Skip to main content
SpringerLink
Log in

Selectively-Amortized Resource Bounding

  • Conference paper
  • First Online:
Static Analysis (SAS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12913))

Included in the following conference series:

Abstract

We consider the problem of automatically proving resource bounds. That is, we study how to prove that an integer-valued resource variable is bounded by a given program expression. Automatic resource-bound analysis has recently received significant attention because of a number of important applications (e.g., detecting performance bugs, preventing algorithmic-complexity attacks, identifying side-channel vulnerabilities), where the focus has often been on developing precise amortized reasoning techniques to infer the most exact resource usage. While such innovations remain critical, we observe that fully precise amortization is not always necessary to prove a bound of interest. And in fact, by amortizing selectively, the needed supporting invariants can be simpler, making the invariant inference task more feasible and predictable. We present a framework for selectively-amortized analysis that mixes worst-case and amortized reasoning via a property decomposition and a program transformation. We show that proving bounds in any such decomposition yields a sound resource bound in the original program, and we give an algorithm for selecting a reasonable decomposition.

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. Abadi, M., Banerjee, A., Heintze, N., Riecke, J.G.: A core calculus of dependency. In: Principles of Programming Languages (POPL), pp. 147–160 (1999). https://doi.org/10.1145/292540.292555

  2. Albert, E., Arenas, P., Genaim, S., Puebla, G., Zanardini, D.: Cost analysis of Java bytecode. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 157–172. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71316-6_12

    Chapter  Google Scholar 

  3. Albert, E., Genaim, S., Gómez-Zamalloa, M.: Heap space analysis for Java bytecode. In: International Symposium on Memory Management (ISMM), pp. 105–116 (2007). https://doi.org/10.1145/1296907.1296922

  4. Albert, E., Genaim, S., Gómez-Zamalloa, M.: Live heap space analysis for languages with garbage collection. In: International Symposium on Memory Management (ISMM), pp. 129–138 (2009). https://doi.org/10.1145/1542431.1542450

  5. Albert, E., Genaim, S., Masud, A.N.: More precise yet widely applicable cost analysis. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 38–53. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-18275-4_5

    Chapter  Google Scholar 

  6. Alonso-Blas, D.E., Genaim, S.: On the limits of the classical approach to cost analysis. In: Miné, A., Schmidt, D. (eds.) SAS 2012. LNCS, vol. 7460, pp. 405–421. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33125-1_27

    Chapter  Google Scholar 

  7. Atkey, R.: Amortised resource analysis with separation logic. In: Gordon, A.D. (ed.) ESOP 2010. LNCS, vol. 6012, pp. 85–103. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11957-6_6

    Chapter  Google Scholar 

  8. Brockschmidt, M., Emmes, F., Falke, S., Fuhs, C., Giesl, J.: Alternating runtime and size complexity analysis of integer programs. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 140–155. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54862-8_10

    Chapter  Google Scholar 

  9. Cadek, P., Danninger, C., Sinn, M., Zuleger, F.: Using loop bound analysis for invariant generation. In: Formal Methods in Computer Aided Design (FMCAD), pp. 1–9 (2018). https://doi.org/10.23919/FMCAD.2018.8603005

  10. Carbonneaux, Q., Hoffmann, J., Shao, Z.: Compositional certified resource bounds. In: Programming Language Design and Implementation (PLDI), pp. 467–478 (2015). https://doi.org/10.1145/2737924.2737955

  11. Carbonneaux, Q., Hoffmann, J., Reps, T., Shao, Z.: Automated resource analysis with Coq proof objects. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 64–85. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63390-9_4

    Chapter  Google Scholar 

  12. Chatterjee, K., Fu, H., Goharshady, A.K., Goharshady, E.K.: Polynomial invariant generation for non-deterministic recursive programs. In: Programming Language Design and Implementation (PLDI), pp. 672–687 (2020). https://doi.org/10.1145/3385412.3385969

