ON ALGORITHMIC SOLVABILITY OF THE PROBLEM OF GENERALIZED CONJUGACY OF SUBGROUPS IN COXETER GROUPS WITH A TREE STRUCTURE

The main algorithmic problems of combinatorial group theory posed by M. Den and G. Titze at the beginning of the twentieth century are the problems of word, word conjugacy and of group isomorphism. However, these problems, as follows from the results of P.S. Novikov and S.I. Adyan, turned out to be unsolvable in the class of finitely defined groups. Therefore, algorithmic problems began to be considered in specific classes of groups. The word conjugacy problem allows for two generalizations. On the one hand, we consider the problem of conjugacy of subgroups, that is, the problem of constructing an algorithm that allows for any two finitely generated subgroups to determine whether they are conjugate or not. On the other hand, the problem of generalized conjugacy of words is posed, that is, the problem of constructing an algorithm that allows for any two finite sets of words to determine whether they are conjugated or not. Combining both of these generalizations into one, we obtain the problem of generalized conjugacy of subgroups. Coxeter groups were introduced in the 30s of the last century, and the problems of equality and conjugacy of words are algorithmically solvable in them. To solve other algorithmic problems, various subclasses are distinguished. This is partly due to the unsolvability in Coxeter groups of another important problem – the problem of occurrence, that is, the problem of the existence of an algorithm that allows for any word and any finitely generated subgroup of a certain group to determine whether this word belongs to this subgroup or not. The paper proves the algorithmic solvability of the problem of generalized conjugacy of subgroups in Coxeter groups with a tree structure.

Computability logic: Giving Caesar what belongs to Caesar

The present article is a brief informal survey o$\textit {computability logic}$ (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic. In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems – understood in their most general, interactive sense – are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool. CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics.