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Counting solutions to Presburger formulas: how and why

Author:

William PughAuthors Info & Claims

ACM SIGPLAN Notices, Volume 29, Issue 6

Pages 121 - 134

https://doi.org/10.1145/773473.178254

Published: 01 June 1994 Publication History

Abstract

We describe methods that are able to count the number of integer solutions to selected free variables of a Presburger formula, or sum a polynomial over all integer solutions of selected free variables of a Presburger formula. This answer is given symbolically, in terms of symbolic constants (the remaining free variables in the Presburger formula).

For example, we can create a Presburger formula who's solutions correspond to the iterations of a loop. By counting these, we obtain an estimate of the execution time of the loop.

In more complicated applications, we can create Presburger formulas who's solutions correspond to the distinct memory locations or cache lines touched by a loop, the flops executed by a loop, or the array elements that need to be communicated at a particular point in a distributed computation. By counting the number of solutions, we can evaluate the computation/memory balance of a computation, determine if a loop is load balanced and evaluate message traffic and allocate message buffers.

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Cited By

Molavi ASchneider TDowning MBang L(2020)MCBAT: Model Counting for Constraints over Bounded Integer ArraysSoftware Verification10.1007/978-3-030-63618-0_8(124-143)Online publication date: 6-Dec-2020
https://doi.org/10.1007/978-3-030-63618-0_8
Jing RMoreno Maza M(2020)The Z_Polyhedra Library in MapleMaple in Mathematics Education and Research10.1007/978-3-030-41258-6_10(132-144)Online publication date: 28-Feb-2020
https://doi.org/10.1007/978-3-030-41258-6_10
WOODS K(2015)PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALSThe Journal of Symbolic Logic10.1017/jsl.2015.480:2(433-449)Online publication date: 22-Apr-2015
https://doi.org/10.1017/jsl.2015.4
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Counting solutions to Presburger formulas: how and why

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Published In

cover image ACM SIGPLAN Notices

ACM SIGPLAN Notices Volume 29, Issue 6

June 1994

360 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/773473

Guest Editor:
Brent T. Hailpern

Issue’s Table of Contents

PLDI '94: Proceedings of the ACM SIGPLAN 1994 conference on Programming language design and implementation
August 1994
360 pages
ISBN:089791662X
DOI:10.1145/178243
Chairmen:
Vivek Sarkar
IBM Santa Teresa Lab.
,
Barbara Ryder
Rutgers Univ., New Brunswick, NJ
,
Mary Lou Soffa
Univ. of Pittsburgh, Pittsburgh, PA

Copyright © 1994 ACM.

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Association for Computing Machinery

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Publication History

Published: 01 June 1994

Published in SIGPLAN Volume 29, Issue 6

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Cited By

Molavi ASchneider TDowning MBang L(2020)MCBAT: Model Counting for Constraints over Bounded Integer ArraysSoftware Verification10.1007/978-3-030-63618-0_8(124-143)Online publication date: 6-Dec-2020
https://doi.org/10.1007/978-3-030-63618-0_8
Jing RMoreno Maza M(2020)The Z_Polyhedra Library in MapleMaple in Mathematics Education and Research10.1007/978-3-030-41258-6_10(132-144)Online publication date: 28-Feb-2020
https://doi.org/10.1007/978-3-030-41258-6_10
WOODS K(2015)PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALSThe Journal of Symbolic Logic10.1017/jsl.2015.480:2(433-449)Online publication date: 22-Apr-2015
https://doi.org/10.1017/jsl.2015.4
Woods K(2013)Presburger arithmetic, rational generating functions, and quasi-polynomialsProceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II10.1007/978-3-642-39212-2_37(410-421)Online publication date: 8-Jul-2013
https://dl.acm.org/doi/10.1007/978-3-642-39212-2_37
Balasa FZhu HLuican I(2012)Computation of Data Storage Requirements for Affine Algorithmic SpecificationsEnergy-Aware Memory Management for Embedded Multimedia Systems10.1201/b11418-4(67-116)Online publication date: 4-Jan-2012
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Bygde SErmedahl ALisper B(2011)An efficient algorithm for parametric WCET calculationJournal of Systems Architecture: the EUROMICRO Journal10.1016/j.sysarc.2010.06.00957:6(614-624)Online publication date: 1-Jun-2011
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Shrivastava ALee JJeyapaul R(2010)Cache vulnerability equations for protecting data in embedded processor caches from soft errorsACM SIGPLAN Notices10.1145/1755951.175591045:4(143-152)Online publication date: 13-Apr-2010
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Kejariwal ANicolau ASaito HTian XGirkar MBanerjee UPolychronopoulos CGibbons PVishkin U(2006)A general approach for partitioning N-dimensional parallel nested loops with conditionalsProceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures10.1145/1148109.1148117(49-58)Online publication date: 30-Jul-2006
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Burgstaller BScholz BBlieberger J(2006)Symbolic analysis of imperative programming languagesProceedings of the 7th joint conference on Modular Programming Languages10.1007/11860990_12(172-194)Online publication date: 13-Sep-2006
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