  13. Colón, M.A., Sankaranarayanan, S., Sipma, H.B.: Linear invariant generation using non-linear constraint solving. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 420–432. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45069-6_39

    Chapter  Google Scholar 

  14. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

    Chapter  Google Scholar 

  15. Defense Advanced Research Projects Agency (DARPA): Space/time analysis for cybersecurity (STAC) (2019). https://www.darpa.mil/program/space-time-analysis-for-cybersecurity

  16. Dillig, I., Dillig, T., Li, B., McMillan, K.L.: Inductive invariant generation via abductive inference. In: Object-Oriented Programming Systems, Languages, and Applications (OOPSLA), pp. 443–456 (2013). https://doi.org/10.1145/2509136.2509511

  17. Guéneau, A., Charguéraud, A., Pottier, F.: A fistful of dollars: formalizing asymptotic complexity claims via deductive program verification. In: Ahmed, A. (ed.) ESOP 2018. LNCS, vol. 10801, pp. 533–560. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89884-1_19

    Chapter  Google Scholar 

  18. Guéneau, A., Jourdan, J.-H., Charguéraud, A., Pottier, F.: Formal proof and analysis of an incremental cycle detection algorithm. In: Interactive Theorem Proving (ITP), vol. 141, pp. 18:1–18:20 (2019). https://doi.org/10.4230/LIPIcs.ITP.2019.18

  19. Gulwani, S., Zuleger, F.: The reachability-bound problem. In: Programming Language Design and Implementation (PLDI), pp. 292–304 (2010). https://doi.org/10.1145/1806596.1806630

  20. Gulwani, S., Jain, S., Koskinen, E.: Control-flow refinement and progress invariants for bound analysis. In: Programming Language Design and Implementation (PLDI), pp. 375–385 (2009a). https://doi.org/10.1145/1542476.1542518

  21. Gulwani, S., Mehra, K.K., Chilimbi, T.M.: SPEED: precise and efficient static estimation of program computational complexity. In: Principles of Programming Languages (POPL), pp. 127–139 (2009). https://doi.org/10.1145/1480881.1480898

  22. Hoffmann, J., Aehlig, K., Hofmann, M.: Multivariate amortized resource analysis. In: Principles of Programming Languages (POPL), pp. 357–370 (2011). https://doi.org/10.1145/1926385.1926427

  23. Hoffmann, J., Das, A., Weng, S.-C.: Towards automatic resource bound analysis for OCaml. In: Principles of Programming Languages (POPL), pp. 359–373 (2017). https://doi.org/10.1145/3009837.3009842

  24. Hrushovski, E., Ouaknine, J., Pouly, A., Worrell, J.: Polynomial invariants for affine programs. In: Logic in Computer Science (LICS), pp. 530–539 (2018). https://doi.org/10.1145/3209108.3209142

  25. Kincaid, Z., Breck, J., Boroujeni, A.F., Reps, T.W.: Compositional recurrence analysis revisited. In: Programming Language Design and Implementation (PLDI), pp. 248–262 (2017). https://doi.org/10.1145/3062341.3062373

  26. Kincaid, Z., Cyphert, J., Breck, J., Reps, T.W.: Non-linear reasoning for invariant synthesis. Proc. ACM Program. Lang. 2(POPL), 54:1–54:33 (2018). https://doi.org/10.1145/3158142

  27. Kincaid, Z., Breck, J., Cyphert, J., Reps, T.W.: Closed forms for numerical loops. Proc. ACM Program. Lang. 3(POPL), 55:1–55:29 (2019). https://doi.org/10.1145/3290368

  28. Lu, T., Černý, P., Chang, B.-Y.E., Trivedi, A.: Type-directed bounding of collections in reactive programs. In: Enea, C., Piskac, R. (eds.) VMCAI 2019. LNCS, vol. 11388, pp. 275–296. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11245-5_13

    Chapter  Google Scholar 

  29. Lu, T., Chang, B.-Y.E., Trivedi, A.: Selectively-amortized resource bounding (extended version) (2021). https://arxiv.org/abs/2108.08263

  30. Lu, T., Chang, B.-Y.E., Trivedi, A.: Selectively-amortized resource bounding (artifact) (2021). https://zenodo.org/record/5140586

  31. Manna, Z., Pnueli, A.: Completing the temporal picture. Theor. Comput. Sci. 83(1), 91–130 (1991). https://doi.org/10.1016/0304-3975(91)90041-Y

    Article  MATH  Google Scholar 

  32. Sharma, R., Dillig, I., Dillig, T., Aiken, A.: Simplifying loop invariant generation using splitter predicates. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 703–719. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_57

    Chapter  Google Scholar 

  33. Sinn, M., Zuleger, F., Veith, H.: A simple and scalable static analysis for bound analysis and amortized complexity analysis. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 745–761. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_50

    Chapter  Google Scholar 

  34. Sinn, M., Zuleger, F., Veith, H.: Difference constraints: an adequate abstraction for complexity analysis of imperative programs. In: Formal Methods in Computer Aided Design (FMCAD), pp. 144–151 (2015)

    Google Scholar 

  35. Sinn, M., Zuleger, F., Veith, H.: Complexity and resource bound analysis of imperative programs using difference constraints. J. Autom. Reason. 59(1), 3–45 (2017). https://doi.org/10.1007/s10817-016-9402-4

    Article  MathSciNet  MATH  Google Scholar 

  36. Tarjan, R.E.: Amortized computational complexity. SIAM J. Alg. Discrete Methods 6(2), 306–318 (1985)

    Article  MathSciNet  Google Scholar 

  37. Weiser, M.: Program slicing. IEEE Trans. Software Eng. 10(4), 352–357 (1984). https://doi.org/10.1109/TSE.1984.5010248

    Article  MATH  Google Scholar 

  38. Wilhelm, R., et al.: The worst-case execution-time problem - overview of methods and survey of tools. ACM Trans. Embed. Comput. Syst. 7(3):36:1–36:53 (2008). https://doi.org/10.1145/1347375.1347389

  39. Zuleger, F., Gulwani, S., Sinn, M., Veith, H.: Bound analysis of imperative programs with the size-change abstraction. In: Yahav, E. (ed.) SAS 2011. LNCS, vol. 6887, pp. 280–297. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23702-7_22

    Chapter  Google Scholar 

Download references

We thank Pavol Černý for his valuable contributions in the early stages of this research. We also thank the anonymous reviewers and members of the CUPLV lab for their helpful reviews and suggestions. This research was supported in part by the Defense Advanced Research Projects Agency under grant FA8750-15-2-0096, and also by the National Science Foundation under grant CCF-2008369.

Author information

Authors and Affiliations

  1. University of Colorado Boulder, Boulder, USA

    Tianhan Lu, Bor-Yuh Evan Chang & Ashutosh Trivedi

Authors
  1. Tianhan Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bor-Yuh Evan Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ashutosh Trivedi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Tianhan Lu , Bor-Yuh Evan Chang or Ashutosh Trivedi .

Editor information

Editors and Affiliations

  1. Informal Systems, Inria, ENS, DI - Ecole Normale Supérieure, Paris, France

    Cezara Drăgoi

  2. Microsoft Corporation, Redmond, WA, USA

    Suvam Mukherjee

  3. Nokia Bell Labs, Murray Hill, NJ, USA

    Kedar Namjoshi

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

Lu, T., Chang, BY.E., Trivedi, A. (2021). Selectively-Amortized Resource Bounding. In: Drăgoi, C., Mukherjee, S., Namjoshi, K. (eds) Static Analysis. SAS 2021. Lecture Notes in Computer Science(), vol 12913. Springer, Cham. https://doi.org/10.1007/978-3-030-88806-0_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-88806-0_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-88805-3

  • Online ISBN: 978-3-030-88806-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation