Language Processors F29LP2, Lecture 5

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1 Lnguge Pocessos F29LP2, Lectue 5 Jmie Gy Feuy 2, / 1

2 Nondeteministic Finite Automt (NFA) NFA genelise deteministic finite utomt (DFA). They llow sevel (0, 1, o moe thn 1) outgoing tnsitions with the sme lel. An NFA ccets wod if some choice of tnsitions tkes the mchine to finl stte (othe choices my led to non-finl stte; we don t ce).,, Note: two tnsitions out of with inut lette, going to nd. Note: no tnsitions fom with inut lette. The nume of tnsitions my e zeo, one o moe. 2 / 1

3 Nondeteministic Finite Automt (NFA) NFA my hve diffeent comuttions fo the sme inut wod.,, Conside w =. The fist inut lette gives two choices: go to o go to. If we go to thee is no tnsition with ; we get stuck. If we go to then the second inut lette tkes the mchine to. Then nd cn tke the mchine to, then. Thee exists th to finl stte; is cceted; nd L(A). 3 / 1

4 Nondeteministic Finite Automt (NFA) A comuttion tee summises comuttions with inut : We see tht the NFA ccets exctly those wods tht contin s suwod, so the NFA is euivlent to the DFA of lectue 4 slide 7. 4 / 1

5 Nondeteministic Finite Automt (NFA),,, Cll two utomt euivlent when they ecognise the sme lnguge. 5 / 1

6 Pecise definition of NFA An NFA A = (Q, Σ, δ, 0, F ) is secified y: Stte set Q, inut lhet Σ, tnsition function δ, initil stte 0 nd the finl stte set F. δ is defined diffeently thn fo DFAs. It gives fo ech stte nd inut lette set δ(, ) Q of ossile next sttes. Using the owe set nottion 2 Q = {S S Q} we cn wite δ : Q Σ 2 Q. 6 / 1

7 Pecise definition of NFA δ fo the NFA on slide 2 is given s follows:, {, } {} {} {} {}, 7 / 1

8 Extending δ Extend δ to ˆδ s we did in DFA. Define ˆδ : Q Σ 2 Q such tht ˆδ(, w) is the set of ll sttes the mchine cn ech fom stte eding inut wod w. A ecusive definition goes s follows: 1. Fo evey stte, ˆδ(, ε) = {}. 2. Fo evey stte, wod w nd lette, ˆδ(, w) = { Q ˆδ(, w) : δ(, )} = ˆδ(,w) δ(, ). w 8 / 1

9 Extending δ On single inut lettes, δ nd ˆδ e eul: δ(, ) = ˆδ(, ). So thee s no confusion if we do the ht nd wite δ insted of ˆδ. In the NFA of slide 2 δ(, ) = {, }, δ(, ) = {}, δ(, ) = {, }, δ(, ) = {,, }.,, 9 / 1

10 Lnguge ecognised y n NFA The lnguge ecognised y NFA A = (Q, Σ, δ, 0, F ) is L(A) = {w Σ δ( 0, w) F }. Tht is, L(A) contins the w such tht some finl stte is echle fom 0 using w. Thee might lso e non-finl sttes echle fom 0, ut we don t ce. 10 / 1

11 Some execises: Constuct NFAs ove Σ = {, } tht ecognise the following lnguges: 1. Wods ending in. 2. Wods contining s suwod. 3. Wods stting with nd ending with. 4. Wods contining two s seted y n even nume of s. Convesely, detemine (nd descie in English) the lnguges ecognised y the following NFA:,,,,, 11 / 1

12 Conveting n NFA to DFA How to check whethe this NFA ccets w =? We check whethe ny comuttion ths fo w end in finl (cceting) stte. This cn gow exonentilly with the length of the inut! Bette to scn the inut once, keeing tck of the set of ossile sttes. 12 / 1

13 Conveting n NFA to DFA Fo exmle, with the exmle of the evious slide nd inut w = we hve {} {, } {, } {, } {, } {} {} {} so δ(, w) = {}, nd wod w is not cceted ecuse is not finl stte. Testing is deteministic (even though the NFA is nondeteministic). 13 / 1

14 Conveting n NFA to DFA Mkes it ovious tht ny NFA cn e conveted to DFA. The sttes of the DFA e sets of sttes of the NFA; tnsitions e such tht the stte of the DFA fte inut w is δ( 0, w) the set of sttes tht cn e eched in the NFA with inut w. So NFAs e just convenient shothnd fo (ossily much lge) DFA. NFA only ecognise egul lnguges. 14 / 1

15 Conveting n NFA to DFA Conside,, Initilly the NFA is in stte so the coesonding DFA is initilly in stte {}. With inut lette the NFA my move to eithe stte o, so fte inut the DFA will e in stte {, }. With inut the NFA emins in stte : {} {,} 15 / 1

16 Conveting n NFA to DFA Next figue out the tnsitions fom stte {, }. If the DFA is in stte {, } it mens tht the NFA cn e in eithe stte o. With inut the NFA cn move to o (if it ws in stte ) o to (if it ws in stte ). Theefoe, with inut lette the mchine cn move to o o, nd so the DFA must move to {,, }. With inut the only tnsition fom sttes nd is into. Tht is why DFA hs tnsition fom {, } ck into {}: {} {,} {,,} 16 / 1

17 Conveting n NFA to DFA Conside {,, }. With inut the NFA cn ech ny stte so the DFA loos to {,, }. With inut the NFA cn move to (if it ws t ) o to (if it ws t ) so the DFA hs tnsition fom {,, } into {, }: {} {,} {,,} {,} We gin hve new stte {, } to ocess. Fom nd the NFA cn go to ny stte with inut, nd to nd with inut : {} {,} {,,} {,} 17 / 1

18 Conveting n NFA to DFA No new sttes intoduced! All necessy tnsitions in lce! We still hve to clculte finl sttes. The NFA wod w if it leds to t lest one finl stte. The DFA should ccet w when it ends to stte S contining t lest one finl stte of the NFA. Ou smle NFA hs only one finl stte, so evey set contining is finl: 18 / 1

19 Conveting n NFA to DFA The constuction is comlete. We hve DFA tht ccets the sme lnguge s the oiginl NFA. This constuction is clled oweset constuction. It is lusile (nd tue) tht the oweset constuction cn e lied to ny NFA to convet it into DFA. Thee my e othe DFA cceting the sme lnguge, some of which my e simle thn the oweset DFA. We don t ce: we only wnted to demonstte tht thee exists t lest one such DFA. 19 / 1

20 ε-moves Suisingly useful nd convenient: extend NFA y llowing sontneous tnsitions. When n NFA executes sontneous tnsition, known s n ε-move, it chnges its stte without eding ny inut lette. Any nume of ε-moves e llowed / 1

21 ε-moves Wod w = is cceted s follows: The fist kees the mchine in stte 1. An ε-move to stte 2 is executed without eding ny inut. Next is consumed though sttes 3 nd 2, followed y nothe ε-move to stte 4. The lst kees the utomton in the cceting stte 4. The utomton of this exmle ccets ny seuence of s followed y ny eetition of s followed y ny nume of s: L(A) = { i () j k i, j, k 0}. 21 / 1

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