Tag system

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A tag system is a deterministic computational model published by Emil Leon Post in 1943 as a simple form of string rewriting system. A tag system may also be viewed as an abstract machine, called a Post tag machine (not to be confused with Post-Turing machines) -- briefly, a finite state machine whose only tape is a FIFO queue of unbounded length, such that in each transition the machine reads the symbol at the head of the queue, deletes a fixed number of symbols from the head, and may append symbols to the tail.

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[edit] Definition

A tag system is a triplet (m, A, P), where

  • m is a positive integer, called the deletion number.
  • A is a finite alphabet of symbols, one of which is a special halting symbol. Finite (possibly empty) strings on A are called words.
  • P is a set of production rules, assigning a word P(x) (called a production) to each symbol x in A. The production (say P(H)) assigned to the halting symbol is seen below to play no role in computations, but for convenience is taken to be P(H) = 'H'.

The term m-tag system is often used to emphasise the deletion number. Definitions vary somewhat in the literature [1][2][3][4][5], the one presented here being that of Rogozhin[5].

  • A halting word is a word that either begins with the halting symbol or whose length is less than m.
  • A transformation t is defined on the set of non-halting words, such that if x denotes the leftmost symbol of a word S, then t(S) is the result of deleting the leftmost m symbols of S and appending the word P(x) on the right.
  • A computation by a tag system is a finite sequence of words produced by iterating the transformation t, starting with an initially given word and halting when a halting word is produced. (A computation is not considered to exist unless a halting word is produced in finitely-many iterations.)

For each m > 1, the set of m-tag systems is Turing-complete. In particular, Rogozhin [5] established the universality of the class of 2-tag systems with alphabet {a1, ..., an, H} and corresponding productions {ananW1, ..., ananWn-1, anan, H}, where the Wk are nonempty words.

The use of a halting symbol in the present definition allows the output of a computation to be encoded in the final word, whereas otherwise the sequence of deleted symbols would be used for that purpose.

[edit] Example

2-tag system 
    Alphabet: {a,b,c,H} 
    Production rules:
         a  -->  ccbaH
         b  -->  cca
         c  -->  cc

Computation
    Initial word: baa
                    acca
                      caccbaH
                        ccbaHcc
                          baHcccc
                            Hcccccca (halt).

[edit] Historical note on the definition of tag system

The above definition differs from that of Post [1], whose tag systems use no halting symbol, but rather halt only on the empty word, with the tag operation t being defined as follows:

  • If x denotes the leftmost symbol of a nonempty word S, then t(S) is the operation consisting of first appending the word P(x) to the right end of S, and then deleting the leftmost m symbols of the result — deleting all if there be less than m symbols.

The above remark concerning the Turing-completeness of the set of m-tag systems, for any m > 1, applies also to these tag systems as originally defined by Post.


[edit] Cyclic Tag Systems

A cyclic tag system is a modification of the original tag system. The alphabet consists of only two symbols, 0 and 1, and the production rules comprise a list of productions considered sequentially, cycling back to the beginning of the list after considering the "last" production on the list. For each production, the leftmost symbol of the word is examined -- if the symbol is 1, the current production is appended to the right end of the word; if the symbol is 0, no characters are appended to the word; in either case, the leftmost symbol is then deleted. The system halts if and when the word becomes empty.

[edit] Example

Cyclic Tag System
 Productions: (010, 000, 1111)

Computation
 Initial Word: 11001
    Production         Word
    ----------         --------------
       010             11001
       000              1001010
       1111              001010000
       010                01010000
       000                 1010000
       1111                 010000000
       010                   10000000
        .                      .
        .                      .

Cyclic tag systems were created by Matthew Cook under the employ of Stephen Wolfram, and were used in Cook's demonstration that the Rule 110 cellular automaton is universal. A key part of the demonstration was that cyclic tag systems can emulate a Turing-complete class of tag systems.

[edit] Emulation of tag systems by cyclic tag systems

An m-tag system with alphabet {a1, ..., an} and corresponding productions {P1, ..., Pn} is emulated by a cyclic tag system with m*n productions (Q1, ..., Qn, -, -, ..., -), where all but the first n productions are the empty string (denoted by '-'). The Qk are encodings of the respective Pk, obtained by replacing each symbol of the tag system alphabet by a length-n binary string as follows (these are to be applied also to the initial word of a tag system computation):

a1 = 100...00
a2 = 010...00
.
.
.
an = 000...01

That is, ak is encoded as a binary string with a 1 in the kth position from the left, and 0's elsewhere. Successive lines of a tag system computation will then occur encoded as every (m*n)th line of its emulation by the cyclic tag system.

[edit] Example

This is a very small example to illustrate the emulation technique.

2-tag system
    Production rules: (a --> bb, b --> abH, H --> H)
    Alphabet encoding: a = 100, b = 010, H = 001 
    Production encodings: (bb = 010 010, abH = 100 010 001, H = 001)

Cyclic tag system 
    Productions: (010 010, 100 010 001, 001, -, -, -)

Tag system computation
    Initial word: ba
                    abH
                      Hbb (halt)

Cyclic tag system computation
    Initial word: 010 100 (=ba)

    Production       Word
    ----------       -------------------------------
  * 010 010          010 100  (=ba)
    100 010 001       10 100
    001                0 100 100 010 001
    -                    100 100 010 001
    -                     00 100 010 001
    -                      0 100 010 001
  * 010 010                  100 010 001  (=abH)
    100 010 001               00 010 001 010 010
    001                        0 010 001 010 010
    -                            010 001 010 010
    -                             10 001 010 010
    -                              0 001 010 010
  * 010 010       emulated halt -->  001 010 010  (=Hbb)
    100 010 001                       01 010 010
    001                                1 010 010
    -                                    010 010 001
    ...                                    ...

Every sixth line (marked by '*') produced by the cyclic tag system is the encoding of a corresponding line of the tag system computation, until the emulated halt is reached.

[edit] References

  • [1] Post, E.: "Formal reductions of the combinatorial decision problem", American Journal of Mathematics, 65 (2), 197-215 (1943). (Tag systems are introduced on p.203ff.)
  • [2] Wang, H.: "Tag Systems and Lag Systems", Math. Annalen 152, 65-74, 1963.
  • [3] Cocke, J., and Minsky, M.: "Universality of Tag Systems with P=2", J. Assoc. Comput. Mach. 11, 15-20, 1964.
  • [4] Uspensky, V.A.: "A Post Machine" (in Russian), Moscow, "Nauka", 1979.
  • [5] Rogozhin, Yu.: "Small Universal Turing Machines", Theoret. Comput. Sci. 168, 215-240, 1996.

[edit] External links

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