Finite Automata. Reading: Chapter 2

Transcription

1 Fiite Automata Readig: Chapter 2

2 Fiite Automato (FA) Iformally, a state diagram that comprehesively captures all possible states ad trasitios that a machie ca take while respodig to a stream or sequece of iput symbols Recogizer for Regular Laguages Determiistic Fiite Automata (DFA) The machie ca exist i oly oe state at ay give time No-determiistic Fiite Automata (NFA) The machie ca exist i multiple states at the same time 2

3 Determiistic Fiite Automata - Defiitio A Determiistic Fiite Automato (DFA) cosists of: Q ==> a fiite set of states ==> a fiite set of iput symbols (alphabet) q 0 ==> a start state F ==> set of acceptig states δ ==> a trasitio fuctio, which is a mappig betwee Q x ==> Q A DFA is defied by the 5-tuple: {Q,, q 0,F, δ } 3

4 What does a DFA do o readig a iput strig? Iput: a word w i * Questio: Is w acceptable by the DFA? Steps: Start at the start state q 0 For every iput symbol i the sequece w do Compute the ext state from the curret state, give the curret iput symbol i w ad the trasitio fuctio If after all symbols i w are cosumed, the curret state is oe of the acceptig states (F) the accept w; Otherwise, reject w. 4

5 Regular Laguages Let L(A) be a laguage recogized by a DFA A. The L(A) is called a Regular Laguage. Locate regular laguages i the Chomsky Hierarchy 5

6 The Chomsky Hierachy A cotaimet hierarchy of classes of formal laguages Regular (DFA) Cotextfree (PDA) Cotextsesitive (LBA) Recursivelyeumerable (TM) 6

7 Example # Build a DFA for the followig laguage: L = {w w is a biary strig that cotais 0 as a substrig} Steps for buildig a DFA to recogize L: = {0,} Decide o the states: Q Desigate start state ad fial state(s) δ: Decide o the trasitios: Fial states == same as acceptig states Other states == same as o-acceptig states 7

8 Regular expressio: (0+)*0(0+)* DFA for strigs cotaiig 0 What makes this DFA determiistic? Q = {q 0,q,q 2 } start 0, 0 q 0 0 q What if the laguage allows empty strigs? q 2 Acceptig state states = {0,} start state = q 0 F = {q 2 } Trasitio table symbols 0 q 0 q *q 2 q q q 2 q 0 q 2 q 2 8

9 Example #2 Clampig Logic: A clampig circuit waits for a iput, ad turs o forever. However, to avoid clampig o spurious oise, we ll desig a DFA that waits for two cosecutive s i a row before clampig o. Build a DFA for the followig laguage: L = { w w is a bit strig which cotais the substrig } State Desig: q 0 : start state (iitially off), also meas the most recet iput was ot a q : has ever see but the most recet iput was a q 2 : has see at least oce 9

10 Example #3 Build a DFA for the followig laguage: L = { w w is a biary strig that has eve umber of s ad eve umber of 0s}? 0

11 Extesio of trasitios (δ) to Paths (δ) δ (q,w) = destiatio state from state q o iput strig w δ (q,wa) = δ (δ(q,w), a) Work out example #3 usig the iput sequece w=000, a=: δ (q 0,wa) =?

12 Laguage of a DFA A DFA A accepts strig w if there is a path from q 0 to a acceptig (or fial) state that is labeled by w i.e., L(A) = { w δ(q 0,w) F } I.e., L(A) = all strigs that lead to a acceptig state from q 0 2

13 No-determiistic Fiite Automata (NFA) A No-determiistic Fiite Automato (NFA) is of course o-determiistic Implyig that the machie ca exist i more tha oe state at the same time Trasitios could be o-determiistic q i q j q k Each trasitio fuctio therefore maps to a set of states 3

14 No-determiistic Fiite Automata (NFA) A No-determiistic Fiite Automato (NFA) cosists of: Q ==> a fiite set of states ==> a fiite set of iput symbols (alphabet) q 0 ==> a start state F ==> set of acceptig states δ ==> a trasitio fuctio, which is a mappig betwee Q x ==> subset of Q A NFA is also defied by the 5-tuple: {Q,, q 0,F, δ } 4

15 How to use a NFA? Iput: a word w i * Questio: Is w acceptable by the NFA? Steps: Start at the start state q 0 For every iput symbol i the sequece w do Determie all possible ext states from all curret states, give the curret iput symbol i w ad the trasitio fuctio If after all symbols i w are cosumed ad if at least oe of the curret states is a fial state the accept w; Otherwise, reject w. 5

16 Regular expressio: (0+)*0(0+)* NFA for strigs cotaiig 0 Why is this o-determiistic? start 0, 0, q 0 0 q What will happe if at state q a iput of 0 is received? q 2 Fial state states Q = {q 0,q,q 2 } Σ = {0,} start state = q 0 F = {q 2 } Trasitio table symbols 0 q 0 {q 0,q } {q 0 } q Φ {q 2 } *q 2 {q 2 } {q 2 } 6

17 Note: Omittig to explicitly show error states is just a matter of desig coveiece (oe that is geerally followed for NFAs), ad i.e., this feature should ot be cofused with the otio of o-determiism. What is a error state? A DFA for recogizig the key word while q 0 w q h q 2 i q 3 l q 4 e q 5 Ay other iput symbol q err Ay symbol A NFA for the same purpose: q 0 w q h q 2 i q 3 l q 4 e q 5 Trasitios ito a dead state are implicit 7

18 Example #2 Build a NFA for the followig laguage: L = { w w eds i 0}? Other examples Keyword recogizer (e.g., if, the, else, while, for, iclude, etc.) Strigs where the first symbol is preset somewhere later o at least oce 8

19 Extesio of δ to NFA Paths Basis: δ (q,ε) = {q} Iductio: Let δ (q 0,w) = {p,p 2,p k } δ (p i,a) = S i for i=,2...,k The, δ (q 0,wa) = S U S 2 U U S k 9

20 Laguage of a NFA A NFA accepts w if there exists at least oe path from the start state to a acceptig (or fial) state that is labeled by w L(N) = { w δ(q 0,w) F Φ } 20

21 Advatages & Caveats for NFA Great for modelig regular expressios Strig processig - e.g., grep, lexical aalyzer Could a o-determiistic state machie be implemeted i practice? Probabilistic models could be viewed as extesios of odetermiistic state machies (e.g., toss of a coi, a roll of dice) They are ot the same though A parallel computer could exist i multiple states at the same time 2

22 Techologies for NFAs Micro s Automata Processor (itroduced i 203) 2D array of MISD (multiple istructio sigle data) fabric w/ thousads to millios of processig elemets. iput symbol = fed to all states (i.e., cores) No-determiism usig circuits 22

23 But, DFAs ad NFAs are equivalet i their power to capture lagauges!! Differeces: DFA vs. NFA DFA. All trasitios are determiistic Each trasitio leads to exactly oe state 2. For each state, trasitio o all possible symbols (alphabet) should be defied 3. Accepts iput if the last state visited is i F 4. Sometimes harder to costruct because of the umber of states 5. Practical implemetatio is feasible NFA. Some trasitios could be o-determiistic A trasitio could lead to a subset of states 2. Not all symbol trasitios eed to be defied explicitly (if udefied will go to a error state this is just a desig coveiece, ot to be cofused with odetermiism ) 3. Accepts iput if oe of the last states is i F 4. Geerally easier tha a DFA to costruct 5. Practical implemetatios limited but emergig (e.g., Micro automata processor) 23

24 Equivalece of DFA & NFA Should be true for ay L Theorem: A laguage L is accepted by a DFA if ad oly if it is accepted by a NFA. Proof:. If part: Prove by showig every NFA ca be coverted to a equivalet DFA (i the ext few slides ) 2. Oly-if part is trivial: Every DFA is a special case of a NFA where each state has exactly oe trasitio for every iput symbol. Therefore, if L is accepted by a DFA, it is accepted by a correspodig NFA. 24

25 Proof for the if-part If-part: A laguage L is accepted by a DFA if it is accepted by a NFA rephrasig Give ay NFA N, we ca costruct a DFA D such that L(N)=L(D) How to covert a NFA ito a DFA? Observatio: I a NFA, each trasitio maps to a subset of states Idea: Represet: each subset of NFA_states è a sigle DFA_state Subset costructio 25

