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

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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

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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

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Equal and concatenation Definition 6.4. Two strings w 1 =x 1 x 2 …x n + and w 2 =y 1 y 2 …y m + are equal, written as w 1 =w 2, if n=m and x i =y i for 1 i n. Definition 6.5. Let w=x 1 x 2 …x n +. The length of w, denoted as w , is n. Definition 6.6. Let x=x 1 x 2 …x n + and y=y 1 y 2 …y m + The concatenation of x and y, xy, is x 1 x 2 …x n y 1 y 2 …y m. 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.

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Power, prefix and postfix Definition 6.7. The power of x is defined as: x 0 =, x 1 =x, and x n+1 =x x n. 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 .

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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=w 1 w 2 =w 3 w 4, then (1) w 1 is a prefix of w 3, or w 3 is a prefix of w 1 ; and (2) w 2 is a suffix of w 4, or w 4 is a suffix of w 2. Let w=(abb)(cc)=(a)(bbcc)

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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.

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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)

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Closure Ex 6.11. A={x}, then (1) A 0 ={ }; (2) A n ={x n }; (3) A + ={x n n 1}; (4)A*={x n n 0} Ex 6.13. A={, x, x 3, x 4,…} and B={x n n 0}. Then A 2 =B 2 but A B.

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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} * ?

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Properties Lemma 6.1. Let A, B *. If A B, then for all n Z +, A n B n. 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*)*.

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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.

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

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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.

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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

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Ex 6. 17

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Ex 6.18

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Ex 6.19

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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.

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

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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).

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Ex 6.23. It is impossible to have a finite state machine to represent A={0 i 1 i i Z + }. Suppose we can and let S =n 1. Table 6.8 shows the state transition for string 0 n+1 1 n+1. Since S =n, there must have two states s i and s j, where i< j, such that s i =s j. Removing the loop from s i+1 to s j, 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) 1 n+1. This is a contradiction.

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Ex 6.24 One-unit delay machine. If x= x 1 x 2 …x n-1 x n, then the output will be 0 x 1 x2…x n-1.

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

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Definition 6.14. For s i, s j S, s j is reachable from s i if s i =s j or if there is an input string x such v(s i, x)=s j. 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 S 1 S and IA 1 IA. If v 1 =v S1 IA1: S 1 IA 1 S has its range within S 1. A machine is strongly connected if for any states s i, s j S, s j is reachable from s i.

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Transfer sequence For a machine M, let s i, s j S. An input string x is called a transfer sequence from s i to s j if (a) v(s i, x)=s j, (b) for any y with v(s i, y)=s j y x .

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