research-article

Symbolic optimization with SMT solvers

Authors:

Aws Albarghouthi,

Zachary Kincaid,

Arie Gurfinkel,

Marsha ChechikAuthors Info & Claims

POPL '14: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

Pages 607 - 618

https://doi.org/10.1145/2535838.2535857

Published: 08 January 2014 Publication History

Abstract

The rise in efficiency of Satisfiability Modulo Theories (SMT) solvers has created numerous uses for them in software verification, program synthesis, functional programming, refinement types, etc. In all of these applications, SMT solvers are used for generating satisfying assignments (e.g., a witness for a bug) or proving unsatisfiability/validity(e.g., proving that a subtyping relation holds). We are often interested in finding not just an arbitrary satisfying assignment, but one that optimizes (minimizes/maximizes) certain criteria. For example, we might be interested in detecting program executions that maximize energy usage (performance bugs), or synthesizing short programs that do not make expensive API calls. Unfortunately, none of the available SMT solvers offer such optimization capabilities.

In this paper, we present SYMBA, an efficient SMT-based optimization algorithm for objective functions in the theory of linear real arithmetic (LRA). Given a formula φ and an objective function t, SYMBA finds a satisfying assignment of φthat maximizes the value of t. SYMBA utilizes efficient SMT solvers as black boxes. As a result, it is easy to implement and it directly benefits from future advances in SMT solvers. Moreover, SYMBA can optimize a set of objective functions, reusing information between them to speed up the analysis. We have implemented SYMBA and evaluated it on a large number of optimization benchmarks drawn from program analysis tasks. Our results indicate the power and efficiency of SYMBA in comparison with competing approaches, and highlight the importance of its multi-objective-function feature.

Supplementary Material

MP4 File (d3_left_t8.mp4)

Download
314.85 MB

References

[1]

Competition On Software Verification - (SV-COMP). http://sv-comp.sosy-lab.org.

[2]

Yices SMT Solver. http://yices.csl.sri.com.

[3]

Z3 Source Code Repository. http://z3.codeplex.com/.

[4]

A. Albarghouthi and K. L. McMillan. Beautiful Interpolants. In Proc. of CAV'13, pages 313--329, 2013.

Digital Library

[5]

A. Albarghouthi, A. Gurfinkel, and M. Chechik. UFO: A Framework for Abstraction- and Interpolation-Based Software Verification. In Proc. of CAV'12, pages 672--678, 2012.

Digital Library

[6]

E. Balas. Disjunctive programming: Properties of the convex hull of feasible points. Discrete Applied Mathematics, 89(1-3):3--44, 1998.

Digital Library

[7]

T. Ball and S. Rajamani. The SLAM Toolkit. In Proc. of CAV'01, volume 2102 of LNCS, pages 260--264, 2001.

Digital Library

[8]

C. Barrett, A. Stump, and C. Tinelli. The SMT-LIB Standard: Version 2.0. Technical report, Department of Computer Science, The University of Iowa, 2010. Available at www.SMT-LIB.org.

[9]

C. Barrett, C. Conway, M. Deters, L. Hadarean, D. Jovanović, T. King, A. Reynolds, and C. Tinelli. CVC4. In Proc. of CAV'11, pages 171--177, 2011.

Digital Library

[10]

C.W. Barrett, R. Sebastiani, S. A. Seshia, and C. Tinelli. Satisfiability modulo theories. In Handbook of Satisfiability, pages 825--885. 2009.

[11]

P. Barth. A Davis-Putnam based enumeration algorithm for linear pseudo-Boolean optimization. Technical Report MPI-I-95-2-003, Max-Planck-Institute für Informatik, 1995.

[12]

D. Beyer, A. Cimatti, A. Griggio, M. E. Keremoglu, and R. Sebastiani. Software Model Checking via Large-Block Encoding. In Proc. of FMCAD'09, pages 25--32, 2009.

[13]

G. M. Bierman, A. D. Gordon, C. Hritcu, and D. E. Langworthy. Semantic Subtyping with an SMT Solver. J. Funct. Program., 22(1): 31--105, 2012.

Digital Library

[14]

N. Bjorner, K. McMillan, and A. Rybalchenko. Program Verification as Satisfiability Modulo Theories. In Proc. of SMT'12, 2012.

[15]

C. Cadar, D. Dunbar, and D. R. Engler. KLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. In Proc. of OSDI'08, pages 209--224, 2008.

Digital Library

[16]

A. Chaganty, A. Lal, A. Nori, and S. Rajamani. Combining Relational Learning with SMT Solvers using CEGAR. In Proc. of CAV'13, pages 447--462, 2013.

Digital Library

[17]

A. Cimatti, A. Franzén, A. Griggio, R. Sebastiani, and C. Stenico. Satisfiability Modulo the Theory of Costs: Foundations and Applications. In Proc. of TACAS'10, pages 99--113, 2010.

Digital Library

[18]

A. Cimatti, A. Griggio, B. J. Schaafsma, and R. Sebastiani. The MathSAT5 SMT Solver. In Proc. of TACAS'13, pages 93--107, 2013.

Digital Library

[19]

E. Clarke, D. Kroening, and F. Lerda. A Tool for Checking ANSI-C Programs. In Proc. of TACAS'04, volume 2988 of LNCS, pages 168--176, March 2004.

[20]

P. Cousot and R. Cousot. Static Determination of Dynamic Properties of Programs. In Proc. of the Colloque sur la Programmation, 1976.

[21]

P. Cousot and R. Cousot. Abstract Interpretation: A Unified Lattice Model For Static Analysis of Programs by Construction or Approximation of Fixpoints. In Proc. of POPL'77, pages 238--252, 1977.

Digital Library

[22]

L. de Moura and N. Bjørner. Z3: An Efficient SMT Solver. In Proc. of TACAS'08, LNCS, pages 337--340, 2008.

Digital Library

[23]

R. DeLine and K. R. M. Leino. BoogiePL: A Typed Procedural Language for Checking Object-oriented Programs. Technical report, Microsoft Research, 2005.

[24]

B. Dutertre and L. de Moura. A Fast Linear-Arithmetic Solver for DPLL(T). In Proc. of CAV'06, pages 81--94, 2006.

Digital Library

[25]

S. Falke, F. Merz, and C. Sinz. LLBMC: Improved Bounded Model Checking of C Programs Using LLVM. In Proc. of TACAS'13, pages 623--626, 2013.

Digital Library

[26]

H. Ganzinger, G. Hagen, R. Nieuwenhuis, A. Oliveras, and C. Tinelli. DPLL(T): Fast Decision Procedures. In Proc. of CAV'04, LNCS, pages 175--188, 2004.

[27]

P. Godefroid, N. Klarlund, and K. Sen. DART: Directed Automated Random Testing. In Proc. of PLDI'05, pages 213--223, 2005.

Digital Library

[28]

P. Godefroid, M. Y. Levin, and D. A. Molnar. SAGE: Whitebox Fuzzing for Security Testing. Commun. ACM, 55(3):40--44, 2012.

Digital Library

[29]

R. E. Gomory. Outline of an Algorithm for Integer Solutions to Linear Programs. Bull. Amer. Math. Soc., 64(5):275--278, 1958.

[30]

S. Grebenshchikov, N. P. Lopes, C. Popeea, and A. Rybalchenko. Synthesizing Software Verifiers from Proof Rules. In Proc. of PLDI'12, pages 405--416, 2012.

Digital Library

[31]

S. Gulwani, S. Jha, A. Tiwari, and R. Venkatesan. Synthesis of Loop-Free Programs. In Proc. of PLDI'11, pages 62--73, 2011.

Digital Library

[32]

A. Gupta, C. Popeea, and A. Rybalchenko. Solving Recursion-Free Horn Clauses over LI+UIF. In Proc. of APLAS'11, pages 188--203, 2011.

Digital Library

[33]

A. Gurfinkel, S. Chaki, and S. Sapra. Efficient Predicate Abstraction of Program Summaries. In Proc. of NFM'11, volume 6617 of LNCS, pages 131--145, 2011.

Digital Library

[34]

W. R. Harris, S. Sankaranarayanan, F. Ivancic, and A. Gupta. Program Analysis via Satisfiability Modulo Path Programs. In Proc. of POPL'10, pages 71--82, 2010.

Digital Library

[35]

T. Henzinger, R. Jhala, R. Majumdar, and G. Sutre. Lazy Abstraction. In Proc. of POPL'02, pages 58--70, 2002.

Digital Library

[36]

T. A. Henzinger, R. Jhala, R. Majumdar, and K. L. McMillan. Abstractions from Proofs. In Proc. of POPL'04, pages 232--244, 2004.

Digital Library

[37]

S. Jha, S. Gulwani, S. A. Seshia, and A. Tiwari. Oracle-Guided Component-Based Program Synthesis. In Proc. of ICSE'10, pages 215--224, 2010.

Digital Library

[38]

A. S. Köksal, V. Kuncak, and P. Suter. Constraints as Control. In Proc. of POPL'12, pages 151--164, 2012.

Digital Library

[39]

K. R. M. Leino. Dafny: An Automatic Program Verifier for Functional Correctness. In Proc. of LPAR'10, pages 348--370, 2010.

Digital Library

[40]

A. Makhorin. The GNU Linear Programming Kit. http://www.gnu.org/software/glpk/, 2000.

[41]

H. Massalin. Superoptimizer: a Look at the Smallest Program. SIGARCH Comput. Archit. News, 15(5):122--126, Oct. 1987.

Digital Library

[42]

A. Miné. The Octagon Abstract Domain. J. Higher-Order and Symbolic Computation, 19(1):31--100, 2006.

Digital Library

[43]

T. M. Mitchell. Machine Learning. McGraw-Hill, Inc., New York, NY, USA, 1997.

Digital Library

[44]

D. Monniaux. Automatic Modular Abstractions for Linear Constraints. In Proc. of POPL'09, pages 140--151, 2009.

Digital Library

[45]

D. Monniaux. Automatic Modular Abstractions for Template Numerical Constraints. Logical Methods in Computer Science, 6(3), 2010.

[46]

R. Nieuwenhuis and A. Oliveras. On SAT Modulo Theories and Optimization Problems. In Proc. of SAT'06, pages 156--169, 2006.

Digital Library

[47]

F. Niu, C. Ré, A. Doan, and J. Shavlik. Tuffy: Scaling Up Statistical Inference in Markov Logic Networks Using an RDBMS. Proc. of VLDB'11, pages 373--384, 2011.

Digital Library

[48]

R. Piskac, T. Wies, and D. Zufferey. Automating Separation Logic using SMT. In Proc. of CAV'13, 2013.

Digital Library

[49]

J. C. Platt. Fast Training of Support Vector Machines Using Sequential Minimal Optimization. In Advances in Kernel Methods, pages 185--208. MIT Press, 1999.

Digital Library

[50]

R. Raman and I. Grossmann. Modelling and Computational Techniques for Logic Based Integer Programming. Computers and Chemical Engineering, 18(7):563--578, 1994.

[51]

T. Reps, M. Sagiv, and G. Yorsh. Symbolic Implementation of the Best Transformer. In Proc. of VMCAI'04, volume 2937 of LNCS, 2004.

[52]

P. M. Rondon, M. Kawaguchi, and R. Jhala. Liquid Types. In Proc. of PLDI'08, pages 159--169, 2008.

Digital Library

[53]

A. Rybalchenko and V. Sofronie-Stokkermans. Constraint Solving for Interpolation. J. Symb. Comput., 45(11):1212--1233, 2010.

Digital Library

[54]

S. Sankaranarayanan, H. B. Sipma, and Z. Manna. Scalable Analysis of Linear Systems using Mathematical Programming. In Proc. of VMCAI'05, pages 25--41, 2005.

Digital Library

[55]

N.W. Sawaya and I. E. Grossmann. A Cutting Plane Method for Solving Linear Generalized Disjunctive Programming Problems. Computers And Chemical Engineering, 29(9):1891--1913, 2005.

