formulasearchengine - User contributions [en]

The Sea Island Mathematical Manual

2014-02-07T18:24:56Z

66.214.33.61: /* Survey of sea island */

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Six-dimensional space

2014-02-04T06:29:39Z

66.214.33.61: /* String theory */

In mathematics, a '''special conformal transformation''' is a member of the [[Conformal symmetry|conformal symmetry group]]. It is given by<ref>{{cite book |last = Di Francesco |coauthors= Mathieu, Sénéchal |title= Conformal field theory |series= Graduate texts in contemporary physics |year= 1997 |publisher= Springer |isbn= 978-0-387-94785-3 |pages= 97–98}}</ref>
:<math> x'^\mu = \frac{x^\mu-b^\mu x^2}{1-2b\cdot x+b^2x^2} \,. </math>
Its [[Generating set of a group|infinitesimal generator]] is
:<math> K_\mu = -i(2x_\mu x^\nu\partial_\nu - x^2\partial_\mu) \,. </math>

Such a transformation can also be written as sequence of an [[Inversive geometry|inversion]] (''x''μ → ''x''μ/x2), a [[translation (physics)|translation]] (''x''μ → ''x''μ − ''b''μ), and an inversion:

:<math> \frac{x'^\mu}{x'^2} = \frac{x^\mu}{x^2} - b^\mu \,. </math>

==References==
<references/>

{{geometry-stub}}

[[Category:Symmetry]]
[[Category:Scaling symmetries]]
[[Category:Conformal field theory]]

White dwarf

2014-02-03T03:54:53Z

66.214.232.231:

In [[mathematics]], especially in the area of [[abstract algebra]] known as [[combinatorial group theory]], the '''word problem''' for a finitely generated group ''G'' is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if ''A'' is a finite set of [[Generating set of a group|generators]] for ''G'' then the word problem is the membership problem for the [[formal language]] of all words in ''A'' and a formal set of inverses that map to the identity under the natural map from the [[free monoid with involution]] on ''A'' to the group ''G''. If ''B'' is another finite generating set for ''G'', then the word problem over the generating set ''B'' is equivalent to the word problem over the generating set ''A''. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group ''G''.

The related but different '''uniform word problem''' for a class ''K'' of recursively presented groups is the algorithmic problem of deciding, given as input a [[presentation of a group|presentation]] ''P'' for a group ''G'' in the class ''K'' and two words in the generators of ''G'', whether the words represent the same element of ''G''. Some authors require the class ''K'' to be definable by a [[recursively enumerable]] set of presentations.

== History ==

Throughout the history of the subject, computations in groups have been carried out using various [[Normal form (abstract rewriting)|normal forms]]. These usually implicitly solve the word problem for the groups in question. In 1911 [[Max Dehn]] proposed that the word problem was an important area of study in its own right, {{harv|Dehn|1911}}, together with the [[conjugacy problem]] and the [[group isomorphism problem]]. In 1912 he gave an algorithm that solves both the word and conjugacy problem for the [[fundamental group]]s of closed orientable two-dimensional manifolds of genus greater than or equal to ''2'', {{harv|Dehn|1912}}. Subsequent authors have greatly extended [[Small cancellation theory#Dehn's algorithm|Dehn's algorithm]] and applied it to a wide range of group theoretic [[decision problem]]s.<ref>{{Citation|last=Greendlinger|first=Martin|date=June 1959|title=Dehn's algorithm for the word problem|journal=Communications on Pure and Applied Mathematics|volume=13|issue=1|pages=67–83|doi=10.1002/cpa.3160130108|postscript=.}}</ref><ref>{{Citation|last=Lyndon|first=Roger C.|authorlink=Roger Lyndon|date=September 1966|title=On Dehn's algorithm|journal=Mathematische Annalen|volume=166|issue=3|pages=208–228|doi=10.1007/BF01361168|postscript=.|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002296799&L=1}}</ref><ref>{{Citation|last=Schupp|first=Paul E.|date=June 1968|title=On Dehn's algorithm and the conjugacy problem|journal=Mathematische Annalen|volume=178|issue=2|pages=119–130|doi=10.1007/BF01350654|postscript=.|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002300036&L=1}}</ref>

