Random van Kampen diagrams and algorithmic problems in groups

Asymmetric cryptography relies on the principle of ease of calculation and complexity of one-sided functions’ inversion. These functions can be easily implemented, but inverting them is computationally difficult. In this context, NP-complete problems are ideal candidates for the role of such functions in asymmetric cryptography, since generating their cases is easy, but finding solutions is difficult. However, the practical application of NP-complete problems has certain limitations, in particular due to difficulties in creating problems that would be complex on average. Although an NP-complete problem may be hard in general, a particular case of it may be solvable, making it unsuitable for cryptography. The article considers classes of NP problems. Basic definitions and concepts are given. The properties of the class of NP-complete problems, the conditions for determining belonging to the set of NP-complete problems, and the current state of difficult to solve problems are analyzed. It turns out that the class of NP-complete problems is hard for quantum computing. The criteria for belonging of the word problem in groups to NP-complete problems are analyzed. Finite non-Abelian groups are defined for which the word problem is NP-complete. The advantages of using non-Abelian groups for cryptographic applications are considered. The rules of change of form, which determine the transformation of equivalent words, are given. The word problem in finite groups is one of the NP-complete problems. The latest research and prospects for the development of cryptographic primitives of asymmetric cryptography using difficult-to-solve problems in finite groups are analyzed.

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.

In this paper we consider irreducible word problems in groups. In particular, we look at results concerning groups whose irreducible word problem lies in some given class of languages (such as the class of finite languages or the class of context-free languages). Introduction In this paper we look at irreducible word problems in groups; see Section 4 below for the definition. We are particularly interested in connections with formal language theory; to be more specific, we consider which types of group can have their irreducible word problem lying in some given class of languages (such as the class of finite languages or the class of context-free languages). We summarize what we need from formal language theory in Section 2. The general question of the connection between irreducible word problems and classes of languages follows on from the analogous question concerning the links between word problems and classes of languages, and we look at some relevant information in Section 3. We come to reduced and irreducible word problems in Section 4, and we talk there about groups with a finite irreducible word problem. We mention some general results about irreducible word problems and languages in Section 5, and then, in Section 6, concentrate on groups whose irreducible word problem is context-free. We finish with some further comments in Section 7.

Word problems of groups: Formal languages, characterizations and decidability

The aim of this article is to survey some connections between formal language theory and group theory with particular emphasis on the word problem for groups and the consequence on the algebraic structure of a group of its word problem belonging to a certain class of formal languages. We define our terms in Section 2 and then consider the structure of groups whose word problem is regular or context-free in Section 3. In Section 4 we look at groups whose wordproblem is a one-counter language, and we move up the Chomsky hierarchy to briefly consider what happens above context-free in Section 5. In Section 6, we see what happens if we consider languages lying in certain complexity classes, and we summarize the situation in Section 7. For general background material on group theory we refer the reader to [25, 26], and for formal language theory to [7, 16, 20].

We adapt the Deutsch-Jozsa algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary output, with the introduction of a more general concept of parity. We provide examples in which properties of the algorithm allow to reduce the number of oracle queries with respect to the deterministic classical case. This has some consequences for the word problem in groups with a particular kind of presentation.KeywordsWord ProblemFree ProductBinary StringAuxiliary InputHadamard GateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Algebraic Invariance of the Word Problem in Groups

This paper considers questions relating formal languages to word problems of groups with a particular emphasis on the decidability of some problems that arise. We investigate the decidability of certain natural conditions that characterize word problems for various classes of languages and we then turn our attention to the question of a language actually being a word problem. We show that this is decidable for the classes of regular and deterministic context-free languages but undecidable for the one-counter languages.

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 word problem in groups of cohomological dimension

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.

We 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.

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.

This chapter is concerned with the question: for which rings R is the theory of R -modules decidable? That is, for which rings is there an effective way of deciding whether or not a given sentence is true in every module? The first section begins with some definitions and discussion, for the benefit of those unfamiliar with decidability questions. Then it is noted that since, for instance, the word problem of a ring is interpretable within the theory of its modules, we should impose some minimum conditions on the ring before the question: “is the theory of modules decidable?” becomes a reasonable one. I suggest such a condition: one should be able to tell effectively whether certain systems of linear equations have solutions in the ring. It is noted that a ring of finite representation type has decidable theory of modules (assuming it satisfies this condition). It is also shown that decidability of the theory of modules is preserved by “effective Morita equivalence”. If the word problem for groups is interpretable within the theory of R -modules, then that theory is undecidable. In §2 we use this fact to establish undecidability of the theory of modules over a variety of rings. It is conjectured that any ring of wild representation type has undecidable theory of modules. In the third section, we turn to decidability. Although the first decidability results were proved “with bare hands”, all present results may be achieved by giving an explicit description of the topology on the space of indecomposable pure-injectives. By a result of Ziegler, that is enough to establish decidability of the theory of R -modules.

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