A universal machine
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A Universal Machine. A universal Turing machine is designed to simulate the computations of an arbitrary Turing machine M. Universal machine U. M halts with w. Accept. R(M) w. Loop. M does not halt with input w.

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A Universal Machine

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A Universal Machine

A universal Turing machine is designed to simulate the computations of an arbitrary Turing machine M.

Universal machine U

M halts with w

Accept

R(M)w

Loop

M does not halt with input w

R(M) represents a Turing machine M that accepts by halting.W represents the input string.


A Universal Machine

Universal machine U

M halts with w

Accept

R(M)w

Loop

M does not halt with input w

If M halts and accepts input w ===>U halts and accepts input W tooIf M does not halt with W ===>Neither does U.


q0

q1

q2

Turing Machine Example with Halts

0/0 L

B/B R

1/1 R

1/1 L

Run JFLAP


Transition Function

en(qi)0en(x)0en(qj)0en(y)0en(d)

Input alphabet {0,1}, x and y belongs to {0,1,B}tape alphabet {0, 1, B}states q0, q1 ….move to left (L) or right (R)


Symbol -- Encoding

Input alphabet {0,1}tape alphabet {0, 1, B}states q0, q1 ….


0/0 L

B/B R

1/1 R

q0

q1

q2

1/1 L

Universal Machine Example

00010111011011101100110101010100….00….000


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

Put q0 as the start state


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

string w

tape 3:


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

string w

tape 3:


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

string w

tape 3:


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

string w

tape 3:


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

string w

tape 3:


Three-tape Deterministic Universal Machine

00010111011011101100110101010100….00….000

tape 1:

current / updated state

tape 2:

string w

tape 3:

U halts accepting the input if there is no such transition!!


CSE355An Introduction to the Theory of Computer Science

Good luck your finals

&

Enjoy your Winter break …


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