Semi-Thue system explained
between fixed strings over the alphabet, called
rewrite rules, denoted by
, an SRS extends the rewriting relation to all strings in which the left- and right-hand side of the rules appear as
substrings, that is
, where
,
,
, and
are strings.
The notion of a semi-Thue system essentially coincides with the presentation of a monoid. Thus they constitute a natural framework for solving the word problem for monoids and groups.
An SRS can be defined directly as an abstract rewriting system. It can also be seen as a restricted kind of a term rewriting system, in which all function symbols have an arity of at most 1. As a formalism, string rewriting systems are Turing complete.[1] The semi-Thue name comes from the Norwegian mathematician Axel Thue, who introduced systematic treatment of string rewriting systems in a 1914 paper.[2] Thue introduced this notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable - this result was obtained independently by Emil Post and A. A. Markov Jr.[3] [4]
Definition
where
is an alphabet, usually assumed finite.
[5] The elements of the set
(* is the
Kleene star here) are finite (possibly empty) strings on
, sometimes called
words in
formal languages; we will simply call them strings here.
is a
binary relation on strings from
, i.e.,
R\subseteq\Sigma* x \Sigma*.
Each element
is called a
(rewriting) rule and is usually written
.
If the relation
is
symmetric, then the system is called a
Thue system.
The rewriting rules in
can be naturally extended to other strings in
by allowing substrings to be rewritten according to
. More formally, the
one-step rewriting relation relation
induced by
on
for any strings
:
if and only if there exist
such that
,
, and
.
Since
is a relation on
, the pair
(\Sigma*,\xrightarrow[R]{})
fits the definition of an
abstract rewriting system. Obviously
is a subset of
. Some authors use a different notation for the arrow in
(e.g.
) in order to distinguish it from
itself (
) because they later want to be able to drop the subscript and still avoid confusion between
and the one-step rewrite induced by
.
Clearly in a semi-Thue system we can form a (finite or infinite) sequence of strings produced by starting with an initial string
and repeatedly rewriting it by making one substring-replacement at a time:
s0 \xrightarrow[R]{} s1 \xrightarrow[R]{} s2 \xrightarrow[R]{} \ldots
A zero-or-more-steps rewriting like this is captured by the reflexive transitive closure of
, denoted by
(see abstract rewriting system#Basic notions). This is called the
rewriting relation or
reduction relation on
induced by
.
Thue congruence
In general, the set
of strings on an alphabet forms a
free monoid together with the
binary operation of
string concatenation (denoted as
and written multiplicatively by dropping the symbol). In a SRS, the reduction relation
is compatible with the monoid operation, meaning that
implies
for all strings
. Since
is by definition a
preorder,
\left(\Sigma*, ⋅ ,\xrightarrow[R]{*}\right)
forms a monoidal preorder.
Similarly, the reflexive transitive symmetric closure of
, denoted
\overset{*}{\undersetR\leftrightarrow}
(see abstract rewriting system#Basic notions), is a
congruence, meaning it is an
equivalence relation (by definition) and it is also compatible with string concatenation. The relation
\overset{*}{\undersetR\leftrightarrow}
is called the
Thue congruence generated by . In a Thue system, i.e. if is symmetric, the rewrite relation
coincides with the Thue congruence
\overset{*}{\undersetR\leftrightarrow}
.
Factor monoid and monoid presentations
Since
\overset{*}{\undersetR\leftrightarrow}
is a congruence, we can define the
factor monoid l{M}R=\Sigma*/\overset{*}{\undersetR\leftrightarrow}
of the
free monoid
by the Thue congruence in the usual manner. If a monoid
is
isomorphic with
, then the semi-Thue system
is called a
monoid presentation of
.
We immediately get some very useful connections with other areas of algebra. For example, the alphabet with the rules, where ε is the empty string, is a presentation of the free group on one generator. If instead the rules are just, then we obtain a presentation of the bicyclic monoid.
The importance of semi-Thue systems as presentation of monoids is made stronger by the following:
Theorem: Every monoid has a presentation of the form
, thus it may be always be presented by a semi-Thue system, possibly over an infinite alphabet.
[6] In this context, the set
is called the
set of generators of
, and
is called the set of
defining relations
. We can immediately classify monoids based on their presentation.
is called
is finite.
- finitely presented if both
and
are finite.
Undecidability of the word problem
Post proved the word problem (for semigroups) to be undecidable in general, essentially by reducing the halting problem[7] for Turing machines to an instance of the word problem.
