80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES. 2.6 Finite State Automata With Output: Transducers

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1 80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES 2.6 Finite Stte Automt With Output: Trnsducers So fr, we hve only considered utomt tht recognize lnguges, i.e., utomt tht do not produce ny output on ny input (except ccept or reject ). It is interesting nd useful to consider input/output finite stte mchines. Such utomt re clled trnsducers. They compute functions or reltions. First, we define deterministic kind of trnsducer. Definition 2.8. A generl sequentil mchine (gsm) is sextuple M =(Q, Σ,,δ,λ,q 0 ), where (1) Q is finite set of sttes, (2) Σ is finite input lphet, (3) is finite output lphet, (4) δ : Q Σ Q is the trnsition function, (5) λ: Q Σ is the output function nd (6) q 0 is the initil (or strt) stte.

2 2.6. FINITE STATE AUTOMATA WITH OUTPUT: TRANSDUCERS 81 If λ(p, ) ϵ, forllp Q nd ll Σ, then M is nonersing. Ifλ(p, ) forllp Q nd ll Σ, we sy tht M is complete sequentil mchine (csm). An exmple of gsm for which Σ = {, } nd = {0, 1, 2} is shown in Figure 2.9. For exmple is converted to /11 /01 /10 /21 /00 /20 2 Figure 2.9: Exmple of gsm In order to define how gsm works, we extend the trnsition nd the output functions. We define δ : Q Σ Q nd λ : Q Σ recursively s follows: For ll p Q, llu Σ nd ll Σ δ (p, ϵ) =p δ (p, u) =δ(δ (p, u),) λ (p, ϵ) =ϵ λ (p, u) =λ (p, u)λ(δ (p, u),).

3 82 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES For ny w Σ,welet M(w) =λ (q 0,w) nd for ny L Σ nd L,let nd M(L) ={λ (q 0,w) w L} M 1 (L )={w Σ λ (q 0,w) L }. Note tht if M is csm, then M(w) = w for ll w Σ. Also, homomorphism is specil kind of gsm it cn e relized y gsm with one stte. We cn use gsm s nd csm s to compute certin kinds of functions. Definition 2.9. Afunctionf :Σ is gsm (resp. csm) mpping iff there is gsm (resp. csm) M so tht M(w) =f(w), for ll w Σ.

4 2.6. FINITE STATE AUTOMATA WITH OUTPUT: TRANSDUCERS 83 Remrk: Ginsurg nd Rose (1966) chrcterized gsm mppings s follows: Afunctionf :Σ is gsm mpping iff () f preserves prefixes, i.e., f(x) isprefixoff(xy); () There is n integer, m, suchthtforllw Σ nd ll Σ, we hve f(w) f(w) m; (c) f(ϵ) =ϵ; (d) For every regulr lnguge, R,thelnguge f 1 (R) ={w Σ f(w) R} is regulr. Afunctionf :Σ is csm mpping iff f stisfies () nd (d), nd for ll w Σ, f(w) = w.

5 84 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES The following proposition is left s homework prolem. Proposition 2.1. The fmily of regulr lnguges (over n lphet Σ) is closed under oth gsm nd inverse gsm mppings. We cn generlize the gsm model so tht (1) the device is nondeterministic, (2) the device hs set of ccepting sttes, (3) trnsitions re llowed to occur without new input eing processed, (4) trnsitions re defined for input strings insted of individul letters. Here is the definition of such model, the -trnsducer. Amuchmorepowerfulmodeloftrnsducerwilleinvestigted lter: the Turing mchine.

6 2.6. FINITE STATE AUTOMATA WITH OUTPUT: TRANSDUCERS 85 Definition An -trnsducer (or nondeterministic sequentil trnsducer with ccepting sttes) is sextuple M =(K, Σ,,λ,q 0,F), where (1) K is finite set of sttes, (2) Σ is finite input lphet, (3) is finite output lphet, (4) q 0 K is the strt (or initil) stte, (5) F K is the set of ccepting (of finl) sttes nd (6) λ K Σ K is finite set of qudruples clled the trnsition function of M. If λ K Σ + K, thenm is ϵ-free Clerly, gsm is specil kind of -trnsducer. An -trnsducer defines inry reltion etween Σ nd,orequivlently,functionm :Σ 2.

