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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 nZ+, we define the power of as follows: (1) 1=; and (2) n+1={xy x, yn}, 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 w1=x1x2 …xn+ and w2=y1y2…ym+ are equal, written as w1=w2, if n=m and xi=yi for 1in. 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.

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**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.

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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=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)

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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 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 For languages A , B in * , the concatenation of A and B, denoted AB, is {abaA, bB}. Note that ABBA and ABBA. Ex 6.10 Theorem 6.1. For A, B, C *, we have (a) A{}={}A=A; (b) (AB)C=A(BC); (c) A(BC)=ABAC; (d) (BC)A=BCCA; (e) A(BC)ABAC; (f) (BC)A BACA. x, xy in A; yz in B; z in C xyz in ABAC But xyz not in A(BC)

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Closure Ex A={x}, then (1) A0={}; (2) An={xn}; (3) A+={xn n1}; (4)A*={xn n0} Ex A={, x, x3, x4,…} and B={xn n0}. Then A2=B2 but AB.

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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 AB, then for all nZ+, AnBn.**

Theorem 6.2. For A, B *, we have (a) AAB*, (b) AB*A; (c) AB A*B*, (d) AB A+B+, (e) AA*=A*A=A+, (f) A*A*=A*=(A*)*=(A*)+=(A+)*; (g) (AB)*=(A*B*)*=(A*B*)*.

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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) 0A, (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) xyA, and (2) (x)A.

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**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=zyn+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.

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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: SIAS; w: SIA 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 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

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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={0i1iiZ+}. Suppose we can and let S=n1. 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.

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

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

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Definition 6.14. For si, sjS, 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 xIA* implies x=. Once leaving, never go back to itself. A state s is sink if v(s, x)=s for xIA*. A submachine of M. Let S1S and IA1IA. If v1=vS1IA1:S1IA1S has its range within S1. A machine is strongly connected if for any states si, sjS, sj is reachable from si.

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Transfer sequence For a machine M, let si, sjS. 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)=sjyx.

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Language and Automata Theory

Language and Automata Theory

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