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And Singular Matrices Decidability of Membership Problems for Flat Rational Subsets of GL¹2, Qº and Singular Matrices Volker Diekert Igor Potapov Pavel Semukhin Formale Methoden der Informatik, Department of Computer Science, Department of Computer Science, Universität Stuttgart University of Liverpool University of Oxford Stuttgart, Germany Liverpool, United Kingfom Oxford, United Kingfom [email protected] [email protected] [email protected] ABSTRACT ACM Reference Format: This work relates numerical problems on matrices over the rationals Volker Diekert, Igor Potapov, and Pavel Semukhin. 2020. Decidability of Membership Problems for Flat Rational Subsets of GL¹2, Qº and Singular to symbolic algorithms on words and finite automata. Using exact Matrices. In International Symposium on Symbolic and Algebraic Computa- algebraic algorithms and symbolic computation, we prove new tion (ISSAC ’20), July 20–23, 2020, Kalamata, Greece. ACM, New York, NY, decidability results for 2 × 2 matrices over Q. Namely, we introduce USA, 8 pages. https://doi.org/10.1145/3373207.3404038 a notion of flat rational sets: if " is a monoid and # ≤ " is its submonoid, then flat rational sets of " relative to # are finite unions of the form !061!1 ···6C !C where all !8 s are rational subsets 1 INTRODUCTION of # and 68 2 ". We give quite general sufficient conditions under Many problems in the analysis of matrix products are inherently which flat rational sets form an effective relative Boolean algebra. difficult to solve even in dimension two, and most of such problems As a corollary, we obtain that the emptiness problem for Boolean become undecidable in general starting from dimension three or combinations of flat rational subsets of GL¹2, Qº over GL¹2, Zº is four. One of these hard questions is the membership problem for ma- decidable. trix semigroups: Given = × = matrices f", "1,...,"< g, determine We also show a dichotomy for nontrivial group extension of whether there exist an integer : ≥ 1 and 81, . , 8: 2 f1, . ,<g GL¹2, Zº in GL¹2, Qº: if 퐺 is a f.g. group such that GL¹2, Zº < 퐺 ≤ such that " = "81 ··· "8: . In other words, determine whether a GL¹2, Qº, then either 퐺 GL¹2, Zº × Z: , for some : ≥ 1, or 퐺 matrix belongs to a finitely generated (f.g. for short) semigroup. contains an extension of the Baumslag-Solitar group BS¹1,@º, with The membership problem has been intensively studied since 1947 @ ≥ 2, of infinite index. It turns out that in the first case themem- when A. Markov showed in [29] that this problem is undecidable bership problem for 퐺 is decidable but the equality problem for for matrices in Z6×6. A natural and important generalization is the rational subsets of 퐺 is undecidable. In the second case, decidability membership problem in rational subsets of a monoid. Rational sets of the membership problem is open for every such 퐺. In the last are those which can be specified by regular expressions. A special section we prove new decidability results for flat rational sets that case is the problem above: membership in the semigroup generated contain singular matrices. In particular, we show that the mem- by the matrices "1,...,"<. Another difficult question is to decide bership problem is decidable for flat rational subsets of "¹2, Qº 9 2 G1 ··· G< the knapsack problem:“ G1,...,G< N: "1 "< = "?”. Even relative to the submonoid that is generated by the matrices from significantly restricted cases of these problems become undecidable "¹2, Zº with determinants 0, ±1 and the central rational matrices. for high dimensional matrices over the integers [6, 26]; and very few cases are known to be decidable, see [3, 7, 12]. The decidability CCS CONCEPTS of the membership problem remains open even for 2 × 2 matrices • Theory of computation ! Formal languages and automata over integers [11, 14, 21, 25, 33]. theory; • Computing methodologies ! Symbolic and alge- Membership in rational subsets of GL¹2, Zº (the 2 × 2 integer braic algorithms. matrices with determinant ±1) is decidable. Indeed, GL¹2, Zº has a free subgroup of rank 2 and of index 24 by [32]. Hence it is a KEYWORDS f.g. virtually free group, and therefore the family of rational sub- membership problem, rational sets, general linear group sets forms an effective Boolean algebra [38, 40]. Two recent results which extended