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Solving systems of rational equations through strategy iteration

Authors:

Thomas Martin Gawlitza,

Helmut SeidlAuthors Info & Claims

ACM Transactions on Programming Languages and Systems (TOPLAS), Volume 33, Issue 3

Article No.: 11, Pages 1 - 48

https://doi.org/10.1145/1961204.1961207

Published: 04 May 2011 Publication History

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Abstract

We present practical algorithms for computing exact least solutions of equation systems over the reals with addition, multiplication by positive constants, minimum and maximum. The algorithms are based on strategy iteration. Our algorithms can, for instance, be used for the analysis of recursive stochastic games. In the present article we apply our techniques for computing abstract least fixpoint semantics of affine programs over the relational template polyhedra domain. In particular, we thus obtain practical algorithms for computing abstract least fixpoint semantics over the abstract domains of intervals, zones, and octagons.

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Gawlitza Appendix (a11-gawlitza-apndx.pdf)

Online appendix to solving systems of rational equations through strategy iteration on article 11.

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Reviews

Reviewer: Pierre Jouvelot

Advanced compilation techniques that target parallel computing architectures make heavy use of various abstractions of programming constructs to decide, at compile time, whether or not a piece of code can be executed in parallel with another. These choices depend on the data access patterns of the corresponding program statements; among other approaches, such decisions can be seen as checking the satisfiability of affine rational linear equations or inequalities. Solving such systems is thus of key importance, particularly when they are extended to an algebra that includes min and max operators, enabling more precise program analyses. The introduction of min/max operators also makes this framework fit to represent stochastic games in which one player behaves as an outcome maximizer while the other strives to minimize it. Building on this metaphor, the authors propose extending the typical linear-programming-based techniques to solve such systems with two new game-like strategy improvement algorithms. One of the main advantages of this approach, and a key practical contribution of the paper, is that it can be used to perform a quite precise static analysis of programs based on the template polyhedral domain, a powerful data and program abstraction mechanism. Even though the above summary might suggest a terse and dry paper, I enjoyed reading it. The writing is clear, and multiple examples lighten the reading, which is necessary in a paper of this length. The formalism is elegant and the applications are promising. Researchers with game theory or abstract interpretation concerns should find the reading effort quite rewarding. Online Computing Reviews Service

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cover image ACM Transactions on Programming Languages and Systems

ACM Transactions on Programming Languages and Systems Volume 33, Issue 3

April 2011

119 pages

ISSN:0164-0925

EISSN:1558-4593

DOI:10.1145/1961204

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 04 May 2011

Accepted: 01 September 2010

Revised: 01 July 2010

Received: 01 December 2009

Published in TOPLAS Volume 33, Issue 3

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Baldan PKönig BMika-Michalski CPadoan T(2019)Fixpoint games on continuous latticesProceedings of the ACM on Programming Languages10.1145/32903393:POPL(1-29)Online publication date: 2-Jan-2019
https://dl.acm.org/doi/10.1145/3290339
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https://dl.acm.org/doi/10.1016/j.scico.2015.12.005
Gawlitza TSeidl HAdjé AGaubert SGoubault É(2019)Abstract interpretation meets convex optimizationJournal of Symbolic Computation10.1016/j.jsc.2011.12.04847:12(1416-1446)Online publication date: 3-Jan-2019
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Sankaranarayanan SBen Sassi M(2017)Template Polyhedra with a TwistStatic Analysis10.1007/978-3-319-66706-5_16(321-341)Online publication date: 19-Aug-2017
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Abstract interpretation meets convex optimization

How to combine widening and narrowing for non-monotonic systems of equations

Local higher-order fixpoint iteration

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