Article

Termination of integer linear programs

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Mark BravermanAuthors Info & Claims

CAV'06: Proceedings of the 18th international conference on Computer Aided Verification

Pages 372 - 385

https://doi.org/10.1007/11817963_34

Published: 17 August 2006 Publication History

Abstract

We show that termination of a simple class of linear loops over the integers is decidable. Namely we show that termination of deterministic linear loops is decidable over the integers in the homogeneous case, and over the rationals in the general case. This is done by analyzing the powers of a matrix symbolically using its eigenvalues. Our results generalize the work of Tiwari [Tiw04], where similar results were derived for termination over the reals. We also gain some insights into termination of non-homogeneous integer programs, that are very common in practice.

References

[1]

S. Basu, R. Pollack, M.F. Roy, Algorithms in Real Algebraic Geometry, Springer, 2003.

Digital Library

Google Scholar

[2]

M. Bhubaneswar, Algorithmic Algebra, Springer-Verlag, 1993.

Google Scholar

[3]

V.D. Blondel, O. Bournez, P. Koiran, C.H. Papadimitriou, J.N. Tsitsiklis, Deciding stability and mortality of piecewise affine dynamical system. Theoretical Computer Science, 255 (1-2), pp. 687696, 2001.

Digital Library

Google Scholar

[4]

A.R. Bradley, Z. Manna, H.B. Simpa, Linear ranking with reachability, in CAV 2005, pp. 491-504, 2005.

Digital Library

Google Scholar

[5]

A.R. Bradley, Z. Manna, H.B. Simpa, Termination analysis of integer linear loops, in CONCUR 2005, pp. 488-502, 2005.

Digital Library

Google Scholar

[6]

M.A. Colón, S. Sankaranarayanan, H.B. Simpa, Linear invariant generation using non-linear constraint solving, in CAV 2003, LNCS 2725, pp. 420-432, 2003.

Google Scholar

[7]

K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, 2nd ed., 1971.

Google Scholar

[8]

R. Loos, Computing in Algebraic Extensions, in B. Buchberger, G.E. Collins et al. eds., Computer Algebra: Symbolic and Algebraic Computation, 2nd ed., Springer-Verlag, pp. 173-188, 1983.

Google Scholar

[9]

Y. Matiyasevich, Hilbert's Tenth Problem, The MIT Press, Cambridge, London, 1993.

Digital Library

Google Scholar

[10]

A. Podelski, A. Rybalchenko, A complete method for synthesis of linear ranking functions, in B. Steffen and G. Levi (Eds.): VMCAI 2004, LNCS 2937, pp. 239-251, 2004.

Google Scholar

[11]

A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd ed. Berkeley, CA: University of California Press, 1951.

Google Scholar

[12]

A. Tiwari, Termination of linear programs, in R. Alur and D.A. Peled (Eds.): CAV 2004, LNCS 3114, pp. 70-82, 2004.

Google Scholar

[13]

C.K. Yap, Fundamental Problems of Algorithmic Algebra, Oxford University Press, 2000.

Digital Library

Google Scholar

Cited By

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Halava VNiskanen R(2024)On simulating Turing machines with matrix semigroups with integrality testsTheoretical Computer Science10.1016/j.tcs.2024.1146371005:COnline publication date: 24-Jul-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114637
Zhu SKincaid Z(2024)Breaking the Mold: Nonlinear Ranking Function Synthesis Without TemplatesComputer Aided Verification10.1007/978-3-031-65627-9_21(431-452)Online publication date: 24-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-65627-9_21
Dong R(2023)Termination of linear loops under commutative updatesProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597101(236-241)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597101
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CAV'06: Proceedings of the 18th international conference on Computer Aided Verification

August 2006

563 pages

ISBN:354037406X

Editors:
Thomas Ball
Microsoft Research, Redmond, WA,
,
Robert B. Jones
Intel Corporation, RA2-459, 2501 W. 229th Avenue, Hillsboro, OR,

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 17 August 2006

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Halava VNiskanen R(2024)On simulating Turing machines with matrix semigroups with integrality testsTheoretical Computer Science10.1016/j.tcs.2024.1146371005:COnline publication date: 24-Jul-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114637
Zhu SKincaid Z(2024)Breaking the Mold: Nonlinear Ranking Function Synthesis Without TemplatesComputer Aided Verification10.1007/978-3-031-65627-9_21(431-452)Online publication date: 24-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-65627-9_21
Dong R(2023)Termination of linear loops under commutative updatesProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597101(236-241)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597101
Lommen NGiesl J(2023)Targeting Completeness: Using Closed Forms for Size Bounds of Integer ProgramsFrontiers of Combining Systems10.1007/978-3-031-43369-6_1(3-22)Online publication date: 20-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-43369-6_1
Kovács LVaronka A(2023)What Else is Undecidable About Loops?Relational and Algebraic Methods in Computer Science10.1007/978-3-031-28083-2_11(176-193)Online publication date: 3-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-28083-2_11
Almagor SChistikov DOuaknine JWorrell J(2022)O-Minimal Invariants for Discrete-Time Dynamical SystemsACM Transactions on Computational Logic10.1145/350129923:2(1-20)Online publication date: 14-Jan-2022
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Karimov TLefaucheux EOuaknine JPurser DVaronka AWhiteland MWorrell J(2022)What’s decidable about linear loops?Proceedings of the ACM on Programming Languages10.1145/34987276:POPL(1-25)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3498727
Luca FOuaknine JWorrell J(2022)Algebraic Model Checking for Discrete Linear Dynamical SystemsFormal Modeling and Analysis of Timed Systems10.1007/978-3-031-15839-1_1(3-15)Online publication date: 12-Sep-2022
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Lommen NMeyer FGiesl J(2022)Automatic Complexity Analysis of Integer Programs via Triangular Weakly Non-Linear LoopsAutomated Reasoning10.1007/978-3-031-10769-6_43(734-754)Online publication date: 8-Aug-2022
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Barthe GJacomme CKremer S(2021)Universal Equivalence and Majority of Probabilistic Programs over Finite FieldsACM Transactions on Computational Logic10.1145/348706323:1(1-42)Online publication date: 22-Nov-2021
https://dl.acm.org/doi/10.1145/3487063
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Extending the QCR method to general mixed-integer programs

Analyzing Infeasible Mixed-Integer and Integer Linear Programs

Projected Chvátal---Gomory cuts for mixed integer linear programs

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