26 NFA to DFA by subset costructio Let N = {Q N,,δ N,q 0,F N } Goal: Build D={Q D,,δ D,{q 0 },F D } s.t. L(D)=L(N) Costructio:. Q D = all subsets of Q N (i.e., power set) 2. F D =set of subsets S of Q N s.t. S F N Φ 3. δ D : for each subset S of Q N ad for each iput symbol a i : δ D (S,a) = U δ N (p,a) p i s 26

27 Idea: To avoid eumeratig all of power set, do lazy creatio of states NFA to DFA costructio: Example NFA: 0, L = {w w eds i 0} 0 q 0 q q 2 DFA: [q 0 ] 0 0 [q 0,q ] 0 [q 0,q 2 ] δ N 0 q 0 {q 0,q } {q 0 } q Ø {q 2 } *q 2 Ø Ø δ D 0 Ø Ø Ø [q 0 ] {q 0,q } {q 0 } [q ] Ø {q 2 } *[q 2 ] Ø Ø [q 0,q ] {q 0,q } {q 0,q 2 } *[q 0,q 2 ] {q 0,q } {q 0 } *[q,q 2 ] Ø {q 2 } *[q 0,q,q 2 ] {q 0,q } {q 0,q 2 } δ D 0 [q 0 ] [q 0,q ] [q 0 ] [q 0,q ] [q 0,q ] [q 0,q 2 ] *[q 0,q 2 ] [q 0,q ] [q 0 ] 0. Eumerate all possible subsets. Determie trasitios 2. Retai oly those states reachable from {q 0 } 27

28 NFA to DFA: Repeatig the example usig LAZY CREATION NFA: 0, L = {w w eds i 0} 0 q 0 q q 2 DFA: [q 0 ] 0 0 [q 0,q ] 0 [q 0,q 2 ] δ N 0 q 0 {q 0,q } {q 0 } q Ø {q 2 } *q 2 Ø Ø δ D 0 [q 0 ] [q 0,q ] [q 0 ] [q 0,q ] [q 0,q ] [q 0,q 2 ] *[q 0,q 2 ] [q 0,q ] [q 0 ] Mai Idea: Itroduce states as you go (o a eed basis) 28

29 Correctess of subset costructio Theorem: If D is the DFA costructed from NFA N by subset costructio, the L(D)=L(N) Proof: Show that δ D ({q 0 },w) δ N (q 0,w}, for all w Usig iductio o w s legth: Let w = xa δ D ({q 0 },xa) δ D ( δ N (q 0,x}, a ) δ N (q 0,w} 29

30 A bad case where #states(dfa)>>#states(nfa) L = {w w is a biary strig s.t., the k th symbol from its ed is a } NFA has k+ states But a equivalet DFA eeds to have at least 2 k states (Pigeo hole priciple) m holes ad >m pigeos => at least oe hole has to cotai two or more pigeos 30

31 Applicatios Text idexig iverted idexig For each uique word i the database, store all locatios that cotai it usig a NFA or a DFA Fid patter P i text T Example: Google queryig Extesios of this idea: PATRICIA tree, suffix tree 3

32 A few subtle properties of DFAs ad NFAs The machie ever really termiates. It is always waitig for the ext iput symbol or makig trasitios. The machie decides whe to cosume the ext symbol from the iput ad whe to igore it. (but the machie ca ever skip a symbol) => A trasitio ca happe eve without really cosumig a iput symbol (thik of cosumig ε as a free toke) if this happes, the it becomes a ε-nfa (see ext few slides). A sigle trasitio caot cosume more tha oe (o-ε) symbol. 32

33 FA with ε-trasitios We ca allow explicit ε-trasitios i fiite automata i.e., a trasitio from oe state to aother state without cosumig ay additioal iput symbol Explicit ε-trasitios betwee differet states itroduce o-determiism. Makes it easier sometimes to costruct NFAs Defiitio: ε -NFAs are those NFAs with at least oe explicit ε-trasitio defied. ε -NFAs have oe more colum i their trasitio table 33

