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A Note on Context[C Sensitive Languages and Word Problems

arXiv:math/9306204v1 [math.GR] 16 Jun 1993A Note on Context[C Sensitive Languages and Word ProblemsMichael Shapiro*In [AS], Anisimov and Seifert show that a group has a regular word problem if and only ifit is finite. Muller and Schupp [MS] (together with Dunwoody’s accessibility result [D])show that a group has context free word problem if and only if it is virtually free.

In thisnote, we exhibit a class of groups where the word problem is as close as possible to being acontext sensitive language. This class includes the automatic groups of [ECHLPT] and isclosed under passing to finitely generated subgroups.

Consequently, it is quite large. Forexample, it contains all finitely generated subgroups of the n-fold product of free groups,F2 × .

. .

× F2. For n = 2, these include groups which are not finitely presented, and forn > 2, these include groups which are FPn but not FPn+1.Let us make clear what we mean by saying that the word problem is as close aspossible to being a context sensitive language.

Recall that a context sensitive languagecannot contain the empty word e. Since the empty word is always an element of the wordproblem, strictly speaking, the word problem can never be a context sensitive language.So we will abuse terminology and say that the word problem is context sensitive if, afterdeleting the empty word, it is context sensitive. We feel that this is not a grievous abuse: inany practical situation where one is trying to either decide or enumerate the word problem,the empty word is the least of one’s problems!There are two ingredients to our Theorem.One is the notion of a asynchronouscombing of a group (see below).

The other is the following characterization of contextsensitive languages. Given a language L, L −{e} is context sensitive if and only if L is thelanguage of a nondeterministic linear bounded Turing machine†.

(See, for example, [HU]. )Thus, to see that the word problem is context sensitive, we must exhibit an algorithmwhich, given w with w = 1, verifies membership in the word problem using an amountof space which is linear in the length of w. In fact, the process we will describe gives adeterministic linear bounded automaton in the case where the combing language is alsothe language of a deterministic linear bounded automaton.

In this case, our algorithm actsto decide the word problem rather than merely verify membership in the word problem.We start by fixing our terminology. Given a group G and a finite monoid generatingset A, we take A∗to be the free monoid on A.

For each w ∈A∗we denote the length of* I wish to thank the NSF for support.† Indeed, as Neumann has pointed out to me, one might wish to cure this mismatch bythe following method. We could change the definition of a context sensitive language toallow a finite number of start words rather than a single start symbol.

If none of these is e,each could be produced from a single start symbol by a single production rule. Thus theresult of this change of definition would be to include L if and only if L −{e} is contextsensitive under the old definition.1

w by ℓ(w). The empty word is the unique word of length 0 and we denote it by e. Wemap A∗to G by the monoid homomorphism which takes each letter of A to its value inG.

We denote this map by w 7→w. We call {w ∈A∗| w = 1} the word problem.

We willassume that A is supplied with an involution denoted by a 7→a−1 and that a−1 = (a)−1for all a ∈A. This allows one to build the Cayley graph Γ of G with respect to A. Thisis the labelled directed graph whose vertices are the elements of G and whose edges are{(g, a, g′) | g, g′ ∈G, a ∈A, g′ = ga}.

Each edge (g, a, g′) is labelled by a. Elements ofA∗are called words, and each word now labels a unique edge path of Γ based at 1 ∈Γ.Declaring each edge isometric to the unit interval induces the word metric d(·, ·) on G anda length function ℓ(g) = d(1, g).

We call a subset of A∗a language. We call a language La normal form if L = G. (We do not demand that this is a bijection.) We call a normalfrom L an asynchronous combing if there is a constant K so that for any w, w′ ∈L withd(w, w′) ≤1, we can find monotone reparameterizations of [0, ∞) t 7→t′ and t 7→t′′ sothat for all t, d(w(t′), (w′′)) ≤K.

We say that D is a departure function for L if for anyw = xyz ∈L, ℓ(y) ≥n whenever ℓ(y) ≥D(n). We say a language L is short if there areλ and ǫ so that if w ∈L then ℓ(w) ≤λℓ(w) + ǫ.

In addition, we say that L consists of(λ, ǫ)-quasigeodesics if for any w = xyz ∈L, ℓ(y) ≤λℓ(y) + ǫ.Theorem. Suppose that H is a finitely generated subgroup of G and suppose that G pos-sesses a short asynchronous context sensitive combing with a departure function.

Then Hhas a context sensitive word problem.Corollary. A finitely generated subgroup of an automatic group has context sensitive wordproblem.Indeed, we may replace “automatic” by the less popular but equally serviceable class“quasigeodesic asynchronously automatic” [N].Proof.

As we remarked above, we need to give a linear bounded algorithm for verifyingmembership in the word problem. Thus, it suffices to see that G has context sensitive wordproblem.

For suppose that H is generated by B = {h1, . .

., hk}. We choose w1, .

. .

, wk ∈A∗so that wi = hi for i −1, . .

., k. Then, given w′ = hi1 . .

. hin ∈B∗, we replace this byw = wi1 .

. .

win ∈A∗. This has increased length by at most a factor of max{ℓ(wi)}.

Wenow appeal our linear bounded algorithm to determine if w = 1, and this will be linearlybounded in the length of our original word w′.Let L be our combing, and suppose that we are given w = a1 . .

. an ∈A∗.

Supposethat for i = 1, . .

., n, ui ∈L and ui = w(i). Then for each i, ℓ(ui) ≤λn + ǫ, where λ andǫ are the constants which assure us that L is short.

Further, w = 1 if and only if un = 1and this happens if and only if un is one of finitely many words (all of length at most ǫ. )So once we have found un, it is easy to determine if w = 1.

Thus it suffices to see thatwe can find un in a linearly bounded manner, and to do this, it will suffice to show thatwe can find each ui+1 from ui and ai+1 in a manner which is linearly bounded in terms ofℓ(ui), since this latter is itself linearly bounded in terms of ℓ(w).To do this, we start enumerating the words of A∗, say in short-lex order, and test eachone to see if it is an element of L. Since L is context sensitive, we can do this in a linearlybounded fashion. When we find a word u in L, we must check to see if it can be taken asui+1.

That is, we must check whether u = w(i + 1) = uiai+1.2

Now L is an asynchronous combing with a departure function. Thus for each a ∈A,one can build an asynchronous two tape finite state automaton which determines whengiven u, u′ ∈L whether or not u = u′a.

(For details see [ECHLPT] or [BGSS].) Thus thedecision as to whether or not to take u as ui+1 can be made using an amount of memorywhich is bounded by a global constant.If we have not found ui+1, we go on to the next element of A∗and discard u. Eventu-ally, we find ui+1, and we need never check any word of length longer than λ(ℓ(ui)+1)+ǫ.Since no ui has length longer than λn+ǫ we shall eventually find un in a linearly boundedway.The Corollary follows by noting that an automatic group has an automatic structurewith uniqueness.

This will consist of quasigeodesics and hence is short and has a departurefunction. We can assume it does not contain e. This, together with the fact that it isregular, ensures that it is context sensitive.In the case where L is an automatic structure with uniqueness, [ECHLPT] showthat ui+1 can be found from ui by a process whose time is linearly bounded in ℓ(ui).

Thisgives a method for solving the word problem in quadratic time. In a similar vein, if G is adirect product of word hyperbolic groups, the word problem for H can be solved in lineartime using pushdown automata [S].References[AS]A.V.

Anisimov and F.D. Seifert, Zur algebraischen charateristik der durch kon-textfreie Sprachen definierten Gruppen, Elektron.

Informationsverarb. Kybernet.11 (1975) 695—702.

[BGSS] G. Baumslag, S.M. Gersten, M. Shapiro and H. Short, Automatic groups andamalgams, Journal of Pure and Applied Algebra 76 (1991), 229—316.[D]W.

Dunwoody, The Accessibility of finitely presented groups, Inventiones Mathe-matica, (1985), 449—457. [ECHLPT] D.B.A.

Epstein, J.W, Cannon, D.F. Holt, S.V.F.

Levi, M.S. Paterson and W.P.Thurston, “Word Processing in Groups,” Jones and Bartlett Publishers, Boston,1992.[HU]J.E.

Hopcroft and J.D. Ullman, “Introduction to Automata Theory, Languages, andComputation,” Addison Wesley, Reading, 1979.[MS]D.E.

Muller and P.E. Schupp, Groups, the theory of ends, and context-free lan-guages, Journal of Computer and System Sciences, 26 (1983), 295—310.[N]W.D.

Neumann, Asynchronous combings of groups, International Journal of Algebraand Computation, 2 (1992), 179—185.[S]H. Short et al., Notes on word hyperbolic groups, in Group Theory from a GeometricViewpoint, E. Ghys, A. Haefliger, A. Verjovsky eds., World Scientific, 1991.3

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