Finite Automata. a) Reading a symbol, b) Transferring to a new instruction, and c) Advancing the tape head one square to the right.


 Milo Craig
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1 Finite Automata Let u begin by removing almot all of the Turing machine' power! Maybe then we hall have olvable deciion problem and till be able to accomplih ome computational tak. Alo, we might be able to gain inight into the nature of computation by examining what computational loe we incur with thi lo of power. If we do t allow writing or twoway operation of the tape head, we have what ha been traditionally called a finite automaton. Thi machine i only allowed to read it input tape and then, on the bai of what it ha read and proceed, accept or reject the input. Thi retricted machine operate by: a) Reading a ymbol, b) Tranferring to a new intruction, and c) Advancing the tape head one quare to the right. When it arrive at the end of it input it then accept or reject depending upon what intruction i being executed. Thi ound very imple. It i merely a oneway, emiliterate Turing machine that jut decide memberhip problem for a living! Let u examine one. In order to depict one, all we need to do i jot down Turing machine intruction in one large table, leave out the write part (that wa t a pun!), and add a te which indicate whether the machine hould accept. Here i an example: Intruction Read Goto Accept? ame next ame next ye ame ame Look cloely at thi machine. It tay on intruction one until it read a one. Then it goe to intruction two and accept any input unle ather one arrive (the ymbol  t ather input). If two or more one appear in the input, then it end up executing intruction three and doe t accept when the input i over. And if one appear in the input the machine remain on intruction one and doe t accept. So, thi machine accept only input that contain exactly one.
2 Finite Automata (N.B. Endmarker are t needed ince the machine jut move to the right. Accepting happen when the machine finihe the input while executing an intruction that call for acceptance. Try thi machine out on the input,,, etc.) Traditionally thee machine have t had intruction but tate. (Recall A. M. Turing' tate of mind.) Ather way to repreent thi ame machine i to put the next intruction or next tate or goto portion under the input in a table like that in figure. There i ather traditional method to decribe finite automata which i extremely intuitive. It i a picture called a tate graph. The tate of the finite automaton appear a vertice of the graph while the tranition from tate to tate under input are the graph edge. The tate graph for the ame machine alo appear in figure. State Input, Accept? ye Figure  Finite Automaton Repreentation (Note that the two circle that urround tate two mean acceptance.) Before continuing let' examine the computation of a finite automaton. Our firt example begin in tate one and read the input ymbol in turn changing tate a neceary. Thu a computation can be characterized by a equence of tate. (Recall that Turing machine configuration needed the tate plu the tape content. Since a finite automaton never write, we alway kw what i on the tape and need only look at a tate a a configuration.) Here i the equence for the input. Input Read: State:
3 Finite Automata Our next example i an elevator controller. Let' imagine an elevator that erve two floor. Input are call to a floor either from inide the elevator or from the floor itelf. Thi make three ditinct input poible, namely:  call  call to floor one  call to floor two The elevator itelf can be going up, going down, or halted at a floor. If it i on a floor it could be waiting for a call or about to go to the other floor. Thi provide u with the ix tate hown in figure along with the tate graph for the elevator controller., W D W Waiting on firt floor,, U About to go up Going up Going down,, W Waiting  econd floor D About to go down W D, Figure  Elevator Control A tate table for the elevator i provided below a table. State W (wait on ) U (tart up) W (wait on ) D (tart down) Input ne call to call to W W W W W U D W W U W D Table  Elevator Control Accepting and rejecting tate are t included in the elevator deign becaue acceptance i t an iue. If we were to deign a more ophiticated elevator, it might have tate that indicated:
4 Finite Automata 4 a) power failure, b) overloading, or c) breakdown In thi cae acceptance and rejection might make ene. Let u make a few mall te about the deign. If the elevator i about to move (i.e. in tate U or D) and it i called to the floor it i preently on it will tay. (Thi may be good  try it next time you are in an elevator.) And if it i moving (up or down) and get called back the other way, it remember the call by going to the U or D tate upon arrival on the next floor. Of coure the elevator doe t do thing like open and cloe door (thee could be tate too) ince that would have added complexity to the deign. Speaking of complexity, imagine having floor. That i our levity for thi ection. Now that we kw what a finite automaton i, we mut (a uual) define it preciely. Definition. A finite automaton M i a quintuple M = (S,I,δ,,F) where: S i a finite et (of tate) I i a finite alphabet (of input ymbol) δ: S I S (next tate function) S (the tarting tate) F S (the accepting tate). We alo need ome additional tation. The next tate function i called the tranition function and the accepting tate are often called final tate. The entire machine i uually defined by preenting a tate table or a tate graph. In thi way, the tate, alphabet, tranition function, and final tate are contructively defined. The tarting tate i uually the lowet numbered tate. Our firt example of a finite automaton i: M = ({,, }, {,}, δ,, { }) where the tranition function δ, i defined explicitly by either a tate table or a tate graph. At thi point we mut make a light detour and examine a very important yet eemingly inignificant input tring called the empty tring. It i a tring without any ymbol in it and i deted a ε. It i t a tring of blank. An example might make thi clear. Look between the bracket in the picture below.
5 Finite Automata 5 A Blank Empty String [ ] [] Let' look again at a computation by our firt finite automaton. For the input, our machine begin in, read a and goe to δ(,) =, then read a and goe to δ(,) =, and end up in δ(,) = after reading the final. All of that can be put together a: δ(δ(δ(,),),) = We call thi tranition on tring δ * and define it a follow. Definition. Let M = (S,I,δ,,F). For any input tring x, input ymbol a, and tate i, the tranition function on tring δ* take the value: δ* (i,(*e) = i δ* (i,a) = δ( i,a) δ* (i,xa) = δ(δ* (i,x),a). That certainly wa tere. But, δ * i really jut what one expect it to be. It merely applie the tranition function to the ymbol in the tring. Let' look at thi for the example in figure. b a a b a,b a,b Thi machine ha a et of tate = {,,, } and operate over the input alphabet {a, b}. It tarting tate i and it et of final or accepting tate, F = { }. The tranition function i fully decribed twice in figure ; once in figure a a a tate table and once in figure b a a tate graph.
6 Finite Automata 6 State Input a b Accept? ye Figure  Finite Automaton If the machine receive the input bbaa it goe through the equence of tate:,,,, while when it get an input uch a abab it goe through the tate tranition:,,,, Now we hall become a bit more abtract. When a finite automaton receive an input tring uch a: x = x x... x n where the x i are ymbol from it input alphabet, it progree through the equence: k, k,..., k n+ where the tate in the equence are defined a: k k k kn+ = = δ( = δ( k k = δ(,x ) = δ(,x ),x ) = δ (! kn,x ) = δ ( n Getting back to a more intuitive reality, the following table provide an aignment of value to the ymbol ued above for an input of bbaba to the finite automaton of figure.,x x,x x )...x n ) i x i b b a b a k i
7 Finite Automata 7 We have mentioned acceptance and rejection but have t talked too much about it. Thi can be made precie alo. Definition. The et (of tring) accepted by the finite automaton M = (S,I,δ,,F) i: T(M) = { x δ* (,x) F } Thi et of accepted tring (named T(M) to mean Tape of M) i merely all of the tring for which M ended up in a final or accepting tate after proceing the tring. For our firt example (figure ) thi wa all tring of ' and ' that contain exactly one. Our lat example (figure..) accepted the et of tring over the alphabet {a, b} which began with exactly two b'.
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