research-article

Ranking Functions for Linear-Constraint Loops

Authors:

Amir M. Ben-Amram,

Samir GenaimAuthors Info & Claims

Journal of the ACM (JACM), Volume 61, Issue 4

Article No.: 26, Pages 1 - 55

https://doi.org/10.1145/2629488

Published: 01 July 2014 Publication History

Abstract

In this article, we study the complexity of the problems: given a loop, described by linear constraints over a finite set of variables, is there a linear or lexicographical-linear ranking function for this loop? While existence of such functions implies termination, these problems are not equivalent to termination. When the variables range over the rationals (or reals), it is known that both problems are PTIME decidable. However, when they range over the integers, whether for single-path or multipath loops, the complexity has not yet been determined. We show that both problems are coNP-complete. However, we point out some special cases of importance of PTIME complexity. We also present complete algorithms for synthesizing linear and lexicographical-linear ranking functions, both for the general case and the special PTIME cases. Moreover, in the rational setting, our algorithm for synthesizing lexicographical-linear ranking functions extends existing ones, because our definition for such functions is more general, yet it has PTIME complexity.

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Index Terms

Ranking Functions for Linear-Constraint Loops
1. Theory of computation

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cover image Journal of the ACM

Journal of the ACM Volume 61, Issue 4

July 2014

259 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2660259

Editor:
Victor Vianu
University of California, San Diego

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Publication History

Published: 01 July 2014

Accepted: 01 March 2014

Revised: 01 February 2014

Received: 01 July 2013

Published in JACM Volume 61, Issue 4

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