It was shown by [[Pyotr Novikov]] in 1955 that there exists a finitely generated (in fact, a finitely presented) group ''G'' such that the word problem for ''G'' is [[Undecidable problem|undecidable]].<ref>{{Citation|last=Novikov|first=P. S.|authorlink=Pyotr Novikov|year=1955|title=On the algorithmic unsolvability of the word problem in group theory|language=Russian| zbl=0068.01301 | journal=[[Proceedings of the Steklov Institute of Mathematics]]|volume=44|pages=1–143}}</ref> It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by [[William Boone (mathematician)|William Boone]] in 1958.<ref>{{Citation|last=Boone|first=William W.| authorlink=William Boone (mathematician) | year=1958|title=The word problem|journal=Proceedings of the National Academy of Sciences|volume=44|issue=10|pages=1061–1065|url=http://www.pnas.org/cgi/reprint/44/10/1061.pdf|format=PDF|doi=10.1073/pnas.44.10.1061|zbl=0086.24701 }}</ref>

The word problem was one of the first examples of an unsolvable problem to be found not in [[mathematical logic]] or the [[theory of algorithms]], but in one of the central branches of classical mathematics, [[abstract algebra|algebra]]. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.

It is important to realize that the word problem is in fact solvable for many groups ''G''. For example, [[polycyclic group]]s have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the [[Todd–Coxeter algorithm]]<ref>J.A. Todd and H.S.M. Coxeter. "A practical method for enumerating coset of a finite abstract group", ''Proc, Edinburgh Math Soc.'' (2), '''5''', 25---34. 1936</ref> and the [[Knuth–Bendix completion algorithm]].<ref>D. Knuth and P. Bendix. "Simple word problems in universal algebras." ''Computational Problems in Abstract Algebra'' (Ed. J. Leech) pages 263--297, 1970.</ref> On the other hand the fact that a particular algorithm does not solve the word problem for a particular group does not show that the group has unsolvable word problem. For instance Dehn's algorithm does not solve the word problem for the fundamental group of the [[torus]]. However this group is the direct product of two infinite cyclic groups and so has solvable word problem.

== A more concrete description ==

In more concrete terms, the uniform word problem can be expressed as a [[rewriting]] question, for [[literal string]]s, {{harv|Rotman|1994}}. For a presentation ''P'' of a group ''G'', ''P'' will specify a certain number of generators

:''x'', ''y'', ''z'', ...

for ''G''. We need to introduce one letter for ''x'' and another (for convenience) for the group element represented by ''x''−1. Call these letters (twice as many as the generators) the alphabet <math>\Sigma</math> for our problem. Then each element in ''G'' is represented in ''some way'' by a product

:''abc ... pqr''

of symbols from <math>\Sigma</math>, of some length, multiplied in ''G''. The string of length 0 ([[Empty string|null string]]) stands for the [[identity element]] ''e'' of ''G''. The crux of the whole problem is to be able to recognise ''all'' the ways ''e'' can be represented, given some relations.

The effect of the ''relations'' in ''G'' is to make various such strings represent the same element of ''G''. In fact the relations provide a list of strings that can be either introduced where we want, or cancelled out whenever we see them, without changing the 'value', i.e. the group element that is the result of the multiplication.

For a simple example, take the presentation {''a'' | ''a''3}. Writing ''A'' for the inverse of ''a'', we have possible strings combining any number of the symbols ''a'' and ''A''. Whenever we see ''aaa'', or ''aA'' or ''Aa'' we may strike these out. We should also remember to strike out ''AAA''; this says that since the cube of ''a'' is the identity element of ''G'', so is the cube of the inverse of ''a''. Under these conditions the word problem becomes easy. First reduce strings to the empty string, ''a'', ''aa'', ''A'' or ''AA''. Then note that we may also multiply by ''aaa'', so we can convert ''A'' to ''aa'' and convert ''AA'' to ''a''. The result is that the word problem, here for the [[cyclic group]] of order three, is solvable.

This is not, however, the typical case. For the example, we have a [[canonical form]] available that reduces any string to one of length at most three, by decreasing the length monotonically. In general, it is not true that one can get a canonical form for the elements, by stepwise cancellation. One may have to use relations to expand a string many-fold, in order eventually to find a cancellation that brings the length right down.

The upshot is, in the worst case, that the relation between strings that says they are equal in ''G'' is not ''[[decidable]]''.

