Tag system

A tag system is a deterministic computational model published by Emil Leon Post in 1943 as a simple form of Post canonical 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 to the tail appends a symbol-string preassigned to the deleted symbol. (Because all of the indicated operations are performed in each transition, a tag machine strictly has only one state.)

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

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.

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.

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

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.

...
Wikipedia