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Estimating the Volume of Solution Space for Satisfiability Modulo Linear Real Arithmetic

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Abstract

Satisfiability Modulo Theories techniques can check if a formula is satisfiable. In many cases, not only the qualitative judgment (satisfiable or not) but also the quantitative judgment (the dimension and size of the solution space) are of practical interest. For instance, the volume of path condition formula reflects the probability of the corresponding program path being taken. However, existing algorithms are not practical because they only work for small instances. Given a formula with Boolean structures, its volume is typically obtained by first decomposing it to a series of conjunctions (of linear constraints) with disjoint solution spaces and then accumulating the volume of each one. For the former step, we propose a BDD-based search algorithm which sharply reduces the number of conjunctions. For the latter one, we propose a Monte-Carlo integration with a ray-based sampling strategy, which approximates the volume efficiently and accurately. Furthermore, degenerate solution spaces, which are not considered by other algorithms, could be handled properly by ours. Experimental results show that our method can handle formulas with up to 20 variables, which will cover many practical cases in software engineering

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Notes

  1. Since linear constraints are by default conjuncted, we use “linear constraints” short for “a conjunction of linear constraints” if no ambiguity is caused.

  2. http://en.wikipedia.org/wiki/IEEE_floating_point

  3. This is done by checking whether the maximal and minimal value of s are both 0. The technique is similar to that used for implicit equation detection, which will be introduced later in Section 4.3.

  4. http://en.wikipedia.org/wiki/Curse_of_dimensionality

  5. http://en.wikipedia.org/wiki/Gaussian_elimination

  6. http://vlsi.colorado.edu/~fabio/CUDD/

  7. http://cgm.cs.mcgill.ca/~avis/C/lrs.html

  8. The reason is that their algorithm contains constants such as 801, 1600 in nested loops. It takes hundreds of seconds to solve a 3-dimensional case.

  9. http://code.google.com/p/rvc/

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Acknowledgments

I would like to thank Feifei Ma, a coauthor of the related work [24], for the insightful suggestions. I benefited from discussing with her when carrying out this work.

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Authors and Affiliations

  1. School of Software, Tsinghua University, Beijing, 100084, China

    Min Zhou, Fei He, Shi He & Ming Gu

  2. Key Laboratory for Information System Security, MOE, Shanghai, China

    Min Zhou, Fei He, Shi He & Ming Gu

  3. Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing, China

    Min Zhou, Fei He, Shi He & Ming Gu

  4. Electrical and Computer Engineering, Portland State University, Portland, OR, 97207, USA

    Xiaoyu Song

  5. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Gangyi Chen

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  1. Min Zhou
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  2. Fei He
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  3. Xiaoyu Song
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  4. Shi He
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Correspondence to Min Zhou.

Additional information

This work was supported by the Chinese National 973 Plan under grant No. 2010CB328003, the NSF of China under grants No. 11326070, 61272001, 60903030, 91218302, the Chinese National Key Technology R&D Program under grant No. SQ2012BAJY4052, the Tsinghua University Initiative Scientific Research Program, and the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions under grant No. YETP0167

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Zhou, M., He, F., Song, X. et al. Estimating the Volume of Solution Space for Satisfiability Modulo Linear Real Arithmetic. Theory Comput Syst 56, 347–371 (2015). https://doi.org/10.1007/s00224-014-9553-9

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  • DOI: https://doi.org/10.1007/s00224-014-9553-9

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