A tag system is a deterministic
computational model
A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning from physics, chemistry and biology to economics, psychology, ...
published by
Emil Leon Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Govern ...
in 1943 as a simple form of a
Post canonical system
A Post canonical system, also known as a Post production system, as created by Emil Post, is a string-manipulation system that starts with finitely-many strings and repeatedly transforms them by applying a finite set j of specified rules of a cert ...
. A tag system may also be viewed as an abstract machine, called a Post tag machine (not to be confused with
Post–Turing machine
A Post–Turing machineRajendra Kumar, ''Theory of Automata'', Tata McGraw-Hill Education, 2010, p. 343. is a "program formulation" of a type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation. Post's mode ...
s)—briefly, a
finite-state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...
whose only tape is a
FIFO queue __NOTOC__
Queue () may refer to:
* Queue area, or queue, a line or area where people wait for goods or services
Arts, entertainment, and media
*''ACM Queue'', a computer magazine
* ''The Queue'' (Sorokin novel), a 1983 novel by Russian author ...
of unbounded length, such that in each transition the machine reads the symbol at the head of the queue, deletes a constant number of symbols from the head, and appends to the tail a symbol-string that depends solely on the first symbol read in this transition.
Because all of the indicated operations are performed in a single transition, a tag machine strictly has only one state.
Definitions
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()'') assigned to the halting symbol is seen below to play no role in computations, but for convenience is taken to be ''P()'' = '.
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. Thus, the system processes the m-symbol head into a tail of variable length, but the generated tail depends solely on the first symbol of the head.
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 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.
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:
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
e 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
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integer ...
.
In the original Collatz sequence, the successor of ''n'' is either (for even ''n'') or 3''n'' + 1 (for odd n). The value 3''n'' + 1 is clearly even for odd ''n'', hence the next term after 3''n'' + 1 is surely . In the sequence computed by the tag system below we skip this intermediate step, hence the successor of ''n'' is 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:
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
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any ...
; 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
In computer science, a universal Turing machine (UTM) is a Turing machine that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simu ...
, 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 and corresponding productions , 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
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...
is among the simplest, most-easily described
undecidable decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm wheth ...
s:
Given an arbitrary positive integer ''n'' and a list of ''n''+1 arbitrary words ''P''
1,''P''
2,...,''P''
''n'',''Q'' on the alphabet , 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'', ''t''
1(''Q''), ''t''
2(''Q''), ''t''
3(''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
Matthew Cook (born February 7, 1970) is a mathematician and computer scientist who is best known for having proved Stephen Wolfram's conjecture that the Rule 110 cellular automaton is Turing-complete.
Biography
Cook was born in Morgantown, West ...
and were used in Cook's demonstration that the
Rule 110 cellular automaton
The Rule 110 cellular automaton (often called simply Rule 110) is an elementary cellular automaton with interesting behavior on the boundary between stability and chaos. In this respect, it is similar to Conway's Game of Life. Like Life, Rule 110 ...
is universal. A key part of the demonstration was that cyclic tag systems can emulate a
Turing-complete
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any ...
class of tag systems.
Emulation of tag systems by cyclic tag systems
An ''m''-tag system with alphabet and corresponding productions is emulated by a cyclic tag system with ''m*n'' productions (''Q
1'', ..., ''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):
''a
1'' = 100...00
''a
2'' = 010...00
.
.
.
''a
n'' = 000...01
That is, ''a
k'' is encoded as a binary string with a in the k
th position from the left, and '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.
See also
*
Queue automaton
A queue machine, queue automaton, or pullup automaton (PUA) is a finite state machine with the ability to store and retrieve data from an infinite-memory queue. It is a model of computation equivalent to a Turing machine, and therefore it can proc ...
References
*
*
*
Marvin Minsky
Marvin Lee Minsky (August 9, 1927 – January 24, 2016) was an American cognitive and computer scientist concerned largely with research of artificial intelligence (AI), co-founder of the Massachusetts Institute of Technology's AI laboratory, ...
1961, ''Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines", ''the Annals of Mathematics, 2nd ser., Vol. 74, No. 3. (Nov., 1961), pp. 437–455. .
* {{cite book , last=Minsky, first=Marvin, author-link=Marvin Minsky, date=1967, title=Computation: Finite and Infinite Machines
, url=https://archive.org/details/computationfinit0000mins/page/267
, publisher=Prentice–Hall, Inc., location=Englewoord Cliffs, N.J., pages=267–273, lccn=67-12342
:: In a chapter 14 titled "Very Simple Bases for Computability", Minsky presents a very readable (and exampled) subsection 14.6 ''The Problem of "Tag" and Monogenic Canonical Systems''
pp. 267–273 (this sub-section is indexed as "tag system"). Minsky relates his frustrating experiences with the general problem: "Post found this (00, 1101) problem "intractable," and so did I, even with the help of a computer." He comments that an "effective way to decide, for any string S, whether this process will ever repeat when started with S" is unknown although a few specific cases have been proven unsolvable. In particular he mentions Cocke's Theorem and Corollary 1964.
*
Post, E.: "Formal reductions of the combinatorial decision problem", ''American Journal of Mathematics'', 65 (2)
197–215(1943). (Tag systems are introduced o
p. 203ff)
* Rogozhin, Yu.: "Small Universal Turing Machines", ''Theoret. Comput. Sci.'' 168, 215–240, 1996.
*
Wang, H.: "Tag Systems and Lag Systems", ''Math. Annalen'' 152, 65–74, 1963.
External links
* https://mathworld.wolfram.com/TagSystem.html
* https://mathworld.wolfram.com/CyclicTagSystem.html
* https://www.wolframscience.com/nks/p95/ (cyclic tag systems)
* https://www.wolframscience.com/nks/p669/ (emulation of tag systems)
Models of computation