C ARACTERÍSTICAS DE LA PARTE UTILIZADA DE LA PLANTA

We have already defined a Nondeterministic Turing Machine. For a Nondeterministic Turing Machine NTM = (Z ,K,r,8 ,q ,F )_{N IN O N} we will now construct a NMA simulating NTM in a manner very similar and analogous to the one employed in the previous chapter concerning their deterministic counterparts.

There will be however some small but basic differences in the corresponding construction. We rewrite below the algorithm MA of the previous chapter, whic h was constructed so as to simulate TM.

n+1 n+2 n+K+1 n+K+2 n+K+3 n+K+4 nH-K+5 B'Ç -X ÇB’ B' -X B

block A n rules corresponding to n TM moves not leading to final states block B P ^ .A } B + C C C A + q B'_o

K rules corresponding to K TM moves leading to final states

the unique terminating rule of MA

whenever this rule will be applied, the next be­ comes applicable and MA will never terminate, applying this rule (C C) for ever) .

initial rule (for setting up the string), the first applicable rule of MA.

6.1 Differences in the corresponding c onstructlons 'i The Deterministic MA was constructed in such a way that it was

capable of doing exactly what TM could do and no more. This was achieved in the following way:

a: For every move of TM, a corresponding rule was incorporated among the MA rules. This rule became applicable whenever, in the case of TM, its corresponding move became executable. But exactly the same can be done in the case of the con­ struction of a Nondeterministic Markov Algorithm simulating a Nondeterministic Turing Machine. That is, we will con­ struct (and include among the NMA rules) a NMA rule for every NTM move. Thus, whenever a move (or possibly, moves) will become executable for NTM, a corresponding rule (or several rules, the same number as the TM corresponding moves) will

simultaneously become applicable for MA. (It is assumed here ^ that NTM and NMA are using the same path of next actions.)

b : In the case of the deterministic models, after the application of an applicable rule or after the completion of the appli­ cability search for an inapplicable rule, the next rule tested is uniquely defined. Thus the rule n+K+5 (A-vq^B’), is applied once only and cannot be tested again in the future. Similarly after the application of rule 1 (B'->B), the rules 0 (B’Ç ÇB’) and 1 can never become applicable in the future. This means that after the unique application of rule 1,

applicable rules can only be found among the ones contained in blocks A and B, or among the rules n+K+2, n+K+3, n+K+4. In that construction (deterministic MA for TM), the rules n+K+5, 0 and 1 do not add anything to the power of MA, but they merely give MA the ability to set up a given string it­ self. This modified input string to the main part of MA

can be regarded as a kind of counterpart of TM's head and tape (or indeed a part of TM’s tape actually). The effect of this operation is that if the initially given string is ”w'^ after the application of the setting up rules the string

is modified to"q wB'.‘ _o

The same effect could be achieved by the construction of

two different Algorithms. The former which would only set ■§ up the string, is defined by:

0 : B'5 + CB' 1 : B ’ ,B 3 : A ->• q B ’_o

It is easy to see that this Algorithm if applied to a

string weZ, will modify it to "q^wB" and terminate thereat. • k The second Algorithm will be constructed from the rules of

blocks A and B and from the rules n+K+2, n+K+3, n+K+4. The input string to this Algorithm will be the string"q^wB", if string w is the one initially given.

Alternatively, the second Algorithm alone can be used, provided that every string "w" is modified to "q^wB" before being given an input to it.

Coming now to the construction of a Genuine Nondeterministic Markov Algorithm (GNMA), equivalent to a Nondeterministic Turing Machine (NTM), it should be noted that if we include the rules which set up the string, then GNMA will not be an exact NTM’s simulator.

For the rule n+K+5 (n ->■ q^B), being always applicable, will be applied whenever it is tested and at any position of the string. This would make possible the appearance, in the string, of some "B" characters which may be followed by "non B" characters and the occurrence of more than one "q^". By successive applications of

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rules corresponding to NTM moves some of the q^'s may be modified to q^’s (i^O<n, where n is the number of TM's states). But cor­ responding to this state of affairs is the impossible situation

of NTM being in several states simultaneously and having several # ,y. heads scanning different squares of the tape.

In order to overcome this problem, we exclude the rules necessary for the setting up of the string and before we give to GNMA a string "w" as input, we modify it to "q^wB".

Thus GNMA will be constituted by the rules of blocks A and B and, in addition by the terminating rule (P -> .A). The rules

'B -> O', and 'C ->• C ' are not to be included, and consequently, a B will exist at the end of ever y output string whenever GNMA ter­ minates. The difference from the deterministic MA model, (con­ structed in the previous chapter) as regards the rules of blocks A and B is that in the case of GNMA there may exist rules in blocks A or B or in both with identical LHS's and different RHS's. This will happen because some "state-character" pairs in NTM may result to more than one combinations of "next state-character printed - direction of head's move". For any of these combinations we should include a corresponding rule for GNMA.

GNMA will be Genuine, since we have introduced no restriction in it.. The fact that the LHS of any of its rules can occur once only in the string, at any one time, does not constitute a proper restriction, because there is only one q^ (0^i<n) in the string at a time and there is a q^ in the LHS of every one of the GNMA rules, except for the terminating one, whose LHS will make a unique ap­ pearance somewhere in the string, precisely when this rule becomes applicable.

On the other hand if, alternatively, we wish to keep the auto­ mation, so that the constructed Markov Algorithm should set up the

string itself, we cannot then have a GNMA, but only a restricted type of a Labelled NMA. In such a case the constructed Algorithm will be constituted as follows:

A q^B’ , 1; B'5 ÇB', 1; B' ->- E, {i/V i Block A 0 1 2 3 n+1 : n+2 : Block B n+K+1 : n+K+2 : P . A ; 3al<n+K+3]

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We see that for every rule, beginning with rule 2 and going up to rule n+K+1, there are n+K labels (all the same) for the next rules to be tested. Thus, after the application of rule 2, the Algorithm can be regarded as a Labelled GNMA.

6.2 The constructed GNMA

We can now display the GNMA constructed as equivalent to NTM. It is not significant which rules should be written first and which next since every rule has the same possibility of being chosen as the next to be tested. We may therefore include first the rules which correspond to NTM moves not leading to final states, then

the rules which correspond to NTM moves leading to final states and finally the terminating rule:

GNMA rules block A 11“ 1 n n+K-1 n+K block B

rules corresponding to NTM moves not leading to final states

rules corresponding to NTM moves leading to final states.

terminating rule

Where n and K are the number of NTM moves respectively, not leading and leading to final states with each different combination of "next state-printed character - head's direction" for the same "state-character read" pair being considered as a separate move.

6.3 Equivalence of GNMA and NTM

The proof is very similar to the one followed in the previous chapter for the deterministic counterparts of GNMA and NTM. Thus we can now state that: "For every Nondeterministic Turing

Machine there is a Genuine Nondeterministic Markov Algorithm capable of doing exactly what the Nondeterministic Turing Machine can do, and nothing more".

7. EQUIVALENCE BETWTEEN NONDETERMINISTIC AND DETERMINISTIC