CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata


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1 CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or wheel is spun) nd numer is generted t rndom. Bsed on the generted numer, the pieces on the ord re rerrnged specified y the rules of the gme. hen, nother child throws or spins nd rerrnges the pieces gin. here is no skill or choice involved  the entire gme is sed on the vlues of the rndom numers. Consider ll possile positions of the pieces on the ord nd cll them sttes. We egin with the initil stte of the strting positions of the pieces on the ord. he gme then chnges from one stte to nother sed on the vlue of the rndom numer. For ech possile numer, there is one nd only one resulting stte given the input of the numer, nd the prior stte. his continues until one plyer wins nd the gme is over. his is clled finl stte. Now consider very simple computer with n input device, processor, some memory nd n output device. We wnt to clculte 3 + 4, so we write simple list of instructions nd feed them into the mchine one t time e.g., SORE 3 O X; SORE 4 O Y; LOAD X; ADD Y; WRIE O OUPU). Ech instruction is executed s it is red. If ll goes well, the mchine outputs 7 nd termintes execution. his process is similr to the ord gme. he stte of the mchine chnges fter ech instruction is executed, nd ech stte is completely determined y the prior stte nd the input instruction thus this mchine is defined s deterministic). No choice or skill is involved; no knowledge of the stte of the mchine 4 instructions go is needed. he mchine simply strts t n initil stte, chnges from stte to stte sed on the instruction nd the prior stte, nd reches the finl stte of writing 7. Finite Automt A generl model of which the previous two exmples re instnces) of this type of mchine is clled Finite Automton; finite ecuse the numer of sttes nd the lphet of input symols is finite; utomton ecuse the structure or mchine s it is more commonly clled) is deterministic, i.e., the chnge of stte is completely governed y the input. A finite utomt hs: 1) A finite set of sttes, one of which is designted the initil stte or strt stte, nd some mye none) of which re designted s finl sttes. 2) An lphet of possile input symols. 3) A finite set of trnsitions tht tell for ech stte nd for ech symol of the input lphet, which stte to go to next.
2 In the simple computer exmple, the strt stte is the originl stte of the mchine efore progrm execution nd the finl stte is 7 written on the output device. he input lphet is the set of strings representing the instructions. 2 Exmple 1 Suppose = {,}, the set of sttes = {x, y, z} with x the strt stte nd z the finl stte, nd we hve the following rules of trnsition: 1) from stte x nd with input, goto stte y. 2) from stte x nd with input, goto stte z. 3) from stte y nd with input, goto stte x. 4) from stte y nd with input, goto stte z. 5) from stte z nd with ny input, sty t stte z. his is perfectly defined FA short for finite utomton) ccording to the definition ove. We now need to exmine wht hppens to vrious input strings when presented to this FA. Consider : we egin in stte x nd goto stte y. he next symol is lso n, so from stte y we go to stte x. hen on the lst, we go from stte x ck to stte y. hts ll the input we hve  since we do not end up in stte z, the finl stte, we hve n unsuccessful end. his exercise illustrtes how n FA cn e used to recognize or ccept strings of prticulr lnguge. he set of ll strings tht leve us in finl stte of n FA is clled the lnguge ccepted or defined y the FA. is not word in the lnguge defined y the FA given ove. his is why FAs re lso clled lnguge recognizers. Now, try : from stte x to stte y, from stte y to stte z, from stte z to stte z, from stte z to stte z. his word is prt of the lnguge defined y this FA. In fct, ny input string tht hs just one or more will e ccepted y this FA. We cn define this y the regulr expression: + )* + )* his is very simple FA. ypiclly, the list of trnsition rules cn e quite long, so n lterntive representtion is frequently used. One method is to use trnsition tle: strt x: y z y: x z finl z: z z his tle hs ll the informtion required to define n FA. Even though it is no more thn tle of symols or list of rules, we consider n FA to e mchine, i.e., we understnd tht n FA hs dynmic cpilities. It moves. It processes input. Something goes from stte to stte s the input is red. One wy to represent n FA tht feels more like mchine is trnsition digrm:
3 3 strt x y z, he vertices re the sttes, nd the edges indicte the input into ech stte. he strt stte there s lwys just one) is explicitly identified, nd the finl sttes lso clled the ccept sttes) re doule circles. his representtion mkes it much esier to see tht this FA ccepts only strings with t lest one in them. echniclly, the sttes themselves don t need to e leled, ut very often it helps to lel the sttes with informtion out wht tht stte represents. ere, the x, y, nd x ren t ll tht useful they re just holdovers from the trnsition tle version of the FA. But we cn lel the sttes with nything we wnt, nd ll the rest of the exmples here will do just tht. Additionl Exmples Construct deterministic finite utomt ccepting ech of the following lnguges:. $ w " + ) * : ech in w is immeditely preceded nd immeditely followed y strt. $ w " +, ) * : w included s sustring, strt
4 4 c. $ w " + ) * : w hs neither nor s sustring,, strt d. $ w " + ) * : w includes oth nd s possily overlpping) sustrings All those strings in the lnguge tht strt with n hve efore. hose strings tht strt egin with the letter hve efore. So I viewed this lnguge, s the union of two lnguges: ) * ) *. Modeling Step By Step Processes hese finite stte mchines re useful to illustrting the vrious degrees of progress one cn mke towrd ny one of numer of gols. he generliztion isn t difficult to follow: rther thn frming computtion in terms of string cceptnce or rejection, you cn decorte trnsitions with rolls of single die, or coin flips, or steps in n lgorithm tht ultimtely termintes. ere s vrition on the Lur nd Ming prolem from our proility dys: he Gme: Aigil, Betrice, nd Christopher keep flipping fir coin until one of their respective ptterns,, or comes up. Drw deterministic finite stte mchine tht trcks the progress of the gme, where ech of three finl sttes corresponds to win for one of the three plyers. You needn t drw edges out of the finl sttes, since t tht point the gme is over.) his relly isn t tht different t ll. It s relly nother string cceptnce prolem where the letters of our lphet re nd. ere s the utomton tht trcks everything for you. By imposing structure on the stepystep process, we re in etter position to nswer questions out the proility tht one will win over the other, or the expected numer of flips needed efore someone wins.
5 5 A strt B C Notice tht the stte lels relly men something this time they re used to convey the most exciting strem of flips for the one plyer closest to winning. A, B, nd C re lels tht denote tht either Aigil, Betrice, or Chris won the gme. And we don t other with trnsitions out of the finl sttes, even though the FA is deterministic everywhere else, ecuse the end of the computtion comes with the end of the gme, not with the end of some string.
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