lecture 18uofh - cosc 3340 - dr. verma 1 cosc 3340: introduction to theory of computation university...

Lecture 18 UofH - COSC 3340 - Dr. Verma1

COSC 3340: Introduction to Theory of Computation

University of Houston

Dr. Verma

Lecture 18


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.


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


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 qs = q and(q, b) = (h, a) for each b

Notation: Wa


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

let q be some state

Notation: L and R


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.


Rules for Combining Machines

M1 M2

M1 M2

M3

a

b

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.


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


Example: R□

R

a

b

R□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)

M = ({q0, q1 }, {a, b}, {a, b, □}, , q0, {})

L□ is very similar to R□

Alphabet () = {a, b}


JFLAP SIMULATION


Example: Machine 1

>L□

R□

R □(R□)2 (L□)2 □

□


Example: Machine 1 (contd.)

□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

□abc□a□ab□□a□ab□□ab□□a□c□abc□ab□□abc□abc□abc□abc□abc□abc□abc□

Let C be the TM, we say C transforms □w□ into □w□w□ (Copying Machine)


Example: Machine 2

>L□

L□

R LR

□

□



□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)


Example: Machine 3

>aL□L

Ra

□ aRa□CaLa□

□

LI

aRa□ IR�L□ □

LI

□R�

R

□La□

a

□I

a

I

□



f(n, m) = n m is computed by the TM

(Multiplication Machine)

□II□II□□II□IIa□II□IIa□II□IIa□I□□IIa□I□□IIa□a□□IIa□a□□IIa□a□□II□□a□□II□II□

□a□□II□IIa□a□□II□IIa□□□□II□IIa□□□□II□IIaa□□□II□IIaa□□□II□IIaa□□□II□II□a□□□II□II□a□□□IIIII□a□□□IIIII□

a□□□IIIII□a□□□IIII□□a□□□IIII□□a□□IIIII□□a□□IIIII□□

.

.

.□IIII□□□□□

f(2,2) = ?


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.


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.


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 18uofh - cosc 3340 - dr. verma 1 cosc 3340: introduction to theory of computation university...

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