7 86 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES We cn explin wht this function is y descriing how n -trnsducer mkes sequence of moves from configurtions to configurtions. The current configurtion of n -trnsducer is descried y triple (p, u, v) K Σ, where p is the current stte, u is the remining input, nd v is some ouput produced so fr. We define the inry reltion M on K Σ s follows: For ll p, q K, u, α Σ, β,v, if (p, u, v, q) λ, then (p, uα, β) M (q, α, βv). Let M e the trnsitive nd reflexive closure of M.

8 2.6. FINITE STATE AUTOMATA WITH OUTPUT: TRANSDUCERS 87 The function M :Σ 2 is defined such tht for every w Σ, M(w) ={y (q 0,w,ϵ) M (f, ϵ, y), f F }. For ny lnguge L Σ let M(L) = w L M(w). For ny y,let M 1 (y) ={w Σ y M(w)} nd for ny lnguge L,let M 1 (L )= y L M 1 (y).

9 88 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES Remrk: Notice tht if w M 1 (L ), then there exists some y L such tht w M 1 (y), i.e., y M(w). This does not imply tht M(w) L,only tht M(w) L. One should relize tht for ny L nd ny - trnsducer, M, thereissome-trnsducer, M,(from to 2 Σ )sothtm (L )=M 1 (L ). The following proposition is left s homework prolem: Proposition 2.2. The fmily of regulr lnguges (over n lphet Σ) is closed under oth -trnsductions nd inverse -trnsductions.

10 2.7. AN APPLICATION OF NFA S: TEXT SEARCH An Appliction of NFA s: Text Serch AcommonproleminthegeoftheWe(ndon-line text repositories) is the following: Given set of words, clled the keywords, findllthe documents tht contin one (or ll) of those words. Serch engines re populr exmple of this process. Serch engines use inverted indexes (for ech word ppering on the We, list of ll the plces where tht word occurs is stored). However, there re pplictions tht re unsuited for inverted indexes, ut re good for utomton-sed techniques. Some text-processing progrms, such s dvnced forms of the UNIX grep commnd (such s egrep or fgrep) re sed on utomton-sed techniques. The chrcteristics tht mke n ppliction suitle for serches tht use utomt re:

11 90 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES (1) The repository on which the serch is conducted is rpidly chnging. (2) The documents to e serched cnnot e ctlogued. For exmple, Amzon.com cretes pges on the fly in response to queries. We cn use n NFA to find occurrences of set of keywords in text. This NFA signls y entering finl stte tht it hs seen one of the keywords. The form of such n NFA is specil. (1) There is strt stte, q 0,withtrnsitiontoitselfon every input symol from the lphet, Σ. (2) For ech keyword, w = w 1 w k (with w i Σ), there re k sttes, q (w) 1,...,q (w) k,ndthereistrnsition from q 0 to q (w) 1 on input w 1,trnsitionfrom q (w) 1 to q (w) 2 on input w 2,ndsoon,untiltrnsition from q (w) k 1 to q(w) k on input w k. The stte q (w) k is n ccepting stte nd indictes tht the keyword w = w 1 w k hs een found. The NFA constructed ove cn then e converted to DFA using the suset construction.

12 2.7. AN APPLICATION OF NFA S: TEXT SEARCH 91 Here is n exmple where Σ = {, } nd the set of keywords is {,, }. q1 q2 q3, 0 q1 q2 q1 q2 Figure 2.10: NFA for the keywords,,. Applying the suset construction to the NFA, we otin the DFA whose trnsition tle is:

13 92 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES ,q1,q ,q ,q1,q2,q ,q1,q1,q ,q1,q1,q2,q3 1 3 The finl sttes re: 3, 4, Figure 2.11: DFA for the keywords,,.

14 2.7. AN APPLICATION OF NFA S: TEXT SEARCH 93 The good news news is tht, due to the very specil structure of the NFA, the numer of sttes of the corresponding DFA is t most the numer of sttes of the originl NFA! We find tht the sttes of the DFA re (check it yourself!): (1) The set {q 0 },ssocitedwiththestrtstteq 0 of the NFA. (2) For ny stte p q 0 of the NFA reched from q 0 long pthcorrespondingtostringu = u 1 u m,the set consisting of: () q 0 () p (c) The set of ll sttes q of the NFA rechle from q 0 y following pth whose symols form nonempty suffix of u, i.e.,stringoftheform u j u j+1 u m. As consequence, we get n efficient (w.r.t. time nd spce) method to recognize set of keywords. In fct, this DFA recognizes leftmost occurrences of keywords in text(wecnstopssoonsweenterfinlstte).

15 94 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES

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