==Examples==
The following groups have a solvable word problem:
*[[Automatic group]]s, including:
**[[Finite group]]s
**[[Polycyclic group]]s
**[[Negatively curved group]]s
**[[Euclidean group]]s
**[[Coxeter group]]s
**[[Braid group]]s
**[[Geometrically finite group]]s
*Finitely generated recursively [[Absolute presentation of a group|absolutely presented group]]s,<ref>H.Simmons, "The word problem for absolute presentations." ''J. London Math. Soc.'' (2) 6, 275-280 1973</ref> including:
**Finitely presented simple groups.
*Finitely presented [[residually finite]] groups
*One relator groups<ref>Roger C. Lyndon, Paul E Schupp, Combinatorial Group Theory, Springer, 2001</ref> (this is a theorem of Magnus), including:
**Fundamental groups of closed orientable two-dimensional manifolds.
*Combable groups

Examples with unsolvable word problems are also known:
*Given a recursively enumerable set ''A'' of positive integers that has insoluble membership problem, ''⟨a,b,c,d | anban = cndcn : n ∈ A⟩'' is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble {{harv|Collins|Zieschang|1990|p=149}}
*Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem {{harv|Collins|Zieschang|1993|loc=Cor. 7.2.6}}
*The number of relators in a finitely presented group with insoluble word problem may be as low as 14 by {{harv|Collins|1969}} or even 12 by {{harv|Borisov|1969}}, {{harv|Collins|1972}}.
*An explicit example of a reasonable short presentation with insoluble word problem is given in {{harv|Collins|1986}}:<ref>We use the corrected version from [http://shell.cas.usf.edu/~eclark/algctlg/groups.html John Pedersen's A Catalogue of Algebraic Systems]</ref>
:<math>\begin{array}{lllll}\langle & a,b,c,d,e,p,q,r,t,k & | & &\\
&p^{10}a = ap, &pacqr = rpcaq, &ra=ar, &\\
&p^{10}b = bp, &p^2adq^2r = rp^2daq^2, &rb=br, &\\
&p^{10}c = cp, &p^3bcq^3r = rp^3cbq^3, &rc=cr, &\\
&p^{10}d = dp, &p^4bdq^4r = rp^4dbq^4, &rd=dr, &\\
&p^{10}e = ep, &p^5ceq^5r = rp^5ecaq^5, &re=er, &\\
&aq^{10} = qa, &p^6deq^6r = rp^6edbq^6, &pt=tp, &\\
&bq^{10} = qb, &p^7cdcq^7r = rp^7cdceq^7, &qt=tq, &\\
&cq^{10} = qc, &p^8ca^3q^8r = rp^8a^3q^8, &&\\
&dq^{10} = qd, &p^9da^3q^9r = rp^9a^3q^9, &&\\
&eq^{10} = qe, &a^{-3}ta^3k = ka^{-3}ta^3 &&\rangle \end{array}</math>

==Partial solution of the word problem==

The word problem for a recursively presented group can be partially solved in the following sense:

::Given a recursive presentation ''P'' = ⟨''X''|''R''⟩ for a group ''G'', define:
:::<math>S=\{\langle u,v \rangle : u \mbox{ and } v \mbox{ are words in } X \mbox{ and } u=v \mbox{ in } G\ \}</math>
::then there is a partial recursive function ''fP'' such that:
:::<math>f_P(\langle u,v \rangle) =
\left\{\begin{matrix}
0 &\mbox{if}\ \langle u,v \rangle \in S \\
\mbox{undefined/does not halt}\ &\mbox{if}\ \langle u,v \rangle \notin S
\end{matrix}\right.</math>

More informally, there is an algorithm that halts if ''u=v'', but does not do so otherwise.

It follows that to solve the word problem for ''P'' it is sufficient to construct a recursive function g such that:
::<math>g(\langle u,v \rangle) =
\left\{\begin{matrix}
0 &\mbox{if}\ \langle u,v \rangle \notin S \\
\mbox{undefined/does not halt}\ &\mbox{if}\ \langle u,v \rangle \in S
\end{matrix}\right.</math>

However ''u=v'' in ''G'' if and only if ''uv−1=1'' in ''G''. It follows that to solve the word problem for ''P'' it is sufficient to construct a recursive function ''h'' such that:
::<math>h(x) =
\left\{\begin{matrix}
0 &\mbox{if}\ x\neq1\ \mbox{in}\ G \\
\mbox{undefined/does not halt}\ &\mbox{if}\ x=1\ \mbox{in}\ G
\end{matrix}\right.</math>

===Example===
The following will be proved as an example of the use of this technique:

:: '''Theorem:''' A finitely presented residually finite group has solvable word problem.

''Proof:'' Suppose ''G'' = ⟨''X''|''R''⟩ is a finitely presented, residually finite group.

Let ''S'' be the group of all permutations of '''N''', the natural numbers, that fixes all but finitely many numbers then:
# ''S'' is [[locally finite group|locally finite]] and contains a copy of every finite group.
# The word problem in ''S'' is solvable by calculating products of permutations.
# There is a recursive enumeration of all mappings of the finite set ''X'' into ''S''.
# Since ''G'' is residually finite, if ''w'' is a word in the generators ''X'' of ''G'' then ''w ≠ 1'' in ''G'' if and only of some mapping of ''X'' into ''S'' induces a homomorphism such that ''w ≠ 1'' in ''S''.

