1. COSC 3340: Introduction to Theory of Computation
University of Houston
Dr. Verma
Lecture 18
Lecture 18
UofH - COSC Dr. Verma

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 TM’s 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.
Lecture 18
UofH - COSC Dr. Verma

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
Lecture 18
UofH - COSC Dr. Verma

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
Lecture 18
UofH - COSC Dr. Verma

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 TM’s 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
Lecture 18
UofH - COSC Dr. Verma

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.
Lecture 18
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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
M1 M2
M1 M2 M3 a b
After M1 halts either M2 or M3 would start depending on the symbol.
Lecture 18
UofH - COSC Dr. Verma

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
Lecture 18
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9. M = ({q0, q1 }, {a, b}, {a, b, □}, , q0, {})
Example: R□
Alphabet () = {a, b}
M = ({q0, q1 }, {a, b}, {a, b, □}, , q0, {}) a R□
State
Symbol
Next state action q0 □
(q1, □, R) a
(q1, a, R) b
(q1, b, R) q1
halt R b
L□ is very similar to R□
Lecture 18
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10. JFLAP SIMULATION
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11. JFLAP SIMULATION
Lecture 18
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12. JFLAP SIMULATION
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13. JFLAP SIMULATION
Lecture 18
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14. JFLAP SIMULATION
Lecture 18
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15. Example: Machine 1 >L□ R□ R □(R□)2 (L□)2   □ □ Lecture 18
UofH - COSC Dr. Verma

16. Example: Machine 1 (contd.)
□abc□a
□ab□□a
□ab□□ab□
□a□c□abc
□ab□□abc
□abc□abc
□abc□abc□
□abc□
□□bc□
□□bc□□
□□bc□a
□abc□a
□abc□a
□a□c□a
□a□c□a□
□a□c□ab
□abc□ab
Let C be the TM,
we say C transforms □w□ into □w□w□ (Copying Machine)
Lecture 18
UofH - COSC Dr. Verma

17. Example: Machine 2   □ >L□ LR R □ L□ Lecture 18
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18. Example: Machine 2 (contd.)
□ab□
aab□
aab□
abb□
ab□
Let SL be the TM,
we say SL transforms □w□ into w□ (Shift-Left Machine)
Lecture 18
UofH - COSC Dr. Verma

19. Example: Machine 3 I >aL□L □ L aRa□CaLa□ I □ □ □ aRa□ LI IRL□ a □
Lecture 18
UofH - COSC Dr. Verma

20. Example: Machine 3 (contd.)
f(2,2) = ?
□II□II□
□II□IIa
□I□□IIa
□a□□IIa
□a□□II□
□a□□II□II□
□a□□II□IIa
□□□□II□IIa
a□□□II□IIa
a□□□II□II□
a□□□IIIII□
a□□□IIIII□
a□□□IIII□□
a□□IIIII□□ .
□IIII□□□□□
f(n, m) = n  m is computed by the TM
(Multiplication Machine)
Lecture 18
UofH - COSC Dr. Verma

21. Turing Machine Models Variants of TM model Two-way infinite tape
Multiple tapes
Multiple heads on each tape
Multi-dimensional tapes, and
Combinations of the above.
Lecture 18
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22. Are the variants more powerful than the basic model?
Ans: No
All "reasonable" extensions including those listed before lead to the same classes of languages or functions. Proved by showing that the basic model can simulate the extensions.
Lecture 18
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23. Techniques for Simulations
The basic TM has
expanded set of states
expanded alphabet
Note: one step in extended model is usually simulated by many steps in the basic model.
Lecture 18
UofH - COSC Dr. Verma