1. Chapter 6 Languages: finite state machines
Yen-Liang Chen
Dept of Information Management
National Central University

2. 6.1 Language: the set theory of strings
We use  to denote a nonempty finite set of symbols, collectively called an alphabet.
Definition 6.1. If  is an alphabet and nZ+, we define the power of  as follows: (1) 1=; and (2) n+1={xy x, yn}, where xy denotes the juxtaposition of x and y.
Ex 6.1

3. Empty string and sentences
Definition 6.2. 0={}, where  denotes the empty string.
(1)Although , ; (2) {} since ; (3) {} because {}=1.
We refer to the elements of + or * as strings, words, sentences
Ex 6.2

4. Equal and concatenation
Definition 6.4. Two strings w1=x1x2 …xn+ and w2=y1y2…ym+ are equal, written as w1=w2, if n=m and xi=yi for 1in.
Definition 6.5. Let w=x1x2 …xn +. The length of w, denoted as w, is n.
Definition 6.6. Let x=x1x2 …xn+ and y=y1y2…ym+ The concatenation of x and y, xy, is x1x2…xny1y2…ym.
The concatenation of x and  is x=x. The concatenation of  and x is x=x. Finally, the concatenation of  and  is . Since x=x=x, the element  is the identity for the operation of concatenation.

5. Power, prefix and postfix
Definition 6.7. The power of x is defined as: x0=, x1=x, and xn+1=x xn.
Ex 6.3
Definition 6.8. If x, y* and w=xy, then x is a prefix of w, and if y, then x is a proper prefix of w. Similarly, y is a suffix of w; it is a proper suffix when x.

6. Examples
Ex 6.4: Consider the string w=abbcc. What are the prefixes, proper prefixes, suffixes and proper suffixes of w?
Ex 6.6, If w=w1w2=w3w4, then (1) w1 is a prefix of w3, or w3 is a prefix of w1; and (2) w2 is a suffix of w4, or w4 is a suffix of w2.
Let w=(abb)(cc)=(a)(bbcc)

7. Substring and language
Definition 6.9. If x, y, z* and w=xyz, then y is called as a substring of w. When at least one of x and z is different from , we call y a proper substring.
Ex 6.7
Definition For a given , any subset of * is called a language over . This includes , the empty language.
Ex 6.8, Ex 6.9.

8. the concatenation of languages
Definition For languages A , B in * , the concatenation of A and B, denoted AB, is {abaA, bB}.
Note that ABBA and ABBA.
Ex 6.10
Theorem 6.1. For A, B, C *, we have (a) A{}={}A=A; (b) (AB)C=A(BC); (c) A(BC)=ABAC; (d) (BC)A=BCCA; (e) A(BC)ABAC; (f) (BC)A BACA.
x, xy in A; yz in B; z in C xyz in ABAC
But xyz not in A(BC)

9. Closure
Ex A={x}, then (1) A0={}; (2) An={xn}; (3) A+={xn n1}; (4)A*={xn n0}
Ex A={, x, x3, x4,…} and B={xn n0}. Then A2=B2 but AB.

10. Examples
Ex 6.12
A={xx, xy, yx, yy}. A* is the language of all strings w in * where the length of w is even.
A={xx, xy, yx, yy} and B={x, y}. What is BA*?
What is {x}{x, y}*? What is {x}{x, y}+?
What is {x, y}*{yy}?
What is {x}*{y}*?
Why {x}*{y}* {x, y}*?

11. Properties Lemma 6.1. Let A, B*. If AB, then for all nZ+, AnBn.
Theorem 6.2. For A, B *, we have
(a) AAB*,
(b) AB*A;
(c) AB  A*B*,
(d) AB  A+B+,
(e) AA*=A*A=A+,
(f) A*A*=A*=(A*)*=(A*)+=(A+)*;
(g) (AB)*=(A*B*)*=(A*B*)*.

12. Examples
Ex Let ={0, 1} and A*, where each word in A contains exactly one occurrence of the symbol 0. Then the language can be defined as: (a) 0A, (b) 1x and x1 is in A, if x is in A.
Ex Let ={(, )} and A*, where A contains those nonempty strings of parentheses that are grammatically correct for algebraic expressions. Then the language can be defined as:
(a) ( ) is in A;
(b) For all x, y in A, we have
(1) xyA, and
(2) (x)A.

13. The reverse of string Ex 6.16
The reverse of x= x1x2 …xn is xR= xn xn-1…x1.
We can define it recursively: (a) R=; and (2) if x=zyn+1, where z in  and y in n, then xR=(zy)R=(yR)z.
Based on this definition, we can show that for x1, x2*, we have (x1x2)R=x2Rx1R.

14. 6.2 Finite state machine: a first course
The machine can be in only one of finitely many sates at a given time.
The machine will accept as input only a finite number of symbols, referred to as the input alphabet.
An output and a next state are determined by each combination of inputs and internal states.
The machine operates in a deterministic manner.

16. Finite state machine
A finite state machine is a five-tuple M=(S, IA, OA, v, w), where S = the set of internal states fro M; IA is the input alphabet for M; OA is the output alphabet; v: SIAS; w: SIA OA

17. Ex 6. 17

18. Ex 6.18

19. Ex 6.19

20. 6.3 Finite state machine: a second encounter
Ex We want to construct a machine that recognizes each occurrence of the sequence 111 as it is encountered in an input string x*. This machine is a recognizer of the language A= {0, 1}*{111}. For example, if x= , then the output is

21. Ex 6.21.
We want to recognize the occurrence of 111 that ends in a position that is a multiple of 3.

22. Ex 6.22.
We want to recognize the occurrence of 0101 in an input string. Figure 6.12(a). We want to recognize the occurrence of 0101 in an input string but its start position is a multiple of four. Figure 6.12(b).

23. Ex 6.23.
It is impossible to have a finite state machine to represent A={0i1iiZ+}. Suppose we can and let S=n1. Table 6.8 shows the state transition for string 0n+11n+1. Since S=n, there must have two states si and sj , where i< j, such that si=sj. Removing the loop from si+1 to sj, we have the table shown in Figure This new sequence means the machine can accept the string x=0(n+1)-(j-i)1n+1. This is a contradiction.

25. Ex 6.24
One-unit delay machine. If x= x1x2…xn-1xn, then the output will be 0 x1x2…xn-1.

26. Ex 6.25.
Two-unit delay machine. If x= x1x2…xn-1xn, then the output will be 0 0x1x2…xn-2.

27. Definition 6.14.
For si, sjS, sj is reachable from si if si=sj or if there is an input string x such v(si, x)=sj.
A state s is transient if v(s, x)=s for xIA* implies x=. Once leaving, never go back to itself.
A state s is sink if v(s, x)=s for xIA*.
A submachine of M. Let S1S and IA1IA. If v1=vS1IA1:S1IA1S has its range within S1.
A machine is strongly connected if for any states si, sjS, sj is reachable from si.

29. Transfer sequence
For a machine M, let si, sjS. An input string x is called a transfer sequence from si to sj if (a) v(si, x)=sj, (b) for any y with v(si, y)=sjyx.