Tag system

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. All 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 (cf References), the one presented here being that of Rogozhin.

A halting word is a word that either begins with the halting symbol or whose length is less than m.

A transformation t (called the tag operation) 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. (By this definition, a computation is not considered to exist unless a halting word is produced in finitely-many iterations. Alternative definitions allow nonhalting computations, for example by using a special subset of the alphabet to identify words that encode output.)

The use of a halting symbol in the above definition allows the output of a computation to be encoded in the final word alone, whereas otherwise the output would be encoded in the entire sequence of words produced by iterating the tag operation.

A common alternative definition uses no halting symbol and treats all words of length less than m as halting words. Another definition is the original one used by Post 1943 (described in the historical note below), in which the only halting word is the empty string.

Example: A simple 2-tag illustration

This is merely to illustrate a simple 2-tag system that uses a halting symbol.

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).

Example: Computation of Collatz sequences

This simple 2-tag system is adapted from [De Mol, 2008]. It uses no halting symbol, but halts on any word of length less than 2, and computes a slightly modified version of the Collatz sequence.

In the original Collatz sequence, the successor of n is either n/2 (for even n) or 3n + 1 (for odd n). The value 3n + 1 is clearly even for odd n, hence the next term after 3n + 1 is surely 3n + 1/2. In the sequence computed by the tag system below we skip this intermediate step, hence the successor of n is 3n + 1/2 for odd n.

In this tag system, a positive integer n is represented by the word aa...a with n a's.

2-tag system Alphabet: {a,b,c} Production rules: a --> bc b --> a c --> aaa Computation Initial word: aaa <--> n=3 abc cbc caaa aaaaa <--> 5 aaabc abcbc cbcbc cbcaaa caaaaaa aaaaaaaa <--> 8 aaaaaabc aaaabcbc aabcbcbc bcbcbcbc bcbcbca bcbcaa bcaaa aaaa <--> 4 aabc bcbc bca aa <--> 2 bc a <--> 1 (halt)

Turing-completeness of m-tag systems

For each m > 1, the set of m-tag systems is Turing-complete; i.e., for each m > 1, it is the case that for any given Turing machine T, there is an m-tag system that emulates T. In particular, a 2-tag system can be constructed to emulate a Universal Turing machine, as was done by Wang 1963 and by Cocke & Minsky 1964.

Conversely, a Turing machine can be shown to be a Universal Turing Machine by proving that it can emulate a Turing-complete class of m-tag systems. For example, Rogozhin 1996 proved the universality of the class of 2-tag systems with alphabet {a₁, ..., a_n, H} and corresponding productions {a_na_nW₁, ..., a_na_nW_n-1, a_na_n, H}, where the W_k are nonempty words; he then proved the universality of a very small (4-state, 6-symbol) Turing machine by showing that it can simulate this class of tag systems.

The 2-tag halting problem

This version of the halting problem is among the simplest, most-easily described undecidable decision problems:

Given an arbitrary positive integer n and a list of n+1 arbitrary words P₁,P₂,...,P_n,Q on the alphabet {1,2,...,n}, does repeated application of the tag operation t: ijX → XP_i eventually convert Q into a word of length less than 2? That is, does the sequence Q, t¹(Q), t²(Q), t³(Q), ... terminate?

Historical note on the definition of tag system

The above definition differs from that of Post 1943, 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.

Origin of the name "tag"

According to a footnote in Post 1943, B. P. Gill suggested the name for an earlier variant of the problem in which the first m symbols are left untouched, but rather a check mark indicating the current position moves to the right by m symbols every step. The name for the problem of determining whether or not the check mark ever touches the end of the sequence was then dubbed the "problem of tag", referring to the children's game of tag.

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.

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 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.

Emulation of tag systems by cyclic tag systems

An m-tag system with alphabet {a₁, ..., a_n} and corresponding productions {P₁, ..., P_n} is emulated by a cyclic tag system with m*n productions (Q₁, ..., Q_n, -, -, ..., -), where all but the first n productions are the empty string (denoted by '-'). The Q_k are encodings of the respective P_k, 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 . . . a_n = 000...01

That is, a_k is encoded as a binary string with a 1 in the k^th 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.

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.