[56]

R. Sebastiani and S. Tomasi. Optimization in SMT with LA(Q) Cost Functions. In Proc. of IJCAR'12, pages 484--498, 2012.

Digital Library

[57]

M. Sheeran and G. Ståmarck. A Tutorial on Stålmarck's Proof Procedure for Propositional Logic. FMSD, 16:23--58, 2000.

Digital Library

[58]

A. Solar-Lezama, L. Tancau, R. Bodík, S. A. Seshia, and V. A. Saraswat. Combinatorial Sketching for Finite Programs. In Proc. of ASPLOS'06, pages 404--415, 2006.

Digital Library

[59]

A. V. Thakur and T. W. Reps. A Method for Symbolic Computation of Abstract Operations. In Proc. of CAV'12, pages 174--192, 2012.

Digital Library

[60]

A. V. Thakur, M. Elder, and T. W. Reps. Bilateral Algorithms for Symbolic Abstraction. In Proc. of SAS'12, pages 111--128, 2012.

Digital Library

[61]

R. Wunderling. Paralleler und Objekt-orientierter Simplex-Algorithmus. PhD thesis, Technische Universität Berlin, 1996.

[62]

D. Monniaux and L. Gonnord. Using Bounded Model Checking to Focus Fixpoint Iterations. In Proc. of SAS'11, pages 369--385, 2011.

Digital Library

Cited By

Jain RTihanyi NNdhlovu MFerrag MCordeiro Ld'Amorim M(2024)Rapid Taint Assisted Concolic Execution (TACE)Companion Proceedings of the 32nd ACM International Conference on the Foundations of Software Engineering10.1145/3663529.3663812(627-631)Online publication date: 10-Jul-2024
https://dl.acm.org/doi/10.1145/3663529.3663812
Zhou ZWang C(2024)ShadowBug: Enhanced Synthetic Fuzzing Benchmark GenerationIEEE Open Journal of the Computer Society10.1109/OJCS.2024.33783845(95-106)Online publication date: 2024
https://doi.org/10.1109/OJCS.2024.3378384
Raskin JTsai Y(2024)Synthesis from LTL with Reward Optimization in Sampled Oblivious EnvironmentsDependable Software Engineering. Theories, Tools, and Applications10.1007/978-981-96-0602-3_1(3-21)Online publication date: 25-Nov-2024
https://doi.org/10.1007/978-981-96-0602-3_1
Show More Cited By

Index Terms

Symbolic optimization with SMT solvers
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
  2. Semantics and reasoning
    1. Program reasoning
      1. Invariants

Recommendations

Symbolic optimization with SMT solvers
POPL '14

The rise in efficiency of Satisfiability Modulo Theories (SMT) solvers has created numerous uses for them in software verification, program synthesis, functional programming, refinement types, etc. In all of these applications, SMT solvers are used for ...
Approximating Quantified SMT-Solving with SAT
SSIRI-C '11: Proceedings of the 2011 Fifth International Conference on Secure Software Integration and Reliability Improvement - Companion

Satisfiability Modulo Theories (SMT) is an extension of SAT towards FOL. SMT solvers have proven highly scalable and efficient for problems based on some ground theorems. However, SMT problems involving quantifiers and combination of theorems is a long-...
JavaSMT 3: Interacting with SMT Solvers in Java
Computer Aided Verification
Abstract
Satisfiability Modulo Theories (SMT) is an enabling technology with many applications, especially in computer-aided verification. Due to advances in research and strong demand for solvers, there are many SMT solvers available. Since different ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

POPL '14: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

January 2014

702 pages

ISBN:9781450325448

DOI:10.1145/2535838

General Chair:
Suresh Jagannathan
Purdue University, USA
,
Program Chair:
Peter Sewell
University of Cambridge, UK

ACM SIGPLAN Notices Volume 49, Issue 1
POPL '14
January 2014
661 pages
ISSN:0362-1340
EISSN:1558-1160
DOI:10.1145/2578855
Editor:
Mark W. Bailey
Hamilton College, Clinton, NY
Issue’s Table of Contents

Copyright © 2014 ACM.

