Skip to main content
SpringerLink
Log in

Bit-Precise Verification of Discontinuity Errors Under Fixed-Point Arithmetic

  • Conference paper
  • First Online:
Software Engineering and Formal Methods (SEFM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13085))

Included in the following conference series:

Abstract

Non-integer arithmetic is prone to numerical errors due to the finite representation of numbers. These errors propagate, possibly non-linearly, throughout the variables of a program and can affect its control flow, altering reachability, and thus safety. We consider the problem of rigorous bit-precise numerical accuracy certification of programs in the presence of control structures and operations under fixed-point arithmetic over (non-deterministic) variables of arbitrary, mixed precision. By applying program transformation, we reduce the problem of assessing whether a given error bound can be exceeded in the initial program to a reachability problem in a bit-vector program. We implement our technique as a pre-processing module that integrates seamlessly with an existing mature BMC-based verification workflow. We present an experimental evaluation of our error certification technique on a set of arithmetic routines commonly used in the industry.

Partially supported by MIUR projects PRIN 2017TWRCNB SEDUCE (Designing Spatially Distributed Cyber-Physical Systems under Uncertainty) and PRIN 2017FTXR7S IT-MATTERS (Methods and Tools for Trustworthy Smart Systems).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
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

References

  1. MISRA-C:2004 — Guidelines for the use of the C language in critical systems. Technical report, MIRA Ltd. (2004)

    Google Scholar 

  2. Programming languages — C — Extensions to support embedded processors. ISO/IEC Technical Report 18037:2008. EEE, New York (2008)

    Google Scholar 

  3. DO-178C/ED-12C, Software considerations in airborne systems and equipment certification. Technical report, RTCA/EUROCAE (2011)

    Google Scholar 

  4. ISO 26262 Road Vehicles - Functional Safety. Technical report. ISO, Geneva, Switzerland (2011)

    Google Scholar 

  5. Abate, A., et al.: Automated formal synthesis of digital controllers for state-space physical plants. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 462–482. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63387-9_23

    Chapter  Google Scholar 

  6. Bessa, I., Abreu, R.B., Filho, J.E.C., Cordeiro, L.C.: SMT-based bounded model checking of fixed-point digital controllers. In: IECON, pp. 295–301. IEEE (2014)

    Google Scholar 

  7. Brillout, A., Kroening, D., Wahl, T.: Mixed abstractions for floating-point arithmetic. In: FMCAD, pp. 69–76. IEEE (2009)

    Google Scholar 

  8. Chaudhuri, S., Gulwani, S., Lublinerman, R.: Continuity and robustness of programs. Commun. ACM 55(8), 107–115 (2012)

    Article  Google Scholar 

  9. Clarke, E., Kroening, D., Lerda, F.: A tool for checking ANSI-C programs. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 168–176. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24730-2_15

    Chapter  MATH  Google Scholar 

  10. Darulova, E., Kuncak, V.: Sound compilation of reals. In: POPL, pp. 235–248. ACM (2014)

    Google Scholar 

  11. Darulova, E., Kuncak, V.: Towards a compiler for reals. ACM Trans. Program. Lang. Syst. 39(2), 8:1–8:28 (2017)

    Google Scholar 

  12. Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24605-3_37

    Chapter  Google Scholar 

  13. Fischer, B., Inverso, O., Parlato, G.: CSeq: a concurrency pre-processor for sequential C verification tools. In: ASE, pp. 710–713. IEEE (2013)

    Google Scholar 

  14. Gadelha, M.R., Cordeiro, L.C., Nicole, D.A.: An efficient floating-point bit-blasting API for verifying C programs. In: Christakis, M., Polikarpova, N., Duggirala, P.S., Schrammel, P. (eds.) NSV/VSTTE -2020. LNCS, vol. 12549, pp. 178–195. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-63618-0_11

    Chapter  Google Scholar 

  15. Ghorbal, K., Goubault, E., Putot, S.: A logical product approach to zonotope intersection. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 212–226. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14295-6_22

    Chapter  Google Scholar 

  16. Giacobbe, M., Henzinger, T.A., Lechner, M.: How many bits does it take to quantize your neural network? In: TACAS 2020. LNCS, vol. 12079, pp. 79–97. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45237-7_5

    Chapter  Google Scholar 

  17. Goubault, E., Putot, S.: Static analysis of finite precision computations. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 232–247. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-18275-4_17

    Chapter  Google Scholar 

  18. Goubault, E., Putot, S.: Robustness analysis of finite precision implementations. In: Shan, C. (ed.) APLAS 2013. LNCS, vol. 8301, pp. 50–57. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03542-0_4

    Chapter  Google Scholar 

  19. Inverso, O., Bemporad, A., Tribastone, M.: Sat-based synthesis of spoofing attacks in cyber-physical control systems. In: ICCPS, pp. 1–9. IEEE/ACM (2018)

    Google Scholar 

  20. Inverso, O., Trubiani, C.: Parallel and distributed bounded model checking of multi-threaded programs. In: PPoPP, pp. 202–216. ACM (2020)

    Google Scholar 

  21. Ivancic, F., Ganai, M.K., Sankaranarayanan, S., Gupta, A.: Numerical stability analysis of floating-point computations using software model checking. In: MEMOCODE, pp. 49–58. IEEE (2010)

    Google Scholar 

  22. Jet Propulsion Laboratory, C.I.o.T.: JPL Institutional Coding Standard for the C Programming Language. Standard (2009)

    Google Scholar 

  23. Martel, M., Najahi, A., Revy, G.: Toward the synthesis of fixed-point code for matrix inversion based on Cholesky decomposition. In: DASIP, pp. 1–8. IEEE (2014)

    Google Scholar 

  24. Martel, M., Najahi, A., Revy, G.: Trade-offs of certified fixed-point code synthesis for linear algebra basic blocks. J. Syst. Archit. 76, 133–148 (2017)

    Article  Google Scholar 

  25. Martinez, A.A., Majumdar, R., Saha, I., Tabuada, P.: Automatic verification of control system implementations. In: EMSOFT, pp. 9–18. ACM (2010)

    Google Scholar 

  26. Narodytska, N., Kasiviswanathan, S.P., Ryzhyk, L., Sagiv, M., Walsh, T.: Verifying properties of binarized deep neural networks. In: AAAI, pp. 6615–6624. AAAI Press (2018)

    Google Scholar 

  27. Navas, J.A., Schachte, P., Søndergaard, H., Stuckey, P.J.: Signedness-agnostic program analysis: precise integer bounds for low-level code. In: Jhala, R., Igarashi, A. (eds.) APLAS 2012. LNCS, vol. 7705, pp. 115–130. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35182-2_9

    Chapter  Google Scholar 

  28. Park, J., Pajic, M., Sokolsky, O., Lee, I.: Automatic verification of finite precision implementations of linear controllers. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 153–169. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54577-5_9

    Chapter  Google Scholar 

  29. Salamati, M., Salvia, R., Darulova, E., Soudjani, S., Majumdar, R.: Memory-efficient mixed-precision implementations for robust explicit model predictive control. ACM Trans. Embed. Comput. Syst. 18(5s), 100:1–100:19 (2019)

    Google Scholar 

  30. Simić, S., Bemporad, A., Inverso, O., Tribastone, M.: Tight error analysis in fixed-point arithmetic. In: Dongol, B., Troubitsyna, E. (eds.) IFM 2020. LNCS, vol. 12546, pp. 318–336. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-63461-2_17

    Chapter  Google Scholar 

  31. Solovyev, A., Baranowski, M.S., Briggs, I., Jacobsen, C., Rakamarić, Z., Gopalakrishnan, G.: Rigorous estimation of floating-point round-off errors with symbolic Taylor expansions. ACM Trans. Program. Lang. Syst. 41(1), 2:1–2:39 (2019)

    Google Scholar 

  32. Stol, J., De Figueiredo, L.H.: Self-validated numerical methods and applications. In: Monograph for 21st Brazilian Mathematics Colloquium, IMPA. Citeseer (1997)

    Google Scholar 

  33. Tabuada, P., Balkan, A., Caliskan, S.Y., Shoukry, Y., Majumdar, R.: Input-output robustness for discrete systems. In: EMSOFT, pp. 217–226. ACM (2012)

    Google Scholar 

  34. Titolo, L., Feliú, M.A., Moscato, M., Muñoz, C.A.: An abstract interpretation framework for the round-off error analysis of floating-point programs. In: VMCAI 2018. LNCS, vol. 10747, pp. 516–537. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73721-8_24

    Chapter  MATH  Google Scholar 

  35. Titolo, L., Moscato, M., Feliu, M.A., Muñoz, C.A.: Automatic generation of guard-stable floating-point code. In: Dongol, B., Troubitsyna, E. (eds.) IFM 2020. LNCS, vol. 12546, pp. 141–159. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-63461-2_8

    Chapter  Google Scholar 

  36. Yates, R.: Fixed-point arithmetic: an introduction. Digital Signal Labs (2009)

    Google Scholar 

  37. Zhao, Y., Shumailov, I., Mullins, R.D., Anderson, R.: To compress or not to compress: understanding the interactions between adversarial attacks and neural network compression. In: MLSys. mlsys.org (2019)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMT School for Advanced Studies, Lucca, Italy

    Stella Simić & Mirco Tribastone

  2. Gran Sasso Science Institute, L’Aquila, Italy

    Omar Inverso

Authors
  1. Stella Simić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omar Inverso
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mirco Tribastone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stella Simić .

Editor information

Editors and Affiliations

  1. University of York, York, UK

    Radu Calinescu

  2. Carnegie Mellon University, Moffett Field, CA, USA

    Corina S. Păsăreanu

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Simić, S., Inverso, O., Tribastone, M. (2021). Bit-Precise Verification of Discontinuity Errors Under Fixed-Point Arithmetic. In: Calinescu, R., Păsăreanu, C.S. (eds) Software Engineering and Formal Methods. SEFM 2021. Lecture Notes in Computer Science(), vol 13085. Springer, Cham. https://doi.org/10.1007/978-3-030-92124-8_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-92124-8_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-92123-1

  • Online ISBN: 978-3-030-92124-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation