Automata and Regular Languages

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1 Chpter 9 Automt n Regulr Lnguges 9. Introution This hpter looks t mthemtil moels of omputtion n lnguges tht esrie them. The moel-lnguge reltionship hs multiple levels. We shll explore the simplest level, n riefly esrie the others t the en. First, we tke rief look t the fountion in sequentil igitl iruits. A new element is e tht n hol its. This pility to pture n rememer vlues gretly inreses wht igitl systems re ple of oing. 9.. Finite Stte Mhines* The igitl iruit shown to the left elow is lle flip-flop. The symol stns for not. The iruit is esrie y the simultneous system of oolen equtions in the enter, whose truth tle is on the right. S R P Q P = S + Q Q = R + P S R P Q In the se tht R = S = the system reues to P = Q Q = P, whih hs two solutions s the truth tle shows. A physil iruit must somehow selet one of these solutions, n it oes so y seleting the solution tht ws present in the previous instnt. In other wors, the flip-flop will hol P s n Q s vlues

2 2 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES inefinitely, until it is fore to hnge y pulse on S, mking P = n Q =, or pulse on R, mking P = n Q =. Simultneous pulses on P n Q n fore the flip-flop into metstle solution, P Q, ut eventully, it will snp to vli stte. Aitionl iruitry is e to minimize this vulnerility. A loke flipflop hs one input presenting the it to e hel. The CLK input is use to ontrol when the next it is pture. DATA CLK D Q We n think of ny igitil system s n instne of finite-stte mhine in n f m out stte k Q S D k next stte k CLK In this rhiteture, Blok f represents purely omintionl oolen system of n + k inputs n m + k outputs. Comintionl tht eh output of f is just oolen omintion of some or ll of the n + k inputs. Blok S represents n rry of simple storge elements, eh ple of storing one it, or, inefinitely. Think of the signl CLK s kin of eletroni metronome, going tik-tok-tik-tok-. On tik, the k one-it storge elements simultneously pture the vlue presente on its input n hols it until the next tik. The perio etween CLK tiks is long enough to llow the iruitry to eletronilly stilize, so tht there re no re onitions where n input is hnging when the tik ours. Copyright 22 Steven D. Johnson Deemer 3, 22

3 9.2. AUTOMATA 3 out O O O 2 O 3 O 4 O i+ = f(s, I ) i i in I I I 2 I 3 I 4 s S S S 2 S 3 S 4 S i+ = ns(s i, I i) s S S 2 S 3 S 4 S 5 CLK 9.2 Automt A finite-stte utomoton (FA) moels the tions of n ielize omputing mhine. Before formlizing this onept, let us look t n exmple epite y the igrm W W X Y X Y,,,, Z Z,,, U,,, The noes of the FA form grph whose eges re lelle y letters of n lphet. The noes S = {W, X, Y, Z} re lle the sttes of the FA n the lphet is A = {,,, }. The FA strts in stte s = W, s inite y the rrow. It is presente with n input wor, w A whih it onsumes one letter t time. If there is n ege (s, s ) lelle with the next input letter, the FA mkes trnsition into stte s onsuming the letter. The proess of mking trnsition n onsuming letter repets until () the input wor is empty, or the FA rehes stte with no trnsition for the next input letter. Copyright 22 Steven D. Johnson Deemer 3, 22

4 4 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES When input wor is empty n the FA is in n epting stte, inite y oule-irle, we sy tht the FA epts w. remrk. Termintion onition () is not stritly neessry. As epite to the right ove, one n lwys eges n trnsition lelle with more thn one letter represents more thn one trnsitions to the trnsistion reltion for missing letters, tking the utomton to e stte tht onsumes ll the remining input letters, n from whih there is no pth to n epting stte. The result is finite-utomton tht eptes extly the sme wors. Exmple 9. For exmple, suppose this FA is presente with the wor the pth it follows is W Y W Y W X X X Z We ll suh pth tre of the FA. It is ustomry to omit the stte nmes, s in to ssert tht tre exists, in this se n epting tre, for. We my t times revite this further to something like 9.2. Formliztion Let us egin to formlize the ie of finite-stte utomton. Definition 9. Let A e n lphet. A finite-stte utomton, or FA over lphet A is struture S, s, F, T onsisting of finite set S is of sttes, single intitl or strting stte s S non-empty suset F S of finl or epting sttes, n A trnsition reltion T (S A) S. Next we efine how FA exeutes. Definition 9.2 Given finite-stte utomoton, FA = S, s, F, T over lphet A n wor w =... n A, we sy FA epts w if () w = n s F, or Copyright 22 Steven D. Johnson Deemer 3, 22

5 9.2. AUTOMATA 5 () there is pth, P = s, s,..., s n in T suh tht s is the strting stte, s n F is n epting stte, n for eh suessive pir (s i, s i+ ), i < n, there is pir ((s i, i ), s i+ ) T. Definition 9.3 Given finite-stte utomton, FA over lphet A the lnguge L(FA) A is the set of ll wors epte y FA Noneterminism The trnsition reltion of Definition 9. rises t lest one importnt question: Wht oes it men when the trnsition reltion is not funtion? When T is funtion, or prtil funtion, it is uniqely etermine wht tion is tken on eh input letter. Suppose to the ontrry tht our FA llows two istint trnsitions from stte on the sme ltter. For instne, elow we hve e seon trnsition uner the letter from stte W. W X Y Z The efinition of eptne still hols; it sks only whether tre exists. However, the question of how tht pth is selete is less ler. How woul physil mehnism eie whih trnsition to tke? Woul it hve to preit wht input letters it will see in the future? If so, oes this pility mke the utomt more powerful in the sense tht they epts roer lss of lnguges? Definition 9.4 A eterministi finite-stte utomton (DFA) is one whose trnsition reltion is (prtil) funtion. A non-eterminsiti finite-stte utomton (NFA) is one whose trnsition reltion is not funtion, tht is, n FA with multiple istit trnsitions from some stte for the sme letter. Copyright 22 Steven D. Johnson Deemer 3, 22

