Chapter 6 Languages: finite state machines

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 6.10. 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 6.11. 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 6.11. A={x}, then (1) A0={}; (2) An={xn}; (3) A+={xn n1}; (4)A*={xn n0} • Ex 6.13. 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 6.14. 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 6.15. 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 6.20. 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=1110101111, then the output is 0010000011.

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 siand sj , where i< j, such that si=sj. Removing the loop from si+1 to sj, we have the table shown in Figure 6.10. 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.