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Non-polynomial Worst-Case Analysis of Recursive Programs

  • Krishnendu Chatterjee
  • Hongfei FuEmail author
  • Amir Kafshdar Goharshady
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10427)

Abstract

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of non-recursive programs. First, we apply ranking functions to recursion, resulting in measure functions, and show that they provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in non-polynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas’ Lemma, and Handelman’s Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form \(\mathcal {O}(n \log n)\) as well as \(\mathcal {O}(n^r)\) where r is not an integer. We present experimental results to demonstrate that our approach can efficiently obtain worst-case bounds of classical recursive algorithms such as Merge-Sort, Closest-Pair, Karatsuba’s algorithm and Strassen’s algorithm.

Keywords

Recursive Programs Worst-case Bound Triple Constraint FLO11 Expression Template Variables 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank all reviewers for valuable comments. The research is partially supported by Vienna Science and Technology Fund (WWTF) ICT15-003, Austrian Science Fund (FWF) NFN Grant No. S11407-N23 (RiSE/SHiNE), ERC Start grant (279307: Graph Games), the Natural Science Foundation of China (NSFC) under Grant No. 61532019 and the CDZ project CAP (GZ 1023).

References

  1. 1.
    Albert, E., Arenas, P., Genaim, S., Gómez-Zamalloa, M., Puebla, G., Ramírez-Deantes, D.V., Román-Díez, G., Zanardini, D.: Termination and cost analysis with COSTA and its user interfaces. Electr. Notes Theor. Comput. Sci. 258(1), 109–121 (2009)CrossRefGoogle Scholar
  2. 2.
    Albert, E., Arenas, P., Genaim, S., Puebla, G.: Automatic inference of upper bounds for recurrence relations in cost analysis. In: Alpuente, M., Vidal, G. (eds.) SAS 2008. LNCS, vol. 5079, pp. 221–237. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-69166-2_15 CrossRefGoogle Scholar
  3. 3.
    Albert, E., Arenas, P., Genaim, S., Puebla, G., Zanardini, D.: Cost analysis of Java bytecode. In: Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 157–172. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-71316-6_12 CrossRefGoogle Scholar
  4. 4.
    Alias, C., Darte, A., Feautrier, P., Gonnord, L.: Multi-dimensional rankings, program termination, and complexity bounds of flowchart programs. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 117–133. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15769-1_8 CrossRefGoogle Scholar
  5. 5.
    Alur, R., Chaudhuri, S.: Temporal reasoning for procedural programs. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 45–60. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-11319-2_7 CrossRefGoogle Scholar
  6. 6.
    Bartle, R.G., Sherbert, D.R.: Introduction to Real Analysis, 4th edn. Wiley, Hoboken (2011)zbMATHGoogle Scholar
  7. 7.
    Bodík, R., Majumdar, R. (eds.): POPL. ACM, New York (2016)zbMATHGoogle Scholar
  8. 8.
    Bournez, O., Garnier, F.: Proving positive almost-sure termination. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 323–337. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-32033-3_24 CrossRefGoogle Scholar
  9. 9.
    Bradley, A.R., Manna, Z., Sipma, H.B.: Linear ranking with reachability. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 491–504. Springer, Heidelberg (2005). doi: 10.1007/11513988_48 CrossRefGoogle Scholar
  10. 10.
    Brockschmidt, M., Emmes, F., Falke, S., Fuhs, C., Giesl, J.: Analyzing runtime and size complexity of integer programs. ACM Trans. Program. Lang. Syst. 38(4), 13:1–13:50 (2016)CrossRefGoogle Scholar
  11. 11.
    Castagna, G., Gordon, A.D. (eds.): POPL. ACM, New York (2017)Google Scholar
  12. 12.
    Chakarov, A., Sankaranarayanan, S.: Probabilistic program analysis with martingales. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 511–526. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_34 CrossRefGoogle Scholar
  13. 13.
    Chatterjee, K., Fu, H.: Termination of nondeterministic recursive probabilistic programs. CoRR abs/1701.02944 (2017). http://arxiv.org/abs/1701.02944
  14. 14.
    Chatterjee, K., Fu, H., Goharshady, A.K.: Termination analysis of probabilistic programs through Positivstellensatzs. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 3–22. Springer, Cham (2016). doi: 10.1007/978-3-319-41528-4_1 Google Scholar
