Deterministic%20Turing%20Machines

1
Reminder
What is it about?
Models of Language Recognition
Models of Language Generation
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Machine Models Automata, that recognize
(understand) the languages
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Deterministic Turing Machines
Read/write tape
q0
Two-way read/write head
Finite State Control

• The Turing machine (TM) is a computational model
  which has a tape of infinite length divided into
  cells and a finite control unit with a two-way
  read/write head. In a move (i.e, a computational
  step) the TM, depending upon the symbol scanned
  by the head and current state of the control,
  does the following
• (1) Writes a symbol on the tape cell under the
  head, replacing what was written,
• (2) changes the state of its control, and
• (3) moves the head to the left, right or does not
  move.

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Deterministic Turing Machines

• Formally a TM is defined as a 6-tuple M ( Q, ?
  , ? , ? , q0 , F ), where
• Q is the finite set of states,
• ? is the set of tape symbols (alphabet)
  including the blank symbol, usually denoted by B
  (a square box in the text),
• ? is the set of input symbols, which is a subset
  of ? ,
• q0 is the start state,
• F is the set of accepting states (F ? Q ), and
• ? is the transition function which maps Q ? ?
  ? Q ? ? ? L, N, R,

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Deterministic Turing Machines

• where L, N, and R denote a move to the left,
  no move, and a move to the right, respectively.
  The transition function ? may be undefined for
  some arguments. In this case we assume that the
  machine halts. It is assumed that initially an
  input string x ? ? is written on the tape and
  the machine is reading the leftmost symbol in the
  start state q0.
• Definition We say a string x is accepted by a
  TM M if M, given x on the tape, enters an
  accepting state after some finite number of
  moves.
• Definition The language recognized (or
  accepted) by a TM M, denoted by L(M), is the set
  of strings accepted by the machine, i.e., L(M)
  x M accepts x.

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Example
Construct a deterministic Turing Machine which
accepts 0n1n n ? 1.

• An idea Traveling back and forth replace "1" by
  Y for each "0" replaced by X. The transition
  function ? is defined as follows.

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Transition Fuction

• ?(q0, 0) (q1, X, R) In q0 , reading 0, change
  it to X,
• ?(q1, 0) (q1, 0, R) move to the right
  searching 1 in q1.
• (q1, 1) (q2, Y, L) Reading 1, replace it by Y,
  and
• (q2, 0) (q2, 0, L) move to the left in q2
  searching X.
• ?(q2, X) (q3, X, R) Reading X, back up in q3 ,
  and
• ?(q3, 0) (q1, X, R) reading 0 repalce it by
  X.
• ?(q1, Y) (q1, Y, R) In q1 , move over Y to the
  right.
• ?(q2, Y) (q2, Y, L) In q2 , move over Y to the
  left.
• ?(q3, Y) (q4, Y, R) Reading Y, instead of 0,
  enter q4,
• ?(q4, Y) (q4, Y, R) and move over Y to the
  right, till
• ?(q4, B) (q5, B, N) there appears blank, and
  accept in q5.

q4
q5
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Transition Table
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Transition Graph
(0,0,L) (Y,Y,L)
(0,0,R) (Y,Y,R)
(Y,Y,R)
q2
(1,Y,L)
q1
(X,X,R)
q4
(0,X,R)
(0,X,R)
(B,B,N)
q0
(Y,Y,R)
q3
start
q5

• Now, the machine is formally defined as follows
• M ( Q, ? , ? , ? , q0 , F ), where Q q0, q1,
  q2, q3, q4, q5, ? 0, 1,
• ? 0, 1, X, Y, B, F q5, and ? is given in
  terms of one of the above forms,
• i.e, state transition function, state transition
  table or state transition graph.

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Restricted Turing Machines (conted)
(1) Deterministic linear bounded automata (DLBA)

• The head cannot cross the left and right boundary
  markers ( and ), i.e., the computation must be
  done within the range of the input string. Other
  operations are the same as in the standard TM.
  Formally a DLBA is defined as a tuple
• M (Q, ? , ? , ? , q0, , ,
  F),
• where, excepts for the boundary markers ' and
  ', all the elements of the tuple are defined
  the same way as for the Turing machines.

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Restricted Turing Machines (conted)
(2) Deterministic pushdown automata (DPDA)

• M ( Q, ? , ? , ? , q0 , Z0, F )
• ? Q ? ( ? ? ? ) ? ? ? Q ? ?

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Restricted Turing Machines (conted)

• Formally a DPDA is a tuple M ( Q, ? , ? , ? ,
  q0 , Z0, F ), where Q, ? , ? , ? , q0 , Z0 and F
  are the set of states, the input alphabet, the
  stack alphabet, the start state, the bottom of
  stack symbol and the set of accepting states,
  respectively. The transition function ? is a
  mapping
• ? Q ? ( ? ? ? ) ? ? ? Q ? ?
• For example ? ( q , a, Z ) ( p, ? ), where q ,
  p ? Q, a ?? , Z ? ? , and ? is a string of
  stack symbols written on top of the stack
  replacing Z. Conventionally, we assume that
  either
• ? 0, i.e., ? ? , means the machine pops
  the stack top symbol,
• ? 1, i.e., rewrites the stack top symbol, or
• ? 2, i.e., pushes a symbol on top of the
  stack.

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Restricted Turing Machines (conted)
Example Two transitions ? ( p , a, Z ) ( q,YX
) and ? ( q, b, Y ) ( r, ? ) in sequence
implies the following.

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Examples
(a) Construct a DPDA which recognizes aibi i gt
0
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Examples (conted)
(b) Construct a DPDA which recognizes aibj i gt
j gt 0
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Examples (conted)
(c) Construct a DPDA which recognizes aibj j gt
i gt 0
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Examples (conted)
(d) Construct a DPDA which recognizes aibkci
k, i gt 0 ? aibkdk k, i gt 0
Notice that this machine needs ?-move, i.e.,
takes a step without reading the input.
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Restricted Turing Machines (conted)

• The input head should move to the right when
  it reads the input. The ? in a
• transition ? (q , ? , Z ) ( p, ? ) means the
  machine does not read the input. We
• call such transition ? -move. The input head
  does not move when it does not
• read. An important restriction is that a DPDA
  cannot have both transitions ? (q , a,
• Z ) and ? (q, ? ,Z) defined. Otherwise, it is a
  nondeterministic PDA, which will be
• discussed later. The language accepted by a DPDA
  M is defined as
• L(M)x M enters an accepting state some
  time after reading the last symbol of
• the input x.
• Notice that, for an input string x to be
  accepted, the PDA should enter an
• accepting state and the whole string must be
  read.

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Restricted Turing Machines (conted)
(3) Deterministic finite state automata (DFA)

• A DFA is defined as a tuple M ( Q, ? , ? , ?
  , q0 , F ). One-way DFA is a DFA whose head is
  not allowed to move to the left. The language
  accepted by a DFA M is defined as
• L(M) x M enters an accepting state after
  reading the last symbol of input x.
• It is proven that one-way FA's and two-way FA's
  are equivalent in the sense that they can accept
  the same languages.

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Example
Construct a DFA which recognizes the following
language x x is a binary number, i.e., x ?
0, 1, which is divisible by 3
State id indicates current remainder.