Skip to main content

Symbolic Computation in Automated Program Reasoning

  • Conference paper
  • First Online:
Formal Methods (FM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14000))

Included in the following conference series:

  • 1074 Accesses

Abstract

We describe applications of symbolic computation towards automating the formal analysis of while-programs implementing polynomial arithmetic. We combine methods from static analysis, symbolic summation and computer algebra to derive polynomial loop invariants, yielding a finite representation of all polynomial equations that are valid before and after each loop execution. While deriving polynomial invariants is in general undecidable, we identify classes of loops for which we automatically can solve the problem of invariant synthesis. We further generalize our work to the analysis of probabilistic program loops. Doing so, we compute higher-order statistical moments over (random) program variables, inferring this way quantitative invariants of probabilistic program loops. Our results yield computer-aided solutions in support of formal software verification, compiler optimization, and probabilistic reasoning.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    As we focus now only on invariant synthesis, we set true to be the loop condition of Fig. 2.

References

  1. Amrollahi, D., Bartocci, E., Kenison, G., Kovács, L., Moosbrugger, M., Stankovic, M.: Solving invariant generation for unsolvable loops. In: Singh, G., Urban, C. (eds.) SAS 2022. LNCS, vol. 13790, pp. 19–43. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22308-2_3

    Chapter  Google Scholar 

  2. Ball, T., Rajamani, S.: The SLAM project: debugging system software via static analysis. In: POPL, pp. 1–3 (2002)

    Google Scholar 

  3. Bartocci, E., Kovács, L., Stankovič, M.: Automatic generation of moment-based invariants for prob-solvable loops. In: Chen, Y.-F., Cheng, C.-H., Esparza, J. (eds.) ATVA 2019. LNCS, vol. 11781, pp. 255–276. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31784-3_15

    Chapter  MATH  Google Scholar 

  4. Buchberger, B.: Symbolic computation (an editorial). J. Symbolic Comput. 1(1), 1–6 (1985)

    Article  MathSciNet  Google Scholar 

  5. Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Logic of Programs, pp. 52–71 (1981)

    Google Scholar 

  6. Cook, B.: Formal reasoning about the security of amazon web services. In: Chockler, H., Weissenbacher, G. (eds.) CAV 2018. LNCS, vol. 10981, pp. 38–47. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96145-3_3

    Chapter  Google Scholar 

  7. Cook, B., Podelski, A., Rybalchenko, A.: Terminator: beyond safety. In: CAV, pp. 415–418 (2006)

    Google Scholar 

  8. Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL, pp. 238–252 (1977)

    Google Scholar 

  9. Floyd, R.W.: Assigning meanings to programs. J. Math. Aspects Comput. Sci. 19, 19–37 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576–580 (1969)

    Article  MATH  Google Scholar 

  11. Hrushovski, E., Ouaknine, J., Pouly, A., Worrell, J.: On strongest algebraic program invariants. J. ACM (2019). To appear

    Google Scholar 

  12. Humenberger, A., Jaroschek, M., Kovács, L.: Automated generation of non-linear loop invariants utilizing hypergeometric sequences. In: ISSAC, pp. 221–228 (2017)

    Google Scholar 

  13. Humenberger, A., Jaroschek, M., Kovács, L.: Aligator.jl – a julia package for loop invariant generation. In: Rabe, F., Farmer, W.M., Passmore, G.O., Youssef, A. (eds.) CICM 2018. LNCS (LNAI), vol. 11006, pp. 111–117. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96812-4_10

    Chapter  Google Scholar 

  14. Humenberger, A., Jaroschek, M., Kovács, L.: Invariant generation for multi-path loops with polynomial assignments. In: VMCAI 2018. LNCS, vol. 10747, pp. 226–246. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73721-8_11

    Chapter  Google Scholar 

  15. Kaminski, B.L., Katoen, J.P., Matheja, C.: On the hardness of analyzing probabilistic programs. Acta Informatica 56(3), 255–285 (2019). https://doi.org/10.1007/s00236-018-0321-1

    Article  MathSciNet  MATH  Google Scholar 

  16. Kauers, M., Zimmermann, B.: Computing the algebraic relations of c-finite sequences and multisequences. J. Symbolic Comput. 43(11), 787–803 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kovács, L.: Aligator: a mathematica package for invariant generation (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 275–282. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71070-7_22

    Chapter  Google Scholar 

  18. Kovács, L.: Reasoning algebraically about p-solvable loops. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 249–264. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_18

    Chapter  Google Scholar 

  19. Moosbrugger, M., Stankovic, M., Bartocci, E., Kovács, L.: This is the moment for probabilistic loops. ACM Program. Lang. 6(OOPSLA2), 1497–1525 (2022)

    Article  Google Scholar 

  20. O’Hearn, P.W.: Continuous reasoning: scaling the impact of formal methods. In: LICS, pp. 13–25 (2018)

    Google Scholar 

  21. Robinson, J.A., Voronkov A. (eds.): Handbook of Automated Reasoning (in 2 volumes). Elsevier, MIT Press; Amsterdam, Cambridge (2001)

    Google Scholar 

  22. Rodríguez-Carbonell, E., Kapur, D: Automatic generation of polynomial loop invariants: algebraic foundations. In: ISSAC, pp. 266–273 (2004)

    Google Scholar 

  23. Sifakis, J.: A unified approach for studying the properties of transition systems. Theor. Comput. Sci. 18, 227–258 (1982)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The work described in this talk is based on joint works with a number of authors, including Daneshvar Amrollahi (TU Wien alumni), Ezio Bartocci (TU Wien), Andreas Humenberger (TU Wien alumni), Maximillian Jaroschek (TU Wien alumni), Tudor Jebelean (RISC-Linz), George Kenison (TU Wien), Marcel Moosbrugger (TU Wien), and Miroslav Stankovic (TU Wien).

The author acknowledges funding and support from the ERC Consolidator Grant 2020 ARTIST 101002685, the ProbInG grant of the Vienna Science and Technology Fund (WWTF) [10.47379/ICT19018], the Austrian FWF project W1255-N23, and the SecInt Doctoral College funded by TU Wien.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Laura Kovács .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kovács, L. (2023). Symbolic Computation in Automated Program Reasoning. In: Chechik, M., Katoen, JP., Leucker, M. (eds) Formal Methods. FM 2023. Lecture Notes in Computer Science, vol 14000. Springer, Cham. https://doi.org/10.1007/978-3-031-27481-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-27481-7_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-27480-0

  • Online ISBN: 978-3-031-27481-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics