Termination proofs for linear simple loops

Chen, Hong Yi; Flur, Shaked; Mukhopadhyay, Supratik

doi:10.1007/s10009-013-0288-8

Cited by 7 publications

(2 citation statements)

References 21 publications

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“…There is a voluminous literature on the topic, albeit largely focussing on heuristics and semi-algorithms (via spectral methods or the synthesis of ranking functions), rather than exact decidability results. Relevant papers include [43,16,11,12,15,17,21,40,41,28,13,10]. Several of these approaches have moreover been implemented in software verification tools, such as Microsoft's Terminator [23,24].…”

Section: Introductionmentioning

confidence: 99%

What's Decidable about Discrete Linear Dynamical Systems?

Karimov¹,

Kelmendi²,

Ouaknine³

et al. 2022

Preprint

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Section: Introductionmentioning

confidence: 99%

What's Decidable about Discrete Linear Dynamical Systems?

Karimov¹,

Kelmendi²,

Ouaknine³

et al. 2022

Preprint

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show abstract

“…There is a voluminous literature on the topic, albeit largely focussing on heuristics and semi-algorithms (via spectral methods or the synthesis of ranking functions), rather than exact decidability results. Relevant papers include [6][7][8][9]13,14,16,21,27,40,41,44]. Several of these approaches have moreover been implemented in software verification tools, such as Microsoft's Terminator [22,23].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Model Checking for Discrete Linear Dynamical Systems

Luca

Ouaknine

Worrell

2022

Lecture Notes in Computer Science

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Model checking infinite-state systems is one of the central challenges in automated verification. In this survey we focus on an important and fundamental subclass of infinite-state systems, namely discrete linear dynamical systems. While such systems are ubiquitous in mathematics, physics, engineering, etc., in the present context our motivation stems from their relevance to the formal analysis and verification of program loops, weighted automata, hybrid systems, and control systems, amongst many others. Our main object of study is the problem of model checking temporal properties on the infinite orbit of a linear dynamical system, and our principal contribution is to show that for a rich class of properties this problem can be reduced to certain classical decision problems on linear recurrence sequences, notably the Skolem Problem. This leads us to discuss recent advances on the latter and to highlight the prospects for further progress on charting the algorithmic landscape of linear recurrence sequences and linear dynamical systems.

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What’s Decidable About Discrete Linear Dynamical Systems?

Karimov

Kelmendi

Ouaknine

et al. 2022

Lecture Notes in Computer Science

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We consider reachability decision problems for linear dynamical systems: Given a linear map on R d , together with source and target sets, determine whether there is a point in the source set whose orbit, obtained by repeatedly applying the linear map, enters the target set. When the source and target sets are semialgebraic, this problem can be reduced to a point-to-polytope reachability question. The latter is generally believed not to be substantially harder than the well-known Skolem and Positivity Problems. The situation is markedly different for multiple reachability, i.e. the question of whether the orbit visits the target set at least m times, for some given positive integer m. In this paper, we prove that when the source set is semialgebraic and the target set consists of a hyperplane, multiple reachability is undecidable; in fact we already obtain undecidability in ambient dimension d = 10 and with fixed m = 9. Moreover, as we observe that procedures for dimensions 3 up to 9 would imply strong results pertaining to effective solutions of Diophantine equations, we mainly focus on the affine plane (R 2 ). We obtain two main positive results. We show that multiple reachability is decidable for halfplane targets, and that it is also decidable for general semialgebraic targets, provided the linear map is a rotation. The latter result involves a new method, based on intersections of algebraic subgroups with subvarieties, due to Bombieri and Zannier.

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Termination proofs for linear simple loops

Cited by 7 publications

References 21 publications

What's Decidable about Discrete Linear Dynamical Systems?

What's Decidable about Discrete Linear Dynamical Systems?

Algebraic Model Checking for Discrete Linear Dynamical Systems

What’s Decidable About Discrete Linear Dynamical Systems?

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