Chapter 6 Languages: finite state machines

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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) ?n1xy? 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 w1x1x2 xn?? and
  w2y1y2ym?? are equal, written as w1w2, if nm
  and xiyi for 1?i?n.
• Definition 6.5. Let wx1x2 xn ??. The length
  of w, denoted as ?w?, is n.
• Definition 6.6. Let xx1x2 xn?? and
  yy1y2ym?? The concatenation of x and y, xy, is
  x1x2xny1y2ym.
• The concatenation of x and ? is x?x. The
  concatenation of ? and x is ?xx. 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?, x1x, and xn1x xn.
• Ex 6.3
• Definition 6.8. If x, y?? and wxy, 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 wabbcc. What are the
  prefixes, proper prefixes, suffixes and proper
  suffixes of w?
• Ex 6.6, If ww1w2w3w4, 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 wxyz, 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??AA (b) (AB)CA(BC) (c) A(B?C)AB?AC
  (d) (B?C)ABC?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. Ax, then (1) A0? (2) Anxn (3)
  Axn? n?1 (4)Axn? n?0
• Ex 6.13. A?, x, x3, x4, and Bxn? n?0. Then
  A2B2 but A?B.

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Examples

• Ex 6.12
• Axx, xy, yx, yy. A is the language of all
  strings w in ? where the length of w is even.
• Axx, xy, yx, yy and Bx, y. What is BA?
• What is xx, y? What is xx, y?
• What is x, yyy?
• What is xy?
• Why xy? x, y?

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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?BA
• (c) A?B ? A?B,
• (d) A?B ? A?B,
• (e) AAAAA,
• (f) AAA(A)(A)(A)
• (g) (A?B)(A?B)(AB).

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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 x1x2 xn is xR xn xn-1x1.
• We can define it recursively (a) ?R? and (2)
  if xzy??n1, 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)Rx2Rx1R.

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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,
  1111. For example, if x1110101111, 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 A0i1i?i?Z. Suppose we can and
  let ?S?n?1. Table 6.8 shows the state transition
  for string 0n11n1. Since ?S?n, there must have
  two states si and sj , where ilt j, such that
  sisj. Removing the loop from si1 to sj, we have
  the table shown in Figure 6.10. This new sequence
  means the machine can accept the string
  x0(n1)-(j-i)1n1. This is a contradiction.

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Ex 6.24

• One-unit delay machine. If x x1x2xn-1xn, then
  the output will be 0 x1x2xn-1.

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Ex 6.25.

• Two-unit delay machine. If x x1x2xn-1xn, then
  the output will be 0 0x1x2xn-2.

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Definition 6.14.

• For si, sj?S, sj is reachable from si if sisj 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
  v1v?S1?IA1S1?IA1?S has its range within S1.
• A machine is strongly connected if for any states
  si, sj?S, sj is reachable from si.

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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?.