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]

Sponsors

SIGPLAN: ACM Special Interest Group on Programming Languages

In-Cooperation

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 January 2014

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

POPL '14

Sponsor:

SIGPLAN

POPL '14: The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

January 22 - 24, 2014

California, San Diego, USA

Acceptance Rates

POPL '14 Paper Acceptance Rate 51 of 220 submissions, 23%;

Overall Acceptance Rate 824 of 4,130 submissions, 20%

Upcoming Conference

POPL '25

Sponsor:
sigplan

The 52nd Annual ACM SIGPLAN Symposium on Principles of Programming Languages

January 19 - 25, 2025

Denver , CO , USA

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

88
Total Citations
View Citations
923
Total Downloads

Downloads (Last 12 months)122
Downloads (Last 6 weeks)15

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Jain RTihanyi NNdhlovu MFerrag MCordeiro Ld'Amorim M(2024)Rapid Taint Assisted Concolic Execution (TACE)Companion Proceedings of the 32nd ACM International Conference on the Foundations of Software Engineering10.1145/3663529.3663812(627-631)Online publication date: 10-Jul-2024
https://dl.acm.org/doi/10.1145/3663529.3663812
Zhou ZWang C(2024)ShadowBug: Enhanced Synthetic Fuzzing Benchmark GenerationIEEE Open Journal of the Computer Society10.1109/OJCS.2024.33783845(95-106)Online publication date: 2024
https://doi.org/10.1109/OJCS.2024.3378384
Raskin JTsai Y(2024)Synthesis from LTL with Reward Optimization in Sampled Oblivious EnvironmentsDependable Software Engineering. Theories, Tools, and Applications10.1007/978-981-96-0602-3_1(3-21)Online publication date: 25-Nov-2024
https://doi.org/10.1007/978-981-96-0602-3_1
Tsiskaridze NBarrett CTinelli C(2024)Generalized Optimization Modulo TheoriesAutomated Reasoning10.1007/978-3-031-63498-7_27(458-479)Online publication date: 3-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-63498-7_27
Wang YLi ZJiang CQiu XRao S(2023)Comparative Synthesis: Learning Near-Optimal Network Designs by QueryProceedings of the ACM on Programming Languages10.1145/35711977:POPL(91-120)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3571197
Sun MYang YWen MWang YZhou YJin HGrundy JPollock LPenta M(2023)Validating SMT Solvers via Skeleton Enumeration Empowered by Historical Bug-Triggering InputsProceedings of the 45th International Conference on Software Engineering10.1109/ICSE48619.2023.00018(69-81)Online publication date: 14-May-2023
https://dl.acm.org/doi/10.1109/ICSE48619.2023.00018
Yao PKe JSun JFu HWu RRen K(2023)Demystifying Template-Based Invariant Generation for Bit-Vector Programs2023 38th IEEE/ACM International Conference on Automated Software Engineering (ASE)10.1109/ASE56229.2023.00069(673-685)Online publication date: 11-Sep-2023
https://doi.org/10.1109/ASE56229.2023.00069
Mufid MAdzkiya DAbate A(2022)SMT-Based Reachability Analysis of High Dimensional Interval Max-Plus Linear SystemsIEEE Transactions on Automatic Control10.1109/TAC.2021.309052567:6(2700-2714)Online publication date: Jun-2022
https://doi.org/10.1109/TAC.2021.3090525
Yao PShi QHuang HZhang C(2021)Program analysis via efficient symbolic abstractionProceedings of the ACM on Programming Languages10.1145/34854955:OOPSLA(1-32)Online publication date: 15-Oct-2021
https://dl.acm.org/doi/10.1145/3485495
Zhang CWagner ROrvalho PGarlan DManquinho VMartins RKang ESpinellis DGousios GChechik MDi Penta M(2021)AlloyMax: bringing maximum satisfaction to relational specificationsProceedings of the 29th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering10.1145/3468264.3468587(155-167)Online publication date: 20-Aug-2021
https://dl.acm.org/doi/10.1145/3468264.3468587
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents