NONDETERMINISTIC FSA


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1 Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is epted only y strting t stte q,nd x = is epted only y strting stte q 2. Tw o strtsttes q nd q 2 : q, q 3 q 2 L(M) ={,,,,,, } (2) Nonunique trnsitions; δ(q i, j )isset of sttes Q. The lnguge L(M) ={x:xtkes M for some hoie of suessive trnsitions from the strtstte to some finlstte nd ll of x is proessed}. δ(q, ) ={q,q 2 } q q q 2 x= n e fully proessed in only 2 wys, nd one of them epts x. For eh NFS M, there is n equivlent deterministi FS M suh tht L(M) =L(M ).
2 6.2 REVERSING N FS MY CRETE NFS, C B M; L(M) = (+)* + ( + )*, C B M r ; L(M r) = ( + )* +(+)* Reversing n FS: Reverse diretion of eh trnsition (my rete nondeterminism). Mke the strtstte the finlstte. Mke eh finlstte strtstte (my rete nondeterminism). Reverse of Lnguge L: L r ={x r :x L}, where x r = k k 2 if x = 2 k k. If L is regulr, then L r is lso regulr. In M: q q 2 q 2 3 k q k k q k In M r : q q 2 q 2 3 k q k k q k Question: If M hs n errorstte, then wht will hppen to it in M r? Cn the reversl proess rete unrehle sttes?
3 6.3 λtrnsition: MULTIPLE STRTSTTES ELIMINTION USING λtrnsitions n FS n hnge stte y using λtrnsition nd without reding n input symol. Elimintion of Multiple Strtsttes: dd new stte s nd mke itthe only strtstte. dd λtrnsition from s to eh of the originl strtstte. Nohnge in finlsttes or other trnsitions. q q 2 q 3 s λ λ q q 2 q 3 q s,, q 2 q 3, (i) Strtsttes = {q, q 2 }. Question:, (ii) n equivlent FS with strtstte nd λmoves., (iii) nother equivlent FS with strtstte.? Give n exmple FS to show tht it is not enough to dd new stte s, mke itthe only strtstte, nd for eh j dd the following trnsitions t s: δ(s, j )= δ(q i, j ), union over ll strtsttes in M. qi (We hve to mke the new strtstte s lso finlstte if one or more the originl strtsttes is finlstte.)? Show the resulting FS when we pply the ove onstrution to the FS shown t the top left. Does it hnge the lnguge?
4 6.4 SUBSETCONSTRUCTION METHOD FOR CONVERTING NFS TO FS The new FS M nnot simulte ll lterntive pths π (x) in the originl FS M for n input string x, euse the numer of π (x) n e exponentilly lrge (in x ) nd M hs finitely mny sttes. Insted, M keeps trk of the end points E(x) ofthe pths π (x); x is epted E(x) ontins one or more finlsttes of M. The endpoints of the pths π (x) form suset of Q in M, nd there re only 2 Q mny different susets. If x = 2 j nd x = x j+,then E(x ) = δ(q i, j+ ). q i E(x) #(pths π (x) for proessing x = n )=n+2. {} {2} {2,3} {2,4} {2,3,4} {2,3,4} Use the susets of Q s the sttes of the new FS.
5 6.5 THE SUBSETCONSTRUCTION v oid onstrution of unrehle sttes: () Choose the set of ll strtsttes in M s the strtstte S of the new FS M. (2) While there is stte S j for whih the trnsitions hve not een determined, do the following: For eh input symol Σin M, (i) Let S = δ (q i, ). (It my hppen tht S =.) (ii) q i S j If S is not lredy stte in M, then dd it s new stte. (iii) dd the trnsition δ(s j, ) =Sin M. (3) Mke eh stte S j in M finlstte if it ontins one or more finlsttes of M 2 4 n NFS 3 {} {2} {4} {2,3} {2,3,4} {2,4} The FS otined y the susetonstrution Note: If we did not hve the dedstte 4 in the ove exmple, then 4 would e removed from ll sttes in the new FS; the stte {4} would now eome.
6 6.6 EXERCISE. Complete the prtil desription of the stte in the finitestte utomton M hs elow for the lnguge L hs (= the inry strings ontining ""), sed on the desriptions of sttes B nd C, to justify the trnsitions to nd from. Note tht eh sttedesription is in terms of the "pst", i.e., the prt of the input whih is proessed to rrive t the stte. M hs :, B = hve not seen "" nd C B = hve not seen "" nd just seen C = seen "" Let M r hs e the nondeterministi utomton otined y pplying the reverslopertion to M hs ; L(M r hs) =L r hs = L hs. () Give suitle desription in English for the sttes of Mhs r tht would justify its trnsitions. Wht is the onnetion etween these desriptions nd the previous desriptions? () Show the FS otined from M r hs y the susetonstrution. lso desrie the sttes of the new FS in simple English in terms of the desriptions in (). 2. Remove the redundnt stte 4 in the NFS in pge 6.4 nd then pply the susetonstrution. How does the result differ from the FS shown ove; dothey ept the sme lnguge? 3. pply the susetonstrution for the NFSs in pge Consider deterministi FS for verifying multiplition of inry numers y 3, with the usul lest signifint it on the right. lso, onsider similr FS for verifying multiplition y 2. The input lphet for these mhines should e {,, 2, 3 }. Now, otin nondeterministi FS for verifying multiplition y either of 2 nd 3; onvert it to deterministi form.
7 6.7 Projetion: PROJECTION OF LNGUGEND λtrnsition IN N FSM If x = x x 2 x 3 x k,where some of x i s n e λ, none of x i ontins, nd k, then the projetion Π (x) =x x 2 x k,whih is simply x minus ll ourrenes of. Π (L) ={Π (x): x L}. Theorem: For ny lnguge L nd the symols, Π (Π (L))) = Π (Π (L))). If L is regulr lnguge, then there is NFSM for Π (L) ontining λtrnsitions. Exmple. M: d 2 d 3 Π d (M): λ 2 λ 3
8 6.8 λtrnsition: ELIMINTION OF λtrnsitions The FS n hnge its stte without reding n input symol. M: λ 2 λ 3 L(M): λ 2 3 L(M): λ 2 3 λ 2 3 Elimintion of λmoves inmgives possily n NFS M : M nd M hve the sme sttes, nd the sme finlsttes. M my hve multiple strtsttes (due to λtrnsitions from strtstte of M) nd nondeterministi trnsitions. Define: λ(q i )={q j :q j is rehle from q i y zero or more λtrnsitions}; q i λ(q i ). lgorithm: () Mke eh stte in λ(q )strtstte in M. (2) For eh δ(q i, j )=q k in M for j λ, let δ(q i, j )=λ(q k )inm. Exmple. For ove M, λ() = {, 2}, λ(2) = {2}, nd λ(3) = {2, 3}., M : 2 3
9 6.9 THE EFFECT OF INTRODUCING ERRORS IN REGULR LNGUGE Lnguge L modified y one replement error: RE (L) ={x:xdiffers from some y L in one position}. L nd RE (L) hve the sme lphet. If L is regulr, then RE (L) islso regulr. Exmple. If L = L div2 = L even,then RE (L) =L odd. L even ={λ,,,,,,,, } L odd = {,,,,,,, } Building n M(RE (L)) from M(L): The onstrution elow pplies to ny FS. M even B NFS for RE (L div2 ) B B trnsitions efore error errortrnsitions trnsitions fter error M(RE (L even )): {, B } {, B} redued M(RE (L even )): ( merged with B )
10 6. EXERCISE. pply the ove method to otin n FS for RE (L hs ). Show ll detils of oversion of NFS to FS nd the detils of stteminimiztion. 2. How will you generlize the ove onstrution for extly k ( 2) replement errors? Illustrte the onstrution using k = 2 nd M div2.(the generliztion to errors is lso esy.) 3. Show tht RE L (L) ={uv w: v L, v = v, nd uvw L} is regulr if oth L nd L re regulr. Note tht v my equl v. (Hint: n NFS for RE L (L) will hve three phses: for the prt u (efore the error), v, nd w (fter the error).) 4. Let DE (L) ={xy: xy L for some x,, nd y} =the set of strings otined y deletion of symol from strings in L. One n show tht DE (L) isregulr y giving method for the onstrution of NFS for DE (L) from n FS for L where the deletion opertion is modeled y λtrnsitions. Illustrte your method y using M hs s n exmple; show the NFS fter the introdution of λtrnsitions (keep the sttes "efore deletion" distint from those "fter deletion" similr to tht for the se of RE (L)). 5. similr result holds for the insertion error. Stte the result lerly.
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