Given these facts, algorithm defined by the following pseudocode:

:For every mapping of ''X'' into ''S''
::If every relator in ''R'' is satisfied in ''S''
:::If ''w ≠ 1'' in ''S''
::::return 0
:::End if
::End if
:End for

defines a recursive function ''h'' such that:

::<math>h(x) =
\left\{\begin{matrix}
0 &\mbox{if}\ x\neq 1\ \mbox{in}\ G \\
\mbox{undefined/does not halt}\ &\mbox{if}\ x=1\ \mbox{in}\ G
\end{matrix}\right. </math>

This shows that ''G'' has solvable word problem.

==Unsolvability of the uniform word problem==

The criterion, given above for the solvability of the word problem in a single group can be extended to a criterion for the uniform solvability of the word problem for a class of finitely presented groups by a straightforward argument. The result is:

::To solve the uniform word problem for a class ''K'' of groups it is sufficient to find a recursive function ''f(P,w)'' that takes a finite presentation ''P'' for a group ''G'' and a word ''w'' in the generators of ''G'' such that whenever in ''G ∈ K'':
:::<math>f(P,w) =
\left\{\begin{matrix}
0 &\mbox{if}\ w\neq1\ \mbox{in}\ G \\
\mbox{undefined/does not halt}\ &\mbox{if}\ w=1\ \mbox{in}\ G
\end{matrix}\right.</math>

:: '''Boone- Rogers Theorem:''' There is no uniform [[partial algorithm]] that solves the word problem in all finitely presented groups with solvable word problem.

In other words the uniform word problem for the class of all finitely presented groups with solvable word problem is unsolvable. This has some interesting consequences. For instance the [[Higman embedding theorem]] can be used to construct a group containing an isomorphic copy of every finitely presented group with solvable word problem. It seems natural to ask whether this group can have solvable word problem. But it is a consequence of the Boone-Rogers result that:

:: '''Corollary:''' There is no universal solvable word problem group. That is, if ''G'' is a finitely presented group that contains an isomorphic copy of every finitely presented group with solvable word problem, then ''G'' itself must have unsolvable word problem.

'''Remark:''' Suppose ''G = ⟨X|R⟩'' is a finitely presented group with solvable word problem and ''H'' is a finite subset ''G''. Let ''H''* = ⟨''H''⟩, be the group generated by ''H''. Then the word problem in ''H''* is solvable: given two words ''h, k'' in the generators ''H'' of ''H''*, write them as words in ''X'' and compare them using the solution to the word problem in ''G''. It is easy to think that this demonstrates a uniform solution the word problem for the class ''K'' (say) of finitely generated groups that can be embedded in ''G''. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers. However, solution just exhibited for the word problem for groups in ''K'' is not uniform. To see this consider a group ''J = ⟨Y|T⟩ ∈ K'', in order to use the above argument to solve the word problem in ''J'', it is first necessary to exhibit a mapping ''e: Y → G'' that extends to an embedding ''e*: J → G''. If there were a recursive function that mapped (finitely generated) presentations of groups in ''K'' to embeddings into ''G'', then a uniform solution the word problem in ''K'' could indeed be constructed. But there is no reason, in general, to suppose that such a recursive function exists. However, it turns out that, using a more sophisicated argument, the word problem in ''J'' can be solved ''without'' using an embedding ''e: J → G''. Instead an ''enumeration of homomorphisms'' is used, and since such enumeration can be constructed uniformly, it results in a uniform solution to the word problem in ''K''.

===Proof that there is no universal solvable word problem group===
Suppose ''G'' were a universal solvable word problem group. Given a finite presentation ''P = ⟨X|R⟩'' of a group ''H'', one can recursively enumerate all homomorphisms ''h: H → G'' by first enumerating all mappings ''h†: X → G''. Not all of these mappings extend to homomorphisms, but, since ''h†(R)'', is finite, it is possible to distinguish between homomorphism and non-homomorphisms by using the solution to the word problem in ''G''. "Weeding out" non-homomorphisms gives the required recursive enumeration: ''h1'', ''h2'', ..., ''hn'', ... .

