The Identity Correspondence Problem and its Applications

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the published journal version of this article, see footnote 3 on page 1

On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a nite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: \Is it decidable for a nitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a nite set of matrices generates a group is also undecidable. We also answer several questions for matrices over di erent number elds. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words

ON THE UNDECIDABILITY OF THE IDENTITY CORRESPONDENCE PROBLEM AND ITS APPLICATIONS FOR WORD AND MATRIX SEMIGROUPS

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several questions for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words

Decidability of the Membership Problem for 2×22\times 2 integer matrices

The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generated by $\{M_1,\dots,M_n\}$. Our algorithm relies on a translation of the numerical problem on matrices into combinatorial problems on words. It also makes use of some algebraical properties of well-known subgroups of $\mathrm{GL}(2,\mathbb{Z})$ and various new techniques and constructions that help to limit an infinite number of possibilities by reducing them to the membership problem for regular languages

Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Erratum: Contrary to a statement in a previous version of the paper, our approach does not show that that the freeness problem for automaton semigroups is undecidable. We discuss this in an erratum at the end of the paper

The Identity Problem in nilpotent groups of bounded class

Let G be a unitriangular matrix group of nilpotency class at most ten. We show that the Identity Problem (does a semigroup contain the identity matrix?) and the Group Problem (is a semigroup a group?) are decidable in polynomial time for finitely generated subsemigroups of G. Our decidability results also hold when G is an arbitrary finitely generated nilpotent group of class at most ten. This extends earlier work of Babai et al. on commutative matrix groups (SODA’96) and work of Bell et al. on SL(2, ℤ) (SODA’17). Furthermore, we formulate a sufficient condition for the generalization of our results to nilpotent groups of class d > 10. For every such d, we exhibit an effective procedure that verifies this condition in case it is true

ON THE UNDECIDABILITY OF THE IDENTITY CORRESPONDENCE PROBLEM AND ITS APPLICATIONS FOR WORD AND MATRIX SEMIGROUPS

Electronic version of an article published as in the International Journal of Foundations of Computer Science [© World Scientific Publishing Company]: http://www.worldscientific.com/doi/abs/10.1142/S0129054110007660In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a nite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: \Is it decidable for a nitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a nite set of matrices generates a group is also undecidable. We also answer several questions for matrices over di erent number elds. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words