initil stte The gol: converting NFAs to DFAs Worked exmple The generl construction At the strt, the NFA cn only be in the initil stte q0. q0,b q2,b q1 String to process: Processed so fr: Next symbol: b ɛ 13 / 23
14 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Stge 1: fter processing The gol: converting NFAs to DFAs Worked exmple The generl construction The NFA could now be in either q0 or q1. q0,b q2,b q1 String to process: Processed so fr: Next symbol: b b 14 / 23
15 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Stge 2: fter processing b The gol: converting NFAs to DFAs Worked exmple The generl construction The NFA could now be in either q1 or q2. q0,b q2,b q1 String to process: Processed so fr: Next symbol: b b 15 / 23
16 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Stge 3: finl stte The gol: converting NFAs to DFAs Worked exmple The generl construction The NFA could now be in q2 or q0. (It could hve got to q2 in two different wys, though we don t need to keep trck of this.) q0,b q2,b q1 String to process: Processed so fr: b b Since we ve reched the end of the input string, nd the set of possible sttes includes the ccepting stte q0, we cn sy tht the string b is ccepted by this NFA. 16 / 23
17 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The key insight The gol: converting NFAs to DFAs Worked exmple The generl construction The process we ve just described is completely deterministic process! Given ny current set of coloured sttes, nd ny input symbol in Σ, there s only one right nswer to the question: Wht should the new set of coloured sttes be? Wht s more, it s finite stte process. A stte is simply choice of coloured sttes in the originl NFA N. If N hs n sttes, there re 2 n such choices. This suggests how n NFA with n sttes cn be converted into n equivlent DFA with 2 n sttes. 17 / 23
18 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The subset construction: exmple The gol: converting NFAs to DFAs Worked exmple The generl construction Our 3-stte NFA gives rise to DFA with 2 3 = 8 sttes. The sttes of this DFA re subsets of {q0, q1, q2}. {q0,q1, q2} b q0,b q2,b q1 b {q0,q1} {q1,q2} {q0,q2} b b {q0} {q1} {q2} b,b b,b {} The ccepting sttes of this DFA re exctly those tht contin n ccepting stte of the originl NFA. 18 / 23
19 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The subset construction in generl The gol: converting NFAs to DFAs Worked exmple The generl construction Given n NFA N = (Q,, S, F ), we cn define n equivlent DFA M = (Q, δ, s, F ) (over the sme lphbet Σ) like this: Q is 2 Q, the set of ll subsets of Q. (Also written P(Q).) δ (A, u) = {q Q q A. (q, u, q ) }. (Set of ll sttes rechble vi u from some stte in A.) s = S. F = {A Q q A. q F }. It s then not hrd to prove mthemticlly tht L(M) = L(N). (See Kozen for detils.) 19 / 23
20 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The subset construction: Summry The gol: converting NFAs to DFAs Worked exmple The generl construction We ve shown tht for ny NFA N, we cn construct DFA M with the sme ssocited lnguge. So n lterntive definition of regulr lnguge would be lnguge recognized by some NFA. Often lnguge cn be specified more concisely by n NFA thn by DFA. We cn utomticlly convert n NFA to DFA ny time we wnt, t the risk of n exponentil blow-up in the number of sttes. In prctice, DFA minimiztion will often mitigte this. 20 / 23
21 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The subset construction: Summry The gol: converting NFAs to DFAs Worked exmple The generl construction We ve shown tht for ny NFA N, we cn construct DFA M with the sme ssocited lnguge. So n lterntive definition of regulr lnguge would be lnguge recognized by some NFA. Often lnguge cn be specified more concisely by n NFA thn by DFA. We cn utomticlly convert n NFA to DFA ny time we wnt, t the risk of n exponentil blow-up in the number of sttes. In prctice, DFA minimiztion will often mitigte this. But not lwys! 20 / 23
22 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Exponentil blow-up: n exmple The gol: converting NFAs to DFAs Worked exmple The generl construction Recll the exmple NFA from erlier: 0,1 1 0,1 0,1 0,1 0,1 q0 q1 q2 q3 q4 q5 Associted lnguge: {x Σ the fifth symbol from the end of x is 1} Any DFA for recognizing this lnguge will need t lest 2 5 = 32 sttes, since in effect such mchine hs to remember the lst five symbols seen. In fct the miniml DFA hs exctly 32 sttes. 21 / 23
23 Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs Simulting n NFA The gol: converting NFAs to DFAs Worked exmple The generl construction In prctice, we do not need to construct the entire stte spce of the corresponding DFA to deterministiclly simulte n NFA. Insted we merely keep trck of the current deterministic stte (i.e., subset of sttes of the NFA). E.g., in the exmple considered on slides 12 16, we store only the current stte, nd updte this s it chnges: {q0} {q0, q1} {q1, q2} {q0, q2} initilly fter reding fter reding b fter reding b This is clled just-in-time simultion. 22 / 23
24 Reding Non-deterministic finite utomt (NFAs) Equivlence of DFAs nd NFAs The gol: converting NFAs to DFAs Worked exmple The generl construction Relevnt reding: Kozen chpters 5 nd 6; J & M section (very brief). Next time: Yet nother wy of specifying regulr lnguges: vi regulr expressions (cf. Inf 1 Computtion & Logic). 23 / 23
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