the border of decidability for the membership prob- Partial support for V. Diekert by the DFG grant DI 435/7, for I. Potapov by the EPSRC lem beyond GL¹2, Zº were [34, 35]. The first one is in case of the grant EP/R018472/1 and for P. Semukhin by the ERC grant AVS-ISS (648701) is greatly acknowledged. semigroups of 2 × 2 nonsingular integer matrices, and the second one is in case of GL¹2, Zº extended by integer matrices with zero Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed determinant. for profit or commercial advantage and that copies bear this notice and the full citation This paper pushes the decidability border even further. First of on the first page. Copyrights for components of this work owned by others than ACM all, we consider membership problems for 2×2 matrices over the ra- must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a tionals whereas [34, 35] deal only with integer matrices. Since decid- fee. Request permissions from [email protected]. ability of the rational membership problem is known for GL¹2, Zº, ISSAC ’20, July 20–23, 2020, Kalamata, Greece we focus on subgroups 퐺 of GL¹2, Qº which contain GL¹2, Zº. © 2020 Association for Computing Machinery. ACM ISBN 978-1-4503-7100-1/20/07...$15.00 In Sec. 4 we prove a dichotomy result. In the first case of the A 0 https://doi.org/10.1145/3373207.3404038 dichotomy, 퐺 is generated by GL¹2, Zº and central matrices 0 A . ISSAC ’20, July 20–23, 2020, Kalamata, Greece Volker Diekert, Igor Potapov, and Pavel Semukhin In that case 퐺 is isomorphic to GL¹2, Zº × Z: for : ≥ 1. It can be emptiness of finite Boolean combinations of sets in Frat¹퐺, 퐻º can derived from known results in the literature about free partially be decided. Thus, we have an abstract general condition to de- commutative monoids and groups that equality test for rational cide such questions for a natural subclass of all rational sets in 퐺 sets in 퐺 is undecidable, but the membership problem in rational where the whole class Rat¹퐺º need not be an effective Boolean subsets is still decidable. So, this is the best we can hope for if a algebra. The immediate application in the present paper concerns group is sitting strictly between GL¹2, Zº and GL¹2, Qº, in general. Frat¹GL¹2, Qº, GL¹2, Zºº, see Thm. 3.3 and Cor. 3.4. For example, If such a group 퐺 is not isomorphic to GL¹2, Zº × Z: , then our GL¹2, Zº × Z appears in GL¹2, Qº and Rat¹GL¹2, Zº × Zº is not an dichotomy states that it contains a Baumslag-Solitar group BS¹1,@º effective Boolean algebra. Still the smaller class of flat rational sets for @ ≥ 2. The Baumslag-Solitar groups BS¹?, @º are defined by two Frat¹GL¹2, Zº × Z , GL¹2, Zºº is a relative Boolean algebra. In or- generators 0 and C with the defining relation C0?C−1 = 0@. They der to apply Thm. 3.3, we need Rat¹퐻º to be an effective relative were introduced in [4] and widely studied since then. It is fairly Boolean algebra. It happens to be an effective Boolean algebra for easy to see (much more is known) that they have no free subgroup virtually free groups and many other groups. This class includes, for of finite index unless ?@ = 0 [18]. As a consequence, in both cases example, all f.g. abelian groups, and it is closed under free products. of the dichotomy, GL¹2, Zº has infinite index in 퐺. Actually, we The power of flat rational sets is even more apparent in the con- A1 0 text of the membership problem for rational subsets of GL¹2, Qº. Let prove more, namely, if 퐺 contains a matrix of the form 0 A with 2 % ¹2, Qº denote the monoid GL¹2, Zº [fℎ 2 GL¹2, Qº j jdet¹ℎºj ¡ 1g; jA j < jA j (which is the second case in the dichotomy), then 퐺 1 2 then Thm. 3.6 states that we can solve the membership problem contains some BS¹1,@º for @ ≥ 2 which has infinite index in 퐺. It is “6 2 '?” for all 6 2 GL¹2, Qº and all ' 2 Frat¹GL¹2, Qº,% ¹2, Qºº. wide open whether the membership to rational subsets of 퐺 can Thm. 3.6 generalizes the main result in [34]. be decided in that second case. For example, let ? ≥ 2 be a prime, Let us summarize the statements about groups 퐺 sitting between 0 0 −1 1 1 1 0 and let 퐺 be generated by 1 0 , 0 1 , and 0 ? . In this case GL¹2, Zº and GL¹2, Qº. Our current knowledge is as follows. There ? 0 0 0 −1 1 1 is some evidence that membership in rational subsets of 퐺 is decid- 0 ?−1 also belongs to 퐺 .
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