If ''H'' has solvable word problem, then at least one of these homomorphism must be an embedding. So given a word ''w'' in the generators of ''H'':

::<math>\mbox{If}\ w\ne 1\ \mbox{in}\ H,\ h_n(w)\ne 1\ \mbox{in}\ G\ \mbox{for some}\ h_n </math>
::<math>\mbox{If}\ w= 1\ \mbox{in}\ H,\ h_n(w)= 1\ \mbox{in}\ G\ \mbox{for all}\ h_n </math>

Consider the algorithm described by the pseudocode:

::Let ''n''=''0''
::Let ''repeat''=TRUE
::while (''repeat''==TRUE)
:::increase ''n'' by ''1''
:::if (solution to word problem in ''G'' reveals ''hn(w) ≠ 1'' in ''G'')
::::Let ''repeat''==FALSE
::output 0.

This describes a recursive function:

::<math>f(w) =
\left\{\begin{matrix}
0 &\mbox{if}\ w\neq1\ \mbox{in}\ H \\
\mbox{undefined/does not halt}\ &\mbox{if}\ w=1\ \mbox{in}\ H.
\end{matrix}\right.</math>

The function ''f'' clearly depends on the presentation ''P''. Considering it to be a function of the two variables, a recursive function ''f(P,w)'' has been constructed that takes a finite presentation ''P'' for a group ''G'' and a word ''w'' in the generators of ''G'' such that whenever ''G'' has soluble word problem:

::<math>f(P,w) =
\left\{\begin{matrix}
0 &\mbox{if}\ w\neq1\ \mbox{in}\ H \\
\mbox{undefined/does not halt}\ &\mbox{if}\ w=1\ \mbox{in}\ H.
\end{matrix}\right.</math>

But this uniformly solves the word problem for the class of all finitely presented groups with solvable word problem contradicting Boone-Rogers. This contradiction proves ''G'' cannot exist.

==Algebraic structure and the word problem==
There are a number of results that relate solvability of the word problem and algebraic structure. The most significant of these is the [[Boone-Higman theorem]]:

::A finitely presented group has solvable word problem if and only if it can be embedded in a [[simple group]] that can be embedded in a finitely presented group.

It is widely believed that it should be possible to do the construction so that the simple group itself is finitely presented. If so one would expect it to be difficult to prove as the mapping from presentations to simple groups would have to be non-recursive.

The following has been proved by [[Bernhard Neumann]] and [[Angus Macintyre]]:

::A finitely presented group has solvable word problem if and only if it can be embedded in every [[algebraically closed group]]

What is remarkable about this is that the algebraically closed groups are so wild that none of them has a recursive presentation.

The oldest result relating algebraic structure to solvability of the word problem is [[Kuznetsov]]'s theorem:

::A recursively presented simple group ''S'' has solvable word problem.

To prove this let ''⟨X|R⟩'' be a recursive presentation for ''S''. Choose ''a ∈ S'' such that ''a ≠ 1'' in ''S''.

If ''w'' is a word on the generators ''X'' of ''S'', then let:

::<math>S_w = \langle X | R\cup \{w\} \rangle.</math>

There is a recursive function <math>f_{\langle X | R\cup \{w\} \rangle}</math> such that:

::<math>f_{\langle X | R\cup \{w\} \rangle}(x) =
\left\{\begin{matrix}
0 &\mbox{if}\ x=1\ \mbox{in}\ S_w\\
\mbox{undefined/does not halt}\ &\mbox{if}\ x\neq 1\ \mbox{in}\ S_w.
\end{matrix}\right.</math>

Write:

::<math>g(w, x) = f_{\langle X | R\cup \{w\} \rangle}(x).</math>

Then because the construction of ''f'' was uniform, this is a recursive function of two variables.

It follows that: ''h(w)=g(w, a)'' is recursive. By construction:

::<math>h(w) =
\left\{\begin{matrix}
0 &\mbox{if}\ a=1\ \mbox{in}\ S_w\\
\mbox{undefined/does not halt}\ &\mbox{if}\ a\neq 1\ \mbox{in}\ S_w.
\end{matrix}\right.</math>

Since ''S'' is a simple group, its only quotient groups are itself and the trivial group. Since ''a ≠ 1'' in ''S'', we see ''a'' = 1 in ''Sw'' if and only if ''Sw'' is trivial if and only if ''w ≠ 1'' in ''S''. Therefore:

::<math>h(w) =
\left\{\begin{matrix}
0 &\mbox{if}\ w\ne 1\ \mbox{in}\ S\\
\mbox{undefined/does not halt}\ &\mbox{if}\ w=1\ \mbox{in}\ S.
\end{matrix}\right.</math>

The existence of such a function is sufficient to prove the word problem is solvable for ''S''.

This proof does not prove the existence of a uniform algorithm for solving the word problem for this class of groups. The non-uniformity resides in choosing a non-trivial element of the simple group. There is no reason to suppose that there is a recursive function that maps a presentation of a simple groups to a non-trivial element of the group. However, in the case of a finitely presented group we know that not all the generators can be trivial (Any individual generator could be, of course). Using this fact it is possible to modify the proof to show:

:The word problem is uniformly solvable for the class of finitely presented simple groups.

==See also==
* [[Combinatorics on words]]
* [[SQ-universal group]]
* [[Word problem (mathematics)]]
* [[Reachability problem]]

== Notes ==
<references/>

== References ==
*{{planetmath reference|id=8301|title=Word problem}}
* W. W. Boone, F. B. Cannonito, and [[Roger Lyndon|R. C. Lyndon]]. ''Word Problems: Decision Problem in Group Theory.'' Netherlands: North-Holland. 1973.
* W. W. Boone and G. Higman, "An algebraic characterization of the solvability of the word problem", ''J. Austral. Math. Soc.'' '''18''', 41-53 (1974)
* W. W. Boone and H. Rogers Jr., "On a problem of J. H. C. Whitehead and a problem of Alonzo Church", ''Math. Scand.'' '''19''', 185-192 (1966).'
*{{Citation | last1=Borisov | first1=V. V. | title=Simple examples of groups with unsolvable word problem | mr=0260851 | year=1969 | journal=Akademiya Nauk SSSR. Matematicheskie Zametki | issn=0025-567X | volume=6 | pages=521–532}}
* {{Citation | last1=Collins | first1=Donald J. | title=Word and conjugacy problems in groups with only a few defining relations | mr=0263903 | year=1969 | journal=Zeitschrift für Mathematische Logik und Grundlagen der Mathematik | volume=15 | issue=20-22 | pages=305–324 | doi=10.1002/malq.19690152001}}
* {{Citation | last1=Collins | first1=Donald J. | title=On a group embedding theorem of V. V. Borisov | mr=0314998 | year=1972 | journal=[[London Mathematical Society|Bulletin of the London Mathematical Society]] | issn=0024-6093 | volume=4 | issue=2 | pages=145–147 | doi=10.1112/blms/4.2.145}}
* {{Citation | last1=Collins | first1=Donald J. | title=A simple presentation of a group with unsolvable word problem | mr=840121 | year=1986 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=30 | issue=2 | pages=230–234}}
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{{DEFAULTSORT:Word Problem For Groups}}
[[Category:Group theory]]
[[Category:Combinatorics on words]]
[[Category:Articles with example pseudocode]]
[[Category:Articles containing proofs]]

Barnsley fern

2013-03-25T23:41:47Z

66.214.64.91: /* Computer generation */

In [[graph theory]], the '''tree-depth''' of a [[connected graph|connected]] [[undirected graph]] ''G'' is a numerical invariant of ''G'', the minimum height of a [[Trémaux tree]] for a [[Glossary of graph theory#Subgraphs|supergraph]] of ''G''. This invariant and its close relatives have gone under many different names in the literature, including vertex ranking number, ordered chromatic number, and minimum elimination tree height; it is also closely related to the [[cycle rank]] of [[directed graph]]s and the [[star height]] of [[regular language]]s.<ref>{{harvtxt|Bodlaender|Deogun|Jansen|Kloks|1998}}; {{harvtxt|Rossman|2008}}; {{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 116.</ref> Intuitively, where the [[treewidth]] graph width parameter measures how far a graph is from being a tree, this parameter measures how far a graph is from being a star.

==Definitions==
The tree-depth of a graph ''G'' may be defined as the minimum height of a [[forest (graph theory)|forest]] ''F'' with the property that every edge of ''G'' connects a pair of nodes that have an ancestor-descendant relationship to each other in ''F''.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Definition 6.1, p. 115.</ref> If ''G'' is connected, this forest must be a single tree; it need not be a subgraph of ''G'', but if it is, it is a [[Trémaux tree]] for ''G''.

The set of ancestor-descendant pairs in ''F'' forms a [[trivially perfect graph]], and the height of ''F'' is the size of the largest [[clique (graph theory)|clique]] in this graph. Thus, the tree-depth may alternatively be defined as the size of the largest clique in a trivially perfect supergraph of ''G'', mirroring the definition of [[treewidth]] as one less than the size of the largest clique in a [[chordal graph|chordal]] supergraph of ''G''.