Concretely, Post devised an encoding as a finite string of the state of a Turing machine plus tape, such that the actions of this machine can be carried out by a string rewrite system acting on this string encoding. The alphabet of the encoding has one set of letters
for symbols on the tape (where
means blank), another set of letters
for states of the Turing machine, and finally three letters
that have special roles in the encoding.
and
are intuitively extra internal states of the Turing machine which it transitions to when halting, whereas
marks the end of the non-blank part of the tape; a machine reaching an
should behave the same as if there was a blank there, and the
was in the next cell. The strings that are valid encodings of Turing machine states start with an
, followed by zero or more symbol letters, followed by exactly one internal state letter
(which encodes the state of the machine), followed by one or more symbol letters, followed by an ending
. The symbol letters are straight off the contents of the tape, and the internal state letter marks the position of the head; the symbol after the internal state letter is that in the cell currently under the head of the Turing machine.
A transition where the machine upon being in state
and seeing the symbol
writes back symbol
, moves right, and transitions to state
is implemented by the rewrite
whereas that transition instead moving to the left is implemented by the rewrite
with one instance for each symbol
in that cell to the left. In the case that we reach the end of the visited portion of the tape, we use instead
,lengthening the string by one letter. Because all rewrites involve one internal state letter
, the valid encodings only contain one such letter, and each rewrite produces exactly one such letter, the rewrite process exactly follows the run of the Turing machine encoded. This proves that string rewrite systems are Turing complete.
The reason for having two halted symbols
and
is that we want all halting Turing machines to terminate at the same
total state, not just a particular
internal state. This requires clearing the tape after halting, so
eats the symbol on it left until reaching the
, where it transitions into
which instead eats the symbol on its right. (In this phase the string rewrite system no longer simulates a Turing machine, since that cannot remove cells from the tape.) After all symbols are gone, we have reached the terminal string
.
A decision procedure for the word problem would then also yield a procedure for deciding whether the given Turing machine terminates when started in a particular total state
, by testing whether
and
belong to the same congruence class with respect to this string rewrite system. Technically, we have the following:
Lemma. Let
be a deterministic Turing machine and
be the string rewrite system implementing
, as described above. Then
will halt when started from the total state encoded as
if and only if
tl{\overset{*}{\underset{R}{\leftrightarrow}}}hqr+2h
(that is to say, if and only if
and
are Thue congruent for
).
That
tl{\overset{*}{\underset{R}{ → }}}hqr+2h
if
halts when started from
is immediate from the construction of
(simply running
until it halts constructs a proof of
tl{\overset{*}{\underset{R}{ → }}}hqr+2h
), but
\overset{*}{\underset{R}{\leftrightarrow}}
also allows the Turing machine
to take steps backwards. Here it becomes relevant that
is deterministic, because then the forward steps are all unique; in a
\overset{*}{\underset{R}{\leftrightarrow}}
walk from
to
the last backward step must be followed by its counterpart as a forward step, so these two cancel, and by induction all backward steps can be eliminated from such a walk. Hence if
does
not halt when started from
, i.e., if we do
not have
tl{\overset{*}{\underset{R}{ → }}}hqr+2h
, then we also do
not have
tl{\overset{*}{\underset{R}{\leftrightarrow}}}hqr+2h
. Therefore, deciding
\overset{*}{\underset{R}{\leftrightarrow}}
tells us the answer to the halting problem for
.
An apparent limitation of this argument is that in order to produce a semigroup
\Sigma*/\overset{*}{\underset{R}{\leftrightarrow}}
with undecidable word problem, one must first have a concrete example of a Turing machine
for which the halting problem is undecidable, but the various Turing machines figuring in the proof of the undecidability of the general halting problem all have as component a hypothetical Turing machine solving the halting problem, so none of those machines can actually exist; all that proves is that there is
some Turing machine for which the decision problem is undecidable. However, that there is some Turing machine with undecidable halting problem means that the halting problem for a
universal Turing machine is undecidable (since that can simulate any Turing machine), and concrete examples of universal Turing machines have been constructed.
Connections with other notions
A semi-Thue system is also a term-rewriting system - one that has monadic words (functions) ending in the same variable as the left- and right-hand side terms,[8] e.g. a term rule
is equivalent to the string rule
.
