On Multiphase-Linear Ranking Functions
Multiphase ranking functions (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M\varPhi $$\end{document}RFs) were prop...
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Published in | Computer Aided Verification pp. 601 - 620 |
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Main Authors | , |
Format | Book Chapter |
Language | English |
Published |
Cham
Springer International Publishing
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Series | Lecture Notes in Computer Science |
Online Access | Get full text |
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Summary: | Multiphase ranking functions (\documentclass[12pt]{minimal}
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\begin{document}$$ M\varPhi $$\end{document}RFs) were proposed as a means to prove the termination of a loop in which the computation progresses through a number of “phases”, and the progress of each phase is described by a different linear ranking function. Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (single-path loops). We provide a complete polynomial-time solution to the problem of existence and of synthesis of \documentclass[12pt]{minimal}
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\begin{document}$$ M\varPhi $$\end{document}RF of bounded depth (number of phases), when variables range over rational or real numbers; a complete solution for the (harder) case that variables are integer, with a matching lower-bound proof, showing that the problem is coNP-complete; and a new theorem which bounds the number of iterations for loops with \documentclass[12pt]{minimal}
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\begin{document}$$ M\varPhi $$\end{document}RFs. Surprisingly, the bound is linear, even when the variables involved change in non-linear way. We also consider a type of lexicographic ranking functions more expressive than types of lexicographic functions for which complete solutions have been given so far. We prove that for the above type of loops, lexicographic functions can be reduced to \documentclass[12pt]{minimal}
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\begin{document}$$ M\varPhi $$\end{document}RFs, and thus the questions of complexity of detection, synthesis, and iteration bounds are also answered for this class. |
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Bibliography: | This work was funded partially by the Spanish MINECO projects TIN2012-38137 and TIN2015-69175-C4-2-R, and by the CM project S2013/ICE-3006. We thank Mooly Sagiv for providing us with a working space at Tel-Aviv University, which was crucial for completing this work. |
ISBN: | 9783319633893 3319633899 |
ISSN: | 0302-9743 1611-3349 |
DOI: | 10.1007/978-3-319-63390-9_32 |