  15. 15.
    Chatterjee, K., Fu, H., Goharshady, A.K.: Non-polynomial worst-case analysis of recursive programs. CoRR abs/1705.00317 (2017). https://arxiv.org/abs/1705.00317
  16. 16.
    Chatterjee, K., Fu, H., Novotný, P., Hasheminezhad, R.: Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs. In: Bodík and Majumdar [7], pp. 327–342Google Scholar
  17. 17.
    Chatterjee, K., Novotný, P., Žikelić, Đ.: Stochastic invariants for probabilistic termination. In: Castagna and Gordon [11], pp. 145–160Google Scholar
  18. 18.
    Chin, W., Khoo, S.: Calculating sized types. Higher-Order Symbolic Comput. 14(2–3), 261–300 (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    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). doi: 10.1007/978-3-540-45069-6_39 CrossRefGoogle Scholar
  20. 20.
    Colón, M.A., Sipma, H.B.: Synthesis of linear ranking functions. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 67–81. Springer, Heidelberg (2001). doi: 10.1007/3-540-45319-9_6 CrossRefGoogle Scholar
  21. 21.
    Cook, B., Podelski, A., Rybalchenko, A.: Termination proofs for systems code. In: Schwartzbach, M.I., Ball, T. (eds.) PLDI, pp. 415–426. ACM (2006)Google Scholar
  22. 22.
    Cook, B., Podelski, A., Rybalchenko, A.: Summarization for termination: no return!. Form. Methods Syst. Des. 35(3), 369–387 (2009)CrossRefzbMATHGoogle Scholar
  23. 23.
    Cook, B., See, A., Zuleger, F.: Ramsey vs. lexicographic termination proving. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 47–61. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-36742-7_4 CrossRefGoogle Scholar
  24. 24.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)zbMATHGoogle Scholar
  25. 25.
    Cousot, P.: Proving program invariance and termination by parametric abstraction, lagrangian relaxation and semidefinite programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 1–24. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30579-8_1 CrossRefGoogle Scholar
  26. 26.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Graham, R.M., Harrison, M.A., Sethi, R. (eds.) POPL, pp. 238–252. ACM (1977)Google Scholar
  27. 27.
    Cousot, P., Cousot, R.: An abstract interpretation framework for termination. In: Field, J., Hicks, M. (eds.) POPL, pp. 245–258. ACM (2012)Google Scholar
  28. 28.
    Farkas, J.: A fourier-féle mechanikai elv alkalmazásai (Hungarian). Mathematikaiés Természettudományi Értesitö 12, 457–472 (1894)Google Scholar
  29. 29.
    Fioriti, L.M.F., Hermanns, H.: Probabilistic termination: soundness, completeness, and compositionality. In: Rajamani, S.K., Walker, D. (eds.) POPL, pp. 489–501. ACM (2015)Google Scholar
  30. 30.
    Flajolet, P., Salvy, B., Zimmermann, P.: Automatic average-case analysis of algorithm. Theor. Comput. Sci. 79(1), 37–109 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Floyd, R.W.: Assigning meanings to programs. Math. Aspects Comput. Sci. 19, 19–33 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Gimenez, S., Moser, G.: The complexity of interaction. In: Bodík and Majumdar [7], pp. 243–255Google Scholar
  33. 33.
    Gödel, K., Kleene, S.C., Rosser, J.B.: On undecidable propositions of formal mathematical systems. Institute for Advanced Study Princeton, NJ (1934)Google Scholar
  34. 34.
    Grobauer, B.: Cost recurrences for DML programs. In: Pierce, B.C. (ed.) ICFP, pp. 253–264. ACM (2001)Google Scholar
  35. 35.
    Gulavani, B.S., Gulwani, S.: A numerical abstract domain based on expression abstraction and max operator with application in timing analysis. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 370–384. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-70545-1_35 CrossRefGoogle Scholar
  36. 36.
    Gulwani, S.: SPEED: symbolic complexity bound analysis. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 51–62. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02658-4_7 CrossRefGoogle Scholar
  37. 37.
    Gulwani, S., Mehra, K.K., Chilimbi, T.M.: SPEED: precise and efficient static estimation of program computational complexity. In: Shao, Z., Pierce, B.C. (eds.) POPL, pp. 127–139. ACM (2009)Google Scholar
  38. 38.
    Handelman, D.: Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132, 35–62 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hesselink, W.H.: Proof rules for recursive procedures. Formal Asp. Comput. 5(6), 554–570 (1993)CrossRefzbMATHGoogle Scholar
