Tag System - Definition

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.