39 research outputs found
Decidability of membership problems for flat rational subsets of and singular matrices
This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove various new decidability results for matrices over . For that, we introduce the concept of flat rational sets: if is a monoid and is a submonoid, then ``flat rational sets of over '' are finite unions of the form where all 's are rational subsets of and . We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of over is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further inside . We also show a dichotomy for nontrivial group extension of in : if is a f.g. group such that , then either , for some , or contains an extension of the Baumslag-Solitar group , with , of infinite index. In the first case of the dichotomy the membership problem for is decidable but the equality problem for rational subsets of is undecidable. In the second case, decidability of the membership problem for rational subsets in is open. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable (in doubly exponential time) for flat rational subsets of over the submonoid that is generated by the matrices from with determinants in
Decidability of membership problems for flat rational subsets of GL(2, Q) and singular matrices
This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N ≤ M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1 L1 ··· gtLt where all Lis are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable. We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if G is a f.g. group such that GL(2, Z) < G ≤ GL(2, Q), then either G ≅ GL(2, Z) × Zk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1, q), with q ≥ 2, of infinite index. It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem is open for every such G. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of M(2, Q) relative to the submonoid that is generated by the matrices from M(2, Z) with determinants 0, ± 1 and the central rational matrices
Recent advances in algorithmic problems for semigroups
In this article we survey recent progress in the algorithmic theory of matrix
semigroups. The main objective in this area of study is to construct algorithms
that decide various properties of finitely generated subsemigroups of an
infinite group , often represented as a matrix group. Such problems might
not be decidable in general. In fact, they gave rise to some of the earliest
undecidability results in algorithmic theory. However, the situation changes
when the group satisfies additional constraints. In this survey, we give an
overview of the decidability and the complexity of several algorithmic problems
in the cases where is a low-dimensional matrix group, or a group with
additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New
On the Identity and Group Problems for Complex Heisenberg Matrices
We study the Identity Problem, the problem of determining if a finitely
generated semigroup of matrices contains the identity matrix; see Problem 3
(Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control
Theory'' by Blondel and Megretski (2004). This fundamental problem is known to
be undecidable for and decidable for . The Identity Problem has been recently shown to be in polynomial
time by Dong for the Heisenberg group over complex numbers in any fixed
dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula.
We develop alternative proof techniques for the problem making a step forward
towards more general problems such as the Membership Problem. We extend our
techniques to show that the fundamental problem of determining if a given set
of Heisenberg matrices generates a group, can also be decided in polynomial
time
Decidability of membership problems for flat rational subsets of and singular matrices
This work relates numerical problems on matrices over the rationals to
symbolic algorithms on words and finite automata. Using exact algebraic
algorithms and symbolic computation, we prove various new decidability results
for matrices over . For that, we introduce the concept
of flat rational sets: if is a monoid and is a submonoid, then ``flat
rational sets of over '' are finite unions of the form where all 's are rational subsets of and .
We give quite general sufficient conditions under which flat rational sets
form an effective relative Boolean algebra. As a corollary, we obtain that the
emptiness problem for Boolean combinations of flat rational subsets of
over is decidable (in singly exponential
time). It is possible that such a strong decidability result cannot be pushed
any further inside .
We also show a dichotomy for nontrivial group extension of
in : if is a f.g. group such that , then either , for
some , or contains an extension of the Baumslag-Solitar group
, with , of infinite index. In the first case of the
dichotomy the membership problem for is decidable but the equality problem
for rational subsets of is undecidable. In the second case, decidability of
the membership problem for rational subsets in is open.
In the last section we prove new decidability results for flat rational sets
that contain singular matrices. In particular, we show that the membership
problem is decidable (in doubly exponential time) for flat rational subsets of
over the submonoid that is generated by the matrices from
with determinants in .Comment: 30 pages + 2 pages appendi
Decidability of membership problems for flat rational subsets of GL (2, Q) and singular matrices
This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N ≤ M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1L1 ··· gtLt where all Lis are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable.
We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if G is a f.g. group such that GL(2, Z) < G ≤ GL(2, Q), then either G ≅ GL(2, Z) × Zk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1, q), with q ≥ 2, of infinite index. It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem is open for every such G. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of M(2, Q) relative to the submonoid that is generated by the matrices from M(2, Z) with determinants 0, ± 1 and the central rational matrices.</p
Reachability games and related matrix and word problems
In this thesis, we study different two-player zero-sum games, where one player, called Eve, has a reachability objective (i.e., aims to reach a particular configuration) and the other, called Adam, has a safety objective (i.e., aims to avoid the configuration). We study a general class of games, called Attacker-Defender games, where the computational environment can vary from as simple as the integer line to n-dimensional topological braids. Similarly, the moves themselves can be simple vector addition or linear transformations defined by matrices. The main computational problem is to decide whether Eve has a winning strategy to reach the target configuration from the initial configuration, or whether the dual holds, that is, whether Adam can ensure that the target is never reached. The notion of a winning strategy is widely used in game semantics and its existence means that the player can ensure that his or her winning conditions are met, regardless of the actions of the opponent. It general, games provide a powerful framework to model and analyse interactive processes with uncontrollable adversaries. We formulated several Attacker-Defender games played on different mathematical domains with different transformations (moves), and identified classes of games, where the checking for existence of a winning strategy is undecidable. In other classes, where the problem is decidable, we established their computational complexity. In the thesis, we investigate four classes of games where determining the winner is undecidable: word games, where the players' moves are words over a group alphabet together with integer weights or where the moves are pairs of words over group alphabets; matrix games on vectors, where players transform a three-dimensional vector by linear transformations defined by 3×3 integer matrices; braid games, where players braid and unbraid a given braid; and last, but not least, games played on two-dimensional Z-VAS, closing the gap between decidable and undecidable cases and answering an existing open problem of the field. We also identified decidable fragments, such as word games, where the moves are over a single group alphabet, games on one-dimensional Z-VASS. For word games, we provide an upper-bound of EXPTIME , while for games on Z-VASS, tight bounds of EXPTIME-complete or EXPSPACE-complete, depending on the state structure. We also investigate single-player systems such as polynomial iteration and identity problem in matrix semigroups. We show that the reachability problem for polynomial iteration is PSPACE-complete while the identity problem for the Heisenberg group is in PTIME for dimension three and in EXPTIME for higher dimensions
Asymptotic invariants, complexity of groups and related problems
We survey results about computational complexity of the word problem in
groups, Dehn functions of groups and related problems.Comment: 86 pages. Preliminary version, comments are welcome. v2: some
references added, misprints fixed, some changes suggested by the readers are
made. 88 pages. v3: more readers' suggestions implemented, index added, the
list of references improved. This version is submitted to a journal. v4: The
paper is accepted in Bulletin of Mathematical Science
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.