6 6 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES 9.3 Composing Automt 9.3. Prllel Composition s s s s2 s s3 s4 L = {inry wors tht en with } L 2 = {inry wors tht strt n en with the sme letter} s4 s2 s s s3 s s2 s s4 s3 L L 2 prout FA S = S S 2 s = ( s, s 2 ) F = (F S 2 ) (F 2 S ) (For L L 2, F = F F 2 ) T = {((s, s 2 ), (, 2 )), (s, s 2)) ((s, ), s ) T n((s 2, 2 ), s 2) T 2 } Copyright 22 Steven D. Johnson Deemer 3, 22

7 9.3. COMPOSING AUTOMATA 7 Remove unrehle (lue) n reunnt (re) sttes s s2 s s4 L L 2 Copyright 22 Steven D. Johnson Deemer 3, 22

8 8 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES Empty Trnsitions Definition 9.5 An empty trnsition is trnsition lelle y. A DFA or NFA with n empty trnsition my noneterminstilly move to the trget stte without onsuming the next input letter. Equivlently, on my think of the input wor s hving one or more s etween ny two letters. Exmple 9.2 s s s s2 s s3 s4 L = {wors tht en with } L 2 = {wors tht strt n en with the sme letter} Contention s s t t t2 t3 t4 L ˆL 2 = {uˆv u L n v L 2 } Copyright 22 Steven D. Johnson Deemer 3, 22

9 9.3. COMPOSING AUTOMATA 9 Proposition 9. Given ny FA with empty trnsitions, A there is n utomton A tht epts the sme lnguge Removing Noneterminism (!?) v s t s t t2 t3 t4 NFA for L L 2 Copyright 22 Steven D. Johnson Deemer 3, 22

10 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES Remove -trnsitions e f g h.6 Power utomton {} {,,h} {,e} {,f}, {e,,h} {,g} {,h} {g,,f}.6 Copyright 22 Steven D. Johnson Deemer 3, 22

11 9.4. REGULAR EXPRESSIONS 9.4 Regulr Expressions Regulr expressions re lnguge for esriing regulr lnguges. Definition 9.6 For given lphet A, the lnguge rexp of regulr expressions n their interprettion s lnguges over A re efine: rexp ( A {,,,, (, ) } ) rexp ::= L[ ] = A R: rexp P(A ) L[ ] = {} L[ x ] = {x} for x A rexp L[ r ] = {u n u L[ r, n N ] rexp rexp L[ r r 2 ] = {uˆv u L[ r, v L[ r 2 ] rexp rexp L[ r r 2 ] = L[ r ] L[ r 2 ] ( rexp ) L[ ( r ) ] = L[ r ] Copyright 22 Steven D. Johnson Deemer 3, 22

12 2 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES Theorem 9.2 There is finite-stte utomton tht epts the lnguge speifie y ny regulr expression. Proof. The proof is y inution on rexp. In eh se n FA is reursively onstrute. The onstrutions re shown informlly elow. : : x A: x r : A 2 A r r 2 : A 2 A A 2 r r 2 : Copyright 22 Steven D. Johnson Deemer 3, 22

13 9.4. REGULAR EXPRESSIONS 3 Theorem 9.3 The lnguge esrie y ny finite-stte utomton n e speifie y regulr expression. proof ie. There re numer of wys to reue FA to regulr expression (Google finite-utomton to regulr expression ). One wy is to perform stte reution on utomt over rexp. For instne, one reution rule might trnslte to * As reution ontinues some of the sttes eome unrehle n n e isre. The utomton ultimtely reues to L(A) n the rexp lelling the single remining ege speifies the lnguge epte y the originl utomton. Like other onstrutions, suh s removing noneterministi trnsitions, the reution of n utomton to regulr expression my involve explosion in the size of the resulting expression. Exmple 9.3 Let A = {,,, }. Is it possile to use regulr expressions to speify lnguge of wors in whih no letter ours twie in suession? Solution The expression elow ws ontriute y Ahijit Mhl, who Copyright 22 Steven D. Johnson Deemer 3, 22

14 4 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES generte it using the JFLAP visuliztion tool ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ) ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ) ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ) remrk. Though vli (proly), this is not something humn oul write, or even omprehen, without it of thought. If you exmine the formul, you will egin to see some repete ptterns. We n ientify some of these suformuls n efine lnguges to go with them, in orer to uil hierrhy. The suformul ( )() ( ) esries lnguge over the lphet {, } in whih neither nor ours twie in suession in ny wor. Nme this lnguge L L = ( )() ( ) The sulnguges L n L some wors tht ontin s n some tht ontin s. L = ( )() ( ) L = ( )() ( ) Using these lnguges, the lrge formul ove reues to L ( L )L L ( L ( L )L ( L ))(L L L ( L )) (L L L L ) This formul omines the sulnguges, ing in some more s n s where neee. You n hek tht wors in this lnguge stisfy the property of no suessive ournes,, or ; ut it is muh more iffiult to etermine whether this formul works for ll suh wors. Copyright 22 Steven D. Johnson Deemer 3, 22

15 9.5. IS THAT ALL THERE IS? Is Tht All There Is? Proposition 9.4 (Pumping Lemm) Let L e regulr lnguge. Then there is whole numer p tht epens only on L suh tht every wor w L of length t lest p is of the form w = xyz where () y > () xy p () for ll i N, xy i z L. Exmple 9.4 {u n v n n N} is not regulr. Copyright 22 Steven D. Johnson Deemer 3, 22

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