For several decision problems about semi-Thue systems, we try to locate the frontier between the decidable and the undecidable from the point of view of the number of rules. We show that the the Termination Problem, the U-Termination Problem, the Accessibility Problem and the Common-Descendant Problem are undecidable for 3 rules semi-Thue systems. As a corollary we obtain the undecidability of the Post-Correspondence Problem for 7 pairs of words.

Decision problems for semi-Thue systems with a few rules

Review: Ju. V. Matijasevic, Sue Walker, Simple Examples of Undecidable Associative Calculi

The history of investigations on the word problem for Thue systems is presented with the emphasis on undecidable systems with a few relations. The best known result, a Thue system with only three relations and undecidable word problem, is presented with details. Bibl. 43 items.

Word Problem for Thue Systems with a Few Relations

A survey of the subject outlined in the heading (with many proof s sketched) is given. A special focus is on the original proofs of the unsolvability theorems of Markov, Post, and Novikov for word problems in semigroups and groups. A method of Shirshov is described, which has led to proof of the main unsolvability theorems for Lie algebras.

Unsolvable algorithmic problems for semigroups, groups, and rings

A monoid is compressible if it admits a presentation in which all words in the defining relations begin and end with some specified word. Associated to any compressible monoid $M$ is a compressed left monoid $L(M)$. We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of compressible monoids. We prove that if $\mathcal{C}$ is a superAFL, then $M$ has word problem in $\mathcal{C}$ if and only if $L(M)$ has word problem in $\mathcal{C}$. In particular, this applies when $\mathcal{C}$ is the class of context-free languages. As a corollary, we deduce that if $L(M)$ has context-free word problem, then the rational subset membership problem for $M$ is decidable. We also consider subspecial monoids, which are the one-relation monoids that can be compressed to special monoids. In this case we extend the above results, and show that a subspecial monoid has context-free word problem if and only if all its maximal subgroups are virtually free. This gives a generalisation of the Muller-Schupp theorem to subspecial monoids.

On the Word Problem for Compressible Monoids

Publisher Summary This chapter discusses result that imposes the condition that ℑactually be embedded in the Thue system that is constructed. The two techniques are described in the chapter. In technique 1, there is a uniform construction applicable to any Thue system ℑ. None of the later Thue systems used for embedding will have the simplicity of the present ℑ, taking into account the length of the defining relations, the nature of the mapping that gives the embedding and the explicitness with which the system is given. There is a prove by a rather messy but basically straightforward induction on the number of applications of defining relations of ℑthat are required to transform X into W. In the second technique, a sharper result is obtained than the Corollary of technique 1.

D.J. Collins

Papers

Embeddings into Semigroups with Only a Few Defining Relations