A semi-Thue system is also a special type of Post canonical system, but every Post canonical system can also be reduced to an SRS. Both formalisms are Turing complete, and thus equivalent to Noam Chomsky's unrestricted grammars, which are sometimes called semi-Thue grammars.[9] A formal grammar only differs from a semi-Thue system by the separation of the alphabet into terminals and non-terminals, and the fixation of a starting symbol amongst non-terminals. A minority of authors actually define a semi-Thue system as a triple
, where
is called the
set of axioms. Under this "generative" definition of semi-Thue system, an unrestricted grammar is just a semi-Thue system with a single axiom in which one partitions the alphabet into terminals and non-terminals, and makes the axiom a nonterminal.
[10] The simple artifice of partitioning the alphabet into terminals and non-terminals is a powerful one; it allows the definition of the
Chomsky hierarchy based on what combination of terminals and non-terminals the rules contain. This was a crucial development in the theory of
formal languages.
In quantum computing, the notion of a quantum Thue system can be developed.[11] Since quantum computation is intrinsically reversible, the rewriting rules over the alphabet
are required to be bidirectional (i.e. the underlying system is a Thue system, not a semi-Thue system).On a subset of alphabet characters
one can attach a Hilbert space
, and a rewriting rule taking a substring to another one can carry out a unitary operation on the tensor product of the Hilbert space attached to the strings; this implies that they preserve the number of characters from the set
. Similar to the classical case one can show that a quantum Thue system is a universal computational model for quantum computation, in the sense that the executed quantum operations correspond to uniform circuit classes (such as those in
BQP when e.g. guaranteeing termination of the string rewriting rules within polynomially many steps in the input size), or equivalently a
Quantum Turing machine.
History and importance
Semi-Thue systems were developed as part of a program to add additional constructs to logic, so as to create systems such as propositional logic, that would allow general mathematical theorems to be expressed in a formal language, and then proven and verified in an automatic, mechanical fashion. The hope was that the act of theorem proving could then be reduced to a set of defined manipulations on a set of strings. It was subsequently realized that semi-Thue systems are isomorphic to unrestricted grammars, which in turn are known to be isomorphic to Turing machines. This method of research succeeded and now computers can be used to verify the proofs of mathematic and logical theorems.
At the suggestion of Alonzo Church, Emil Post in a paper published in 1947, first proved "a certain Problem of Thue" to be unsolvable, what Martin Davis states as "...the first unsolvability proof for a problem from classical mathematics -- in this case the word problem for semigroups."[12]
Davis also asserts that the proof was offered independently by A. A. Markov.[13]
See also
References
Monographs
- Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, .
- Matthias Jantzen, Confluent string rewriting, Birkhäuser, 1988, .
Textbooks
- Martin Davis, Ron Sigal, Elaine J. Weyuker, Computability, complexity, and languages: fundamentals of theoretical computer science, 2nd ed., Academic Press, 1994,, chapter 7
- Elaine Rich, Automata, computability and complexity: theory and applications, Prentice Hall, 2007,, chapter 23.5.
Surveys
- Samson Abramsky, Dov M. Gabbay, Thomas S. E. Maibaum (ed.), Handbook of Logic in Computer Science: Semantic modelling, Oxford University Press, 1995, .
- Grzegorz Rozenberg, Arto Salomaa (ed.), Handbook of Formal Languages: Word, language, grammar, Springer, 1997, .
Landmark papers
- Post . Emil . Recursive Unsolvability of a Problem of Thue . . 1947 . 12 . 1 . 1–11 . 10.2307/2267170 . dead . 2267170 . 30320278 . 2019-09-29 . 2019-09-29 . https://web.archive.org/web/20190929014814/https://projecteuclid.org/euclid.jsl/1183395203 .
Notes and References
- See section "Undecidability of the word problem" in this article.
- Book and Otto, p. 36
- Abramsky et al. p. 416
- Salomaa et al., p.444
- In Book and Otto a semi-Thue system is defined over a finite alphabet through most of the book, except chapter 7 when monoid presentation are introduced, when this assumption is quietly dropped.
- Book and Otto, Theorem 7.1.7, p. 149
- Post, following Turing, technically makes use of the undecidability of the printing problem (whether a Turing machine ever prints a particular symbol), but the two problems reduce to each other. Indeed, Post includes an extra step in his construction that effectively converts printing the watched symbol into halting.
- [Nachum Dershowitz]
- [Daniel I. A. Cohen|D.I.A. Cohen]
- Dan A. Simovici, Richard L. Tenney, Theory of formal languages with applications, World Scientific, 1999, chapter 4
- J. Bausch, T. Cubitt, M. Ozols, The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension, Ann. Henri Poincare 18(11), 2017 pp. 3449-3513
- [Martin Davis (mathematician)|Martin Davis]
- [A. A. Markov]