On the identity problem for the special linear group and the heisenberg group

Sang Ki Ko, Reino Niskanen, Igor Potapov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

15 Scopus citations

Abstract

We study the identity problem for matrices, i.e., whether the identity matrix is in a semigroup generated by a given set of generators. In particular we consider the identity problem for the special linear group following recent NP-completeness result for SL(2, Z) and the undecidability for SL(4, Z) generated by 48 matrices. First we show that there is no embedding from pairs of words into 3 × 3 integer matrices with determinant one, i.e., into SL(3, Z) extending previously known result that there is no embedding into C2×2. Apart from theoretical importance of the result it can be seen as a strong evidence that the computational problems in SL(3, Z) are decidable. The result excludes the most natural possibility of encoding the Post correspondence problem into SL(3, Z), where the matrix products extended by the right multiplication correspond to the Turing machine simulation. Then we show that the identity problem is decidable in polynomial time for an important subgroup of SL(3, Z), the Heisenberg group H(3, Z). Furthermore, we extend the decidability result for H(n, Q) in any dimension n. Finally we are tightening the gap on decidability question for this long standing open problem by improving the undecidability result for the identity problem in SL(4, Z) substantially reducing the bound on the size of the generator set from 48 to 8 by developing a novel reduction technique.

Original languageEnglish
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770767
DOIs
StatePublished - 1 Jul 2018
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 201813 Jul 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume107
ISSN (Print)1868-8969

Conference

Conference45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/TerritoryCzech Republic
CityPrague
Period9/07/1813/07/18

Keywords

  • Decidability
  • Heisenberg group
  • Identity problem
  • Matrix semigroup
  • Special linear group

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