Another definition is the following:

<math>td(G)=\begin{cases}1, & \text{if }|G|=1;\\
1+\min_{v\in V} td(G-v), & \text{if }G\text{ is connected and }|G|>1;\\
\max_{i} td(G_i), &\text{otherwise};
\end{cases}</math>

where the <math>G_i</math> are the connected components of ''G''.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Lemma 6.1, p. 117.</ref>

Tree-depth may also be defined using a form of [[graph coloring]]. A '''centered coloring''' of a graph is a coloring of its vertices with the property that every connected [[induced subgraph]] has a color that appears exactly once. Then, the tree-depth is the minimum number of colors in a centered coloring of the given graph. If ''F'' is a forest of height ''d'' with the property that every edge of ''G'' connects an ancestor and a descendant in the tree, then a centered coloring of ''G'' using ''d'' colors may be obtained by coloring each vertex by its distance from the root of its tree in ''F''.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Section 6.5, "Centered Colorings", pp. 125–128.</ref>

Finally, one can define this in terms of a [[pebble game]], or more precisely as a [[Pursuit-evasion|cops and robber]] game. Consider the following game, played on an undirected graph. There are two players, a robber and a cop. The robber has one pebble he can move along the edges of the given graph. The cop has an unlimited number of pebbles, but she wants to minimize the amount of pebbles she uses. The cop cannot move a pebble after it's been placed on the graph. The game proceeds as follows. The robber places his pebble. The cop then announces where she wants to place a new cop pebble. The robber can then move his pebble along edges, but not through occupied vertices. The game is over when the cop player places a pebble on top of the robber pebble. The tree-depth of the given graph is the minimum number of pebbles needed by the cop to guarantee a win.<ref>{{harvtxt|Gruber|Holzer|2008}}, Theorem 5, {{harvtxt|Hunter|2011}}, Main Theorem.</ref> For a star graph, this is 2 (the strategy is to place at the center vertex, forcing the robber to one arm, and then to place the remaining pebble on the robber). For a path, the robber uses a [[binary search]] strategy, which guarantees a ''log n'' number of pebbles needed.

==Examples==
[[File:Tree-depth.svg|thumb|360px|The tree-depths of the [[complete graph]] ''K''4 and the [[complete bipartite graph]] ''K''3,3 are both four, while the tree-depth of the [[path graph]] ''P''7 is three.]]
The tree-depth of a [[complete graph]] equals its number of vertices, for in this case the only possible forest ''F'' for which every pair of vertices are in an ancestor-descendant relationship is a single path. Similarly, the tree-depth of a [[complete bipartite graph]] ''K''''x'',''y'' is min(''x'',''y'') + 1, for whatever nodes are placed at the leaves of the forest ''F'' must have at least min(''x'',''y'') ancestors in ''F''. A forest achieving this min(''x'',''y'') + 1 bound may be constructed by forming a path for the smaller side of the bipartition, with each vertex on the larger side of the bipartition forming a leaf in ''F'' connected to the bottom vertex of this path.

The tree-depth of a path with ''n'' vertices is exactly <math>\lceil\log_2(n+1)\rceil</math>. A forest ''F'' representing this path with this depth may be formed by placing the midpoint of the path as the root of ''F'' and recursing within the two smaller paths on either side of it.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Formula 6.2, p. 117.</ref>

==Depth of trees and relation to treewidth==
Any ''n''-vertex [[tree (graph theory)|forest]] has tree-depth O(log ''n''). For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2''n''/3 vertices each. By recursively partitioning each of these two subforests, we can easily derive a logarithmic upper bound on the tree-depth. The same technique, applied to a [[tree decomposition]] of a graph, shows that, if the [[treewidth]] of an ''n''-vertex graph ''G'' is ''t'', then the tree-depth of ''G'' is O(''t'' log ''n'').<ref>{{harvtxt|Bodlaender|Gilbert|Hafsteinsson|Kloks|1995}}; {{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Corollary 6.1, p. 124.</ref> Since [[outerplanar graph]]s, [[series-parallel graph]]s, and [[Halin graph]]s all have bounded treewidth, they all also have at most logarithmic tree-depth.

In the other direction, the treewidth of a graph is at most equal to its tree-depth. More precisely, the treewidth is at most the [[pathwidth]], which is at most one less than the tree-depth.<ref>{{harvtxt|Bodlaender|Gilbert|Hafsteinsson|Kloks|1995}}; {{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 123.</ref>

==Graph minors==
A [[minor (graph theory)|minor]] of a graph ''G'' is another graph formed from a subgraph of ''G'' by contracting some of its edges. Tree-depth is monotonic under minors: every minor of a graph ''G'' has tree-depth at most equal to the tree-depth of ''G'' itself.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Lemma 6.2, p. 117.</ref> Thus, by the [[Robertson–Seymour theorem]], for every fixed ''d'' the set of graphs with tree-depth at most ''d'' has a finite set of [[forbidden graph characterization|forbidden minors]].

