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  Theory of Universal Turing machine, Turing Machines

Universal Turing machine:

The universal TM U simulates any random TM M, given its description <M>. The presence of U is frequently employed in proofs of undecidability by stating ‘TM X simulates TM Y, and when Y halts, does so-and-so’. <M> can be considered to be a program for interpreter U. Naturally; U might be a lot slower than TM M it simulates, as U has to run back and forth all along its tape, finding out the suitable instruction from <M>, then functioning it on M’s data.

Whenever designing U, we have to state a code appropriate for explaining random TMs. As U consists of a fixed alphabet A, while arbitrary TMs might have arbitrarily big alphabets, the later should be coded. We suppose this has been completed, and then TM

M to be simulated is given by:

M = (Q, A, f: Q x{0, 1} -> Q x {0, 1} x {L, R, ..}, q0, ..}.

U can be constructed in many distinct ways. For simplicity of understanding, we suppose U have three tapes: T, D and S.

1238_universal TM.jpg

U’s three tapes have the given roles:

A) U’s tape T is at all times a precise copy of M’s tape T, comprising the place of the read or write head.

B) D = <M> is the explanation of M as a sequence of M’s tuples, in some code like #q, a -> q’, b, m#. Here q and q’ are codes for the states in Q. For illustration, qk ∈ Q might be coded as binary representation of k. Likewise, m is a code for M’s tape actions, example: L or R. #, comma, and -> are delimiting markers. In order to build M’s tuples intuitively readable to humans, we have introduced many distinct symbols than essential - a single delimiter, e.g. # is enough. Whatever symbols we introduce define the alphabet A’.

In principle, U just needs read-only access to D, however for purposes of the matching strings of random length it might be convenient to have read or write access, and temporarily transform the symbols on D.

C) The third tape S comprises the pair (q, a) of M’s present state q and the presently scanned symbol a on T. The latter is redundant, as U has this similar information on its own copy of T. However having the pair (q, a) altogether is convenient whenever matching it against the left-hand side of M’s tuples on D.

Therefore, U = (P, A2, g: P x A2 -> P x A2 x LR3, p0) appears somewhat complicated. P is U’s state space; p0 is U’s initial state. A2 = {0, 1} x A’ x A’ is the alphabet, and LR3 = {L, R, ..} x {L, R, ..}x {L, R, ..} is a set of probable tape actions of this 3-tape machine. U begins in an initial configuration comprising of p0, tapes T, D, S initialized with the appropriate content and suitable head positions on all the 3 tapes. The interpreter U consists of the following main loop:

While no halting condition occurs do

begin

 A) Match the pair (q, a) on S to left-hand side of a tuple #q, a -> q’, b, m# on D
 B) Write b to T, execute the tape action m on T, and scan the present symbol c on T
 C) Write the string (q’, c) to S

end.

Halting the conditions based on the accurate definition of M, like entering a halting state or executing the halting transition. In sum up, a universal TM requires nothing more complicated than copying and matching the strings.

Designing the universal TM becomes tricky when we aim at ‘small is beautiful’. There is a continuing competition to design the smallest probable universal TM as measured by state-symbol product |Q| x |A|.

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