COSC 3340: Introduction to Theory of Computation

1
COSC 3340 Introduction to Theory of Computation

• University of Houston
• Dr. Verma
• Lecture 18

2
Combining Turing Machines

• A Turing Machine can be a component of another
  Turing Machine (subroutine)
• We will develop a graphical notation to build
  larger machines for more complex tasks easily.
  The scheme is hierarchical.
• Combination is possible since all TMs are
  designed to be non-hanging so the first
  machine can save something on the leftend of the
  tape.

Reference Elements of the Theory of
Computation by H.R. Lewis and C.H.
Papadimitriou.
3
Combining Turing Machines

• Assumptions, for convenience
• From now on, Turing machines can either write a
  symbol or move head but not both in the same
  move.
• All TMs have only one alphabet ?, containing the
  blank symbol
• All machines start in this position w
• There are two types of basic machines
• Symbol-writing
• Head-moving

4
Basic Machines Symbol-Writing

• There are ? symbol-writing machines, one for
  each symbol in ?. Each TM simply writes a
  specified symbol in the currently scanned tape
  square and halts.
• Formally, the TM which writes a is
• Wa (K, ?, ?, s), where
• K q for some arbitrarily chosen state q
• s q and
• ?(q, b) (h, a) for each b ? ?

Notation Wa
5
Basic Machines Head-Moving

• The head-moving machines simply move the head one
  square to the left or right, and then halt.
• Formally, the TMs are
• VL (q, ?, ?L, q), where
• ?L(q, a) (h, L) for each a ? ?
• VR (q, ?, ?R, q), where
• ?R(q, a) (h, R) for each a ? ?

let q be some state
Notation L and R
let q be some state
6
Rules for Combining Machines

• Machines may be connected just like a Finite
  Automaton.
• If two machine are connected, then the first
  machine has to halt before the other machine
  starts.

7
Rules for Combining Machines
If M1 and M2 are any TM Start in the initial
state of M1 operate as M1 would operate until M1
would halt then initiate M2 and operate as M2
would operate
After M1 halts either M2 or M3 would start
depending on the symbol.
8
Abbreviations

• L/R A TM that moves the head one cell to the
  left/right
• R? - A TM that moves to the Right seeking ?
• L? - A TM that moves to the Left seeking ?
• R? - A TM that moves to the Right seeking a
  nonblank spot
• L? - A TM that moves to the Left seeking nonblank
  spot

9
Example R?

Alphabet (?) a, b
M (q0, q1 , a, b, a, b, ?, ?, q0, )
a
State Symbol Next state action
q0 ? (q1, ?, R)
q0 a (q1, a, R)
q0 b (q1, b, R)
q1 ? halt
q1 a (q1, a, R)
q1 b (q1, b, R)
R?
R
b
L? is very similar to R?
10
JFLAP SIMULATION
11
JFLAP SIMULATION
12
JFLAP SIMULATION
13
JFLAP SIMULATION
14
JFLAP SIMULATION
15
Example Machine 1
16
Example Machine 1 (contd.)
?abc?a ?ab??a ?ab??ab? ?a?c?abc ?ab??abc ?abc?abc
?abc?abc ?abc?abc?
?abc? ?abc? ??bc? ??bc?? ??bc?a ??bc?a ?abc?a
?abc?a ?a?c?a ?a?c?a? ?a?c?ab ?a?c?ab ?abc?ab
Let C be the TM, we say C transforms ?w? into
?w?w? (Copying Machine)
17
Example Machine 2
? ? ?
gtL?
L?R
R
?
L?
18
Example Machine 2 (contd.)
?ab? ?ab? ?ab? ?ab? aab? aab?
aab? aab? abb? abb? ab?
Let SL be the TM, we say SL transforms ?w? into
w? (Shift-Left Machine)
19
Example Machine 3
I
gtaL?L
?
aRa?CaLa?
L
I
?
?
?
aRa?
IR