If ''C'' is a class of graphs closed under taking graph minors, then the graphs in ''C'' have tree-depth <math>O(1)</math> if and only if ''C'' does not include all the [[path graph]]s.<ref name="no12-prop64">{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Proposition 6.4, p. 122.</ref>

==Induced subgraphs==
As well as behaving well under graph minors, tree-depth has close connections to the theory of [[induced subgraph]]s of a graph. Within the class of graphs that have tree-depth at most ''d'' (for any fixed integer ''d''), the relation of being an induced subgraph forms a [[well-quasi-ordering]].<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Lemma 6.13, p. 137.</ref> The basic idea of the proof that this relation is a well-quasi-ordering is to use induction on ''d''; the forests of height ''d'' may be interpreted as sequences of forests of height ''d'' − 1 (formed by deleting the roots of the trees in the height-''d'' forest) and [[Higman's lemma]] can be used together with the induction hypothesis to show that these sequences are well-quasi-ordered.

Well-quasi-ordering implies that any property of graphs that is monotonic with respect to induced subgraphs has finitely many forbidden induced subgraphs, and therefore may be tested in polynomial time on graphs of bounded tree-depth. The graphs with tree-depth at most ''d'' themselves also have a finite set of forbidden induced subgraphs.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 138. Figure 6.6 on p. 139 shows the 14 forbidden subgraphs for graphs of tree-depth at most three, credited to the 2007 Ph.D. thesis of Z. Dvořák.</ref>

If ''C'' is a class of graphs with bounded [[degeneracy (graph theory)|degeneracy]], the graphs in ''C'' have bounded tree-depth if and only if there is a path graph that cannot occur as an induced subgraph of a graph in ''C''.<ref name="no12-prop64"/>

==Complexity==
Computing the tree-depth is computationally hard: the corresponding decision problem is [[NP-complete]].<ref name="p88">{{harvtxt|Pothen|1988}}.</ref> The problem remains NP-complete for [[complement (graph theory)|complement]]s of [[bipartite graph]]s,<ref name="p88" for [[bipartite graph]]s {{harv|Bodlaender|Deogun|Jansen|Kloks|1998}}, as well as for [[chordal graph]]s.<ref>{{harvtxt|Dereniowski|Nadolski|2006}}.</ref>

On the positive side, the problem is solvable in [[polynomial time]] on interval graphs,<ref>{{harvtxt|Aspvall|Heggernes|1994}}.</ref> as well as on permutation, trapezoid, circular-arc, circular permutation graphs, and cocomparability graphs of bounded dimension.<ref>{{harvtxt|Deogun|Kloks|Kratsch|Müller|1999}}.</ref> For undirected trees, the problem is solvable in linear time.<ref>{{harvtxt|Iyer|Ratliff|Vijayan|1988}}; {{harvtxt|Schäffer|1989}}.</ref>

{{harvtxt|Bodlaender|Gilbert|Hafsteinsson|Kloks|1995}} give an [[approximation algorithm]] with approximation ratio <math>O((\log n)^2)</math>.

Because tree-depth is monotonic under graph minors, it is [[Parameterized complexity|fixed-parameter tractable]]: there is an algorithm for computing tree-depth whose time is <math>f(d) n^{O(1)}</math>, where ''d'' is the depth of the given graph and ''n'' is its number of vertices. Thus, for every fixed value of ''d'', the problem of testing whether the depth is at most ''d'' can be solved in [[polynomial time]]. More specifically, the dependence on ''n'' in this algorithm can be made linear, by the following method: compute a depth first search tree, and test whether this tree's depth is greater than 2''d''. If so, the tree-depth is greater than ''d'' and the problem is solved. If not, the shallow depth first search tree can be used to construct a [[tree decomposition]] with bounded width, and standard [[dynamic programming]] techniques for graphs of bounded treewidth can be used to compute the depth in linear time.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 138. A more complicated linear time algorithm based on the planarity of the excluded minors for tree-depth was given earlier by {{harvtxt|Bodlaender|Deogun|Jansen|Kloks|1998}}.</ref>

==Notes==
{{reflist|colwidth=30em}}

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[[Category:Graph coloring]]
[[Category:Graph invariants]]
[[Category:Graph minor theory]]