Word Problem For Groups Research Articles

Word problems of groups: Formal languages, characterizations and decidability

Let G be a group and let φ:Σ⁎→G be a monoid homomorphism from the set of all strings Σ⁎ over some finite alphabet Σ onto the group G. The set Σ is then called a generating set for G and the language {1}φ−1⊆Σ⁎ is called the word problem of G with respect to the generating set Σ (via the homomorphism φ) and is denoted by W(G,Σ).We consider nine conditions that hold in each such language of the form W(G,Σ) and determine which combinations of these conditions are equivalent to the property of the language in question being the word problem of a group. We show that each of these nine conditions is decidable for the family of regular languages but that each is undecidable for the family of one-counter languages (the languages accepted by one-counter pushdown automata). We also show that the property of a language being the word problem of a group is undecidable for the family of one-counter languages but is decidable for the family of deterministic context-free languages (the languages accepted by deterministic pushdown automata).

Calibrating word problems of groups via the complexity of equivalence relations

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

Asymptotic invariants, complexity of groups and related problems

We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.

Random van Kampen diagrams and algorithmic problems in groups

In this paper we study the structure of random van Kampen diagrams over finitely presented groups. Such diagrams have many remarkable properties. In particular, we show that a random van Kampen diagram over a given group is hyperbolic, even though the group itself may not be hyperbolic. This allows one to design new fast algorithms for the Word Problem in groups. We introduce and study a new filling function, the depth of van Kampen diagrams, – a crucial algorithmic characteristic of null-homotopic words in the group.

Space Complexity and Word Problems of Groups

This paper is concerned with the question of determining which groups have their word problems lying in a given complexity class. Our main results give sufficient conditions for the existence of groups whose word problem is contained in some specified space complexity class but is not contained in some other given space complexity class.

The loop problem for monoids and semigroups

AbstractWe propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial group theory. Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We also study the case in which the loop problem is context-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems. We consider right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is the inverse hull of the monoid.

Computability Results Used in Differential Geometry

AbstractTopologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating properties. Although these computability results had been announced earlier, their proofs have been deferred until this paper.Computably enumerable sets have long been used to proveundecidabilityof mathematical problems such as the word problem for groups and Hilbert's Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specificmathematical or geometric complexityof a mathematical structure such as the depth and distribution of local minima.

Computability Theory and Differential Geometry

Abstract. LetMbe a smooth, compact manifold of dimensionn≥ 5 and sectional curvature ∣K∣ ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics onMmodulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machineTe,eϵ ω, there is a sequence (uniformly effective ine) of homologyn-sphereswhich are also hypersurfaces, such thatis diffeomorphic to the standardn-sphereSn(denoted)iffTehalts on inputk, and in this case the connected sum, so, andis associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time ae σe(k)ofTeon inputsy<k.At their request Soare constructed a particular infinite sequence {Ai}ϵωof c.e. sets so that for allithe settling time of the associated Turing machine forAidominates that forAi+1, even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a “fractal” like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as “the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization.” From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structuralcomplexityof the geometry and topology, not merelyundecidabilityresults as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they usenontrivialinformation about c.e. sets, the Soare sequence {Ai}iϵωabove, not merely Gödel's c.e. noncomputable set K of the 1930's; and (3)withoutusing computability theory there is no known proof that local minima exist even for simple manifolds like the torusT5(see §9.5).

GROUPS WITH CONTEXT-FREE REDUCED WORD PROBLEM

ABSTRACT In this paper we consider reduced word problems of groups. We explain the relationship between the word problem and the reduced word problem, and we give necessary and sufficient conditions for a language to be the word problem (or the reduced word problem) of a group. In addition, we show that the reduced word problem is recursive (or recursively enumerable) precisely when the word problem is recursive. We then prove that the groups which have context-free reduced word problem with respect to some finite monoid generating set are exactly the context-free groups, thus proving a conjecture of Haring-Smith. We also show that, if a group G has finite irreducible word problem with respect to a monoid generating set X, then the reduced word problem of G with respect to X is simple; this is a generalization of one direction of a theorem of Haring-Smith.

Small cancellation theory with a weakened small cancellation hypothesis. II. The word problem

In [1] we introduced the small concellation condition W(4) and developed the basic theory for groups having a W(4) presentation. In this paper we solve the word problem for groups with a finite W(4) presentation.

The complexity of Dehn's algorithm for word problems in groups

Given the presentation for a group of Dehn's algorithm, a multitape Turing machine is shown to exist that solves the word problem for that presentation in linear time. Key to this solution is the pattern matching methodology from complexity theory as well as a recent result for confluent Thue systems.

ON THE WORD AND DIVISIBILITY PROBLEMS IN SEMIGROUPS AND GROUPS WITHOUT CYCLES

The author proves the solvability of the right and left divisibility problems, and consequently the word problem in semigroups with a system of defining relations that do not contain cycles. In particular the solvability of the word problem in groups without cycles is proved. Bibliography: 3 titles.

On the Groups of Britton's Theorem A

In his simple proof of the unsolvability of the word problem for groups [1], J. L. Britton proved a normal form theorem (Britton's Lemma) for groups obtained by the HNN construction. In the appendix to that paper he described a generalization of the HNN construction and sketched the proof of a generalization of Britton's Lemma for this new construction, Britton's Theorem A. In this note we demonstrate that all groups obtained by means of this generalized construction are in fact HNN groups; and it will follow that Theorem A is simply a restatement of Britton's Lemma. This argument makes it clear that while Theorem A can be (and has been) useful in various group theoretic situations, in practice, every application of this generalized construction can be replaced by a straightforward application of the HNN construction.

The word problem for division rings

In this paper we prove that the word problem for division rings is recursively unsolvable. Our proof relies on the corresponding result for groups [7], [28], and makes essential use of P. M. Cohn's recent work [11], [13], [15], [16] on division rings.The word problem for groups is usually formulated in terms of group presentations or finitely presented groups, as in [7], [24], [28], [30]. An equivalent formulation, in terms of the universal Horn sentences of group theory, is mentioned in [32]. This formulation makes sense for arbitrary first-order theories, and it is with respect to this formulation that we show that the word problem for division rings has degree 0′.

Hierarchies of Computable groups and the word problem

The word problem for groups was first formulated by M. Dehn [1], who gave a solution for the fundamental groups of a closed orientable surface of genus g ≧ 2. In the following years solutions were given, for example, for groups with one defining relator [2], free groups, free products of groups with a solvable word problem and, in certain cases, free products of groups with amalgamated subgroups [3], [4], [5]. During the period 1953–1957, it was shown independently by Novikov and Boone that the word problem for groups is recursively undecidable [6], [7]; granting Church's Thesis [8], their work implies that the word problem for groups is effectively undecidable.

A calculus for a certain class of word problems in groups

This paper solves the following problems. Let G be a finitely presented group given by generators g 1, g 2, …, g k and relations R 1=R 2=…=R t=I where R i (i= 1, 2, …, t) are give words in g 1, g 2, …, g k and their inverses. Let H be a subgroup of G of finite index, which is generated by a finite set of words, h 1, h 2, …, h r in the generators of G and their inverses. Find a set of representatives σ 1, σ 2, …, σ 3 of the right cosets of H, and furthermore, given a word W in the generators of G and their inverses, express W in the form W=W *σ i where W * is a word in h 1, h 2, … h r and their inverses. The solution to this problem enables one to solve various interesting problems in group theory. The computation is extremely practical and may be programed easily on a high-speed computing machine. It may be pointed out here that this paper extends the results of [3] in that it gives a systematic procedure for expressing the Schreier generators of a subgroup as words in the originally given generators. The process was used in an informal way as far back as 1960, but at that time machine computation was not envisaged.

J. L. Britton. The word problem for groups. Proceedings of the London Mathematical Society, third series, vol. 8 (1958), pp. 493–506. - John L. Britton. The word problem. Annals of mathematics, second series, vol. 77 (1963), pp. 16–32.

J. L. Britton. The word problem for groups. Proceedings of the London Mathematical Society, third series, vol. 8 (1958), pp. 493–506. - John L. Britton. The word problem. Annals of mathematics, second series, vol. 77 (1963), pp. 16–32.

The Word Problem for Groups

The Word Problem for Groups