More on automata. Michael George. March 24 April 7, 2014
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1 More on utomt Michel George Mrch 24 April 7, Automt constructions Now tht we hve forml model of mchine, it is useful to mke some generl constructions. 1.1 DFA Union / Product construction Suppose we hve two mchines nd, nd we wish to construct mchine M tht recognizes L( ) L( ). To e more specific, let s let Q 1 e the set of sttes of, 01 e the strting stte of, A 1 e the ccepting sttes of, nd δ 1 e the trnsition function of. Similrly for. For exmple, we my wnt to ccept the strings tht hve n even numer of s nd nd only one using mchines tht recognizes strings with n even numer of s nd tht recognizes strings with exctly one : 11 12, Wht informtion does M need to know while processing string? If we knew wht sttes nd would e in while processing the string, we could decide whether to ccept (ccept if either one sys ccept ). So, we crete stte of M for ech pir of sttes from nd. The intent is tht if, fter prsing the string x, ends in stte 1 nd ends in stte 2, then M should end in the stte ( 1, 2 ). Thus the set of sttes of M is Q M = Q M1 Q M2 We cn drw these sttes in grid: the sttes of form the x-xis nd the sttes of form the y-xis: 1
2 M , To figure out the strting stte, we sk wht we know out the empty string. We know nd would oth e in their strting sttes (let s cll them 01 nd 02 ). So M should e in the stte ( 01, 02 ). 0M = ( 01, 02 ) If M is in ny stte ( 1, 2 ), nd it sees some chrcter, where should it trnsition? Well, we know would hve een in stte 1, nd would thus hve trnsitioned on to stte δ 1 ( 1, ) (where δ 1 is the trnsition function for ). Similrly, would trnsition to δ 2 ( 2, ). Thus M should trnsition to the stte (δ 1 ( 1, ), δ 2 ( 2, )). δ M (( 1, 2 ), ) = (δ 1 ( 1, ), δ 2 ( 2, )) Finlly, which sttes of M should e ccepting sttes? Since we re trying to ccept the union of L( ) nd L( ), we should ccept the string x if either or would. So the stte ( 1, 2 ) should e n ccept stte if either 1 is n ccept stte of or 2 is n ccept stte of : Here is the complete picture of M: A M = {( 1, 2 ) 1 A 1 or 2 A 2 } 2
3 M , We ve uilt mchine. Cn we prove tht it ccepts the correct lnguge? Our intent is tht if x cuses M to trnsition to the stte ( 1, 2 ), then would e in 1 nd would e in 2. Formlly, we could prove δ M ( 0M, x) = δ 1 ( 01, x), δ 2 ( 02, x) using strightforwrd inductive proof (you will work out the detils for closely relted prolem in the homework). Using this, we cn clculte the lnguge of M: L(M) = {x Σ δ M ( 0M, x) A M } = expnd using definitions, detils in the homework = {x x L( ) or x L( )} = L( ) L( ) which ws our gol. 1.2 NFA Union Constructing n NFA to recognise the union is much esier: we cn simply crete new strt stte with epsilon trnsitions to the strt sttes of the two originl mchines: 3
4 ɛ 11 12, ɛ NFA to DFA conversion It seems s though NFAs re more powerful thn DFAs: we hve more choices when constructing NFAs. It turns out however tht NFAs nd DFAs ccept the sme set of lnguges. Tht is, if lnguge L is recognized y n NFA N, then there is DFA M tht lso recognizes L. Given string x, M should ccept x if ny of the sttes tht N could rech while processing x re ccepting sttes. We cn use this ide to construct M: stte of M will correspond to set of sttes of N. Q M = P(Q N ) Our intent is tht if M is in the M-stte S (which is set of sttes of N), then N could e in ny of the sttes of S. M should ccept in stte S if ny of the N-sttes S re themselves ccept sttes: A M = {S Q M S such tht A N } N strts in its strt stte 0N, ut it cn immeditely perform n epsilon trnsition. Thus fter processing no input, M should e in the stte corresponding to the set of sttes rechle from 0N using epsilon trnsitions: 0M = ɛ N ( 0N ) Finlly, we need to construct the trnsition function δ M to mtch our intended interprettion of the sttes of M. If M is in stte S Q M (which is set of sttes of N), then we know N could hve een in ny of the sttes S. Tht mens tht fter processing n input, N could e in ny of the sttes rechle from. This yields the following definition: δ M (S, ) = S δ N (, ) 4
5 The key property of this construction is tht δ M ( 0M, x) = δ N ( 0N, x) (y n esy inductive proof on x). From this, we cn expnd the definitions of L(M) nd L(N) to show tht they re the sme. 1.4 DFA minimiztion One dvntge of hving cler mchine model is tht we cn reson out optimiztions. One optimiztion we could do for DFAs is to reduce the numer of sttes. For exmple, the following DFA clerly recognizes the lnguge {ɛ}: In sense, the sttes 2 nd 3 re euivlent: if we strt processing string x in either of them, we will lwys get the sme nswer. So we cn lump them together into single ig metstte : 2 1,, 3 We cn generlize this ide. Let e the euivlence reltion on Q defined y 1 2 iff x Σ, δ( 1, x) A δ( 2, x) A This formlizes the ide tht if we strt processing x in 1 or in 2, we will lwys get the sme nswer. If we know, we cn construct n euivlent mchine M min s follows: The sttes Q min re euivlence clsses of sttes of M: Q min = Q M / The ccepting sttes of Q min re the euivlence clsses of ccepting sttes of M. Note tht if 1 A M nd 2 1 then 2 A M (plug ɛ into the definition of ). The initil stte of Q min is just [ 0M ]. The trnsition function δ min is given y δ min ([], ) = [δ M (, )]. This is well-defined (proof y contrdiction). 5
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