34 Example of a ε-nfa L = {w w is empty, or if o-empty will ed i 0} start ε q 0 0, 0 q 0 q δ E 0 ε *q 0 Ø Ø {q 0,q 0 } q 0 {q 0,q } {q 0 } {q 0 } q Ø {q 2 } {q } q 2 ECLOSE(q 0 ) ECLOSE(q 0 ) ECLOSE(q ) ε-closure of a state q, ECLOSE(q), is the set of all states (icludig itself) that ca be reached from q by repeatedly makig a arbitrary umber of ε- trasitios. *q 2 Ø Ø {q 2 } ECLOSE(q 2 ) 34

35 To simulate ay trasitio: Step ) Go to all immediate destiatio states. Step 2) From there go to all their ε-closure states as well. Example of a ε-nfa L = {w w is empty, or if o-empty will ed i 0} start ε q 0 0, 0 q 0 q q 2 Simulate for w=0: q 0 q 0 ε ε q 0 δ E 0 ε *q 0 Ø Ø {q 0,q 0 } q 0 {q 0,q } {q 0 } {q 0 } ECLOSE(q 0 ) ECLOSE(q 0 ) Ø x 0 q 0 q q Ø {q 2 } {q } q 2 *q 2 Ø Ø {q 2 } 35

36 To simulate ay trasitio: Step ) Go to all immediate destiatio states. Step 2) From there go to all their ε-closure states as well. Example of aother ε-nfa ε 0, 0 q 0 q ε q 2 Simulate for w=0:? start q 0 q 3 δ E 0 ε *q 0 Ø Ø {q 0,q 0,q 3 } q 0 {q 0,q } {q 0 } {q 0, q 3 } q Ø {q 2 } {q } *q 2 Ø Ø {q 2 } q 3 Ø {q 2 } {q 3 } 36

37 Equivalecy of DFA, NFA, ε-nfa Theorem: A laguage L is accepted by some ε-nfa if ad oly if L is accepted by some DFA Implicatio: DFA NFA ε-nfa (all accept Regular Laguages) 37

38 Elimiatig ε-trasitios Let E = {Q E,,δ E,q 0,F E } be a ε-nfa Goal: To build DFA D={Q D,,δ D,{q D },F D } s.t. L(D)=L(E) Costructio:. Q D = all reachable subsets of Q E factorig i ε-closures 2. q D = ECLOSE(q 0 ) 3. F D =subsets S i Q D s.t. S F E Φ 4. δ D : for each subset S of Q E ad for each iput symbol a : p i s Let R= U δ E (p,a) // go to destiatio states δ D (S,a) = r U i R ECLOSE(r) // from there, take a uio of all their ε- closures Readig: Sectio i book 38

39 Example: ε-nfa è DFA L = {w w is empty, or if o-empty will ed i 0} 0, ε 0 q 0 q q 2 start q 0 δ E 0 ε *q 0 Ø Ø {q 0,q 0 } q 0 {q 0,q } {q 0 } {q 0 } q Ø {q 2 } {q } *q 2 Ø Ø {q 2 } δ D 0 *{q 0,q 0 } 39

40 Example: ε-nfa è DFA start L = {w w is empty, or if o-empty will ed i 0} ε q 0 0, 0 q 0 q ECLOSE q 2 uio start 0 {q 0, q 0 } 0 {q 0,q } 0 q 0 0 {q 0,q 2 } δ E 0 ε δ D 0 *q 0 Ø Ø {q 0,q 0 } q 0 {q 0,q } {q 0 } {q 0 } q Ø {q 2 } {q } *q 2 Ø Ø {q 2 } *{q 0,q 0 } {q 0,q } {q 0 } {q 0,q } {q 0,q } {q 0,q 2 } {q 0 } {q 0,q } {q 0 } *{q 0,q 2 } {q 0,q } {q 0 } 40

41 Summary DFA Defiitio Trasitio diagrams & tables Regular laguage NFA Defiitio Trasitio diagrams & tables DFA vs. NFA NFA to DFA coversio usig subset costructio Equivalecy of DFA & NFA Removal of redudat states ad icludig dead states ε-trasitios i NFA Pigeo hole priciples Text searchig applicatios 4