Markov algorithm simulates Post machine, Equivalence of TMs, PMs and Markov Algorithms

Markov algorithm simulates Post machine:

Given a random Post machine P we build a Markov algorithm M which simulates P. M consists of one rewrite rule for each transition of P and a few other rules which move newly made characters to the right end of M’s data string, corresponding to the tail of  P’s queue. For simplicity’s sake we limit P as follows:

P’s alphabet is {0, 1, #}, and P’s transitions are of form q, x -> q’, y with |x| ≤ 1 and |y| ≤ 1.

In common, M’s alphabet can be selected as {0, 1, #, q} plus a single marker α. The given algorithm which compresses runs of 0s to a single 0 and runs of 1s to a single 1 serves as an illustration to describe the construction of M’s rewrite rules.

Illustration: Compress runs of similar symbol to a single symbol, illustration: 00011011100# - > 01010#.

Structure of P.  The beginning state q finds out whether the very first run is a run of 0s or of 1s, rotates this symbol, and transfers control to state z, ‘zeros’, or y, ‘ones’, accordingly. The state z recalls that P is presently reading a run of 0s, that this run has already produced a 0 in the output; thus it removes all further 0s of this run.

Analogously for the state y. In any state, # terminates the transformation.

2420_post machine simulation.jpg

For sake of simplicity, we select the alphabet of the Markov algorithm M as {0, 1, #, q, z, y}, where q, z, y are the names of corresponding states. This convention recommends that M’s alphabet expands with the size of Post machine P to be simulated. From the theoretical point of view, though, it is more elegant to encompass a fixed Markov alphabet for any Post machine to be simulated. When P has states q1, q2, .. qs, for illustration, M can code such, for illustration, as q0q, q00q, q0...0q with fixed alphabet {0, 1, #, q}.

The Markov algorithm M which simulates P codes the input and outcome string as:  q00011011100# - > 01010.

The input begins with the identifier of P’s beginning state q. Other states are coded as z and y.

M comprises of 4 sets of rewrite rules, one for all state of P and one for moving newly made characters to the right end of M’s data string. The rewrite rules related with the states are in 1-to-1 correspondence with P’s transitions - therefore, this portion of the Markov algorithm is a mirror image of P. We follow the convention which rules written on similar line can appear in any order, while rules written on various lines should be executed from top to bottom. The set of rules which move a character should appear first, the rules corresponding to P’s states can follow in any order.

Move a newly made character to right:

A) α 0 0 ->  0 α 0, α 0 1 -> 1 α 0, α 0 # ->  # α 0, α 1 0 -> 0 α 1, α 1 1 ->1 α 1, α 1 # ->  # α 1

B) α -> ε

Rules for beginning state q:  C) q 0 -> z α 0, q 1 -> y α 1, q # ¬ # (that is, terminating rule)
Rules for state z:  D)   z 0 -> z, z 1 -> y α 1, z # ¬ # (that is, terminating rule)
Rules for state y:   E) y 1 -> y, y 0 -> z α 0, y # ¬ # (that is, terminating rule)
Initialization:  F) ε -> q

Execution trace on the input 1 1 0 1 1 # with outcome 1 0 1 #.

727_execution trace.jpg

Explanation of the rules. Each transition of the form q, u -> q’, v of a Post machine, if executed, causes a corresponding Markov rule of the form q u -> q’ α v to be performed. The only difference among such two is that the Post machine appends v at right end, while the Markov algorithm inserts v in the direction of left end of data string. Thus, the Markov rule produces a marker α whose task is to push v to far right. The rules including α have top priority, and hence the right shift of v gets terminated before any new Post transition is simulated.

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