Solvability of Matrix-Exponential Equations

We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there e...

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Bibliographic Details
Published in:Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science pp. 798 - 806
Main Authors: Ouaknine, Joel, Pouly, Amaury, Sousa-Pinto, Joao, Worrell, James
Format: Conference Proceeding
Language:English
Published: New York, NY, USA ACM 05.07.2016
Series:ACM Conferences
Subjects:
Computing methodologies
Computing methodologies > Symbolic and algebraic manipulation
Computing methodologies > Symbolic and algebraic manipulation > Symbolic and algebraic algorithms
Computing methodologies > Symbolic and algebraic manipulation > Symbolic and algebraic algorithms > Linear algebra algorithms
Mathematics of computing
Mathematics of computing > Mathematical analysis
Mathematics of computing > Mathematical analysis > Numerical analysis
Mathematics of computing > Mathematical analysis > Numerical analysis > Computations on matrices
matrix logarithms
exponential matrices
commuting matrices
matrix reachability
hybrid automata
ISBN:9781450343916, 1450343910
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Summary:We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, ..., tk such that We show that this problem is undecidable in general, but decidable under the assumption that the matrices A1, ..., Ak commute. Our results have applications to reachability problems for linear hybrid automata. Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.
ISBN:9781450343916
1450343910
DOI:10.1145/2933575.2934538