  40. 40.
    Hoffmann, J., Aehlig, K., Hofmann, M.: Multivariate amortized resource analysis. ACM Trans. Program. Lang. Syst. 34(3), 14 (2012)CrossRefzbMATHGoogle Scholar
  41. 41.
    Hoffmann, J., Aehlig, K., Hofmann, M.: Resource aware ML. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 781–786. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31424-7_64 CrossRefGoogle Scholar
  42. 42.
    Hoffmann, J., Hofmann, M.: Amortized resource analysis with polymorphic recursion and partial big-step operational semantics. In: Ueda, K. (ed.) APLAS 2010. LNCS, vol. 6461, pp. 172–187. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-17164-2_13 CrossRefGoogle Scholar
  43. 43.
    Hoffmann, J., Hofmann, M.: Amortized resource analysis with polynomial potential. In: Gordon, A.D. (ed.) ESOP 2010. LNCS, vol. 6012, pp. 287–306. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-11957-6_16 CrossRefGoogle Scholar
  44. 44.
    Hofmann, M., Jost, S.: Static prediction of heap space usage for first-order functional programs. In: Aiken, A., Morrisett, G. (eds.) POPL, pp. 185–197. ACM (2003)Google Scholar
  45. 45.
    Hofmann, M., Jost, S.: Type-based amortised heap-space analysis. In: Sestoft, P. (ed.) ESOP 2006. LNCS, vol. 3924, pp. 22–37. Springer, Heidelberg (2006). doi: 10.1007/11693024_3 CrossRefGoogle Scholar
  46. 46.
    Hofmann, M., Rodriguez, D.: Efficient type-checking for amortised heap-space analysis. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 317–331. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04027-6_24 CrossRefGoogle Scholar
  47. 47.
    Hughes, J., Pareto, L.: Recursion and dynamic data-structures in bounded space: Towards embedded ML programming. In: Rémi, D., Lee, P. (eds.) ICFP. pp. 70–81. ACM (1999)Google Scholar
  48. 48.
    Hughes, J., Pareto, L., Sabry, A.: Proving the correctness of reactive systems using sized types. In: Boehm, H., Jr., G.L.S. (eds.) POPL. pp. 410–423. ACM Press (1996)Google Scholar
  49. 49.
    Jones, C.: Probabilistic non-determinism. Ph.D. thesis, The University of Edinburgh (1989)Google Scholar
  50. 50.
    Jost, S., Hammond, K., Loidl, H., Hofmann, M.: Static determination of quantitative resource usage for higher-order programs. In: Hermenegildo, M.V., Palsberg, J. (eds.) POPL, pp. 223–236. ACM (2010)Google Scholar
  51. 51.
    Jost, S., Loidl, H.-W., Hammond, K., Scaife, N., Hofmann, M.: “Carbon Credits” for resource-bounded computations using amortised analysis. In: Cavalcanti, A., Dams, D.R. (eds.) FM 2009. LNCS, vol. 5850, pp. 354–369. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-05089-3_23 CrossRefGoogle Scholar
  52. 52.
    Knuth, D.E.: The Art of Computer Programming, vols. I–III. Addison-Wesley, Reading (1973)Google Scholar
  53. 53.
    Kuwahara, T., Terauchi, T., Unno, H., Kobayashi, N.: Automatic termination verification for higher-order functional programs. In: Shao, Z. (ed.) ESOP 2014. LNCS, vol. 8410, pp. 392–411. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54833-8_21 CrossRefGoogle Scholar
  54. 54.
    Lee, C.S.: Ranking functions for size-change termination. ACM Trans. Program. Lang. Syst. 31(3), 10:1–10:42 (2009)CrossRefGoogle Scholar
  55. 55.
    Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termination. In: Hankin, C., Schmidt, D. (eds.) POPL, pp. 81–92. ACM (2001)Google Scholar
  56. 56.
    lp_solve 5.5.2.3 (2016). http://lpsolve.sourceforge.net/5.5/
  57. 57.
    Olmedo, F., Kaminski, B.L., Katoen, J., Matheja, C.: Reasoning about recursive probabilistic programs. In: Grohe, M., Koskinen, E., Shankar, N. (eds.) LICS, pp. 672–681. ACM (2016)Google Scholar
  58. 58.
    Podelski, A., Rybalchenko, A.: A complete method for the synthesis of linear ranking functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 239–251. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24622-0_20 CrossRefGoogle Scholar
  59. 59.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken (1999)zbMATHGoogle Scholar
  60. 60.
    Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  61. 61.
    Shen, L., Wu, M., Yang, Z., Zeng, Z.: Generating exact nonlinear ranking functions by symbolic-numeric hybrid method. J. Syst. Sci. Complex. 26(2), 291–301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Shkaravska, O., Kesteren, R., Eekelen, M.: Polynomial size analysis of first-order functions. In: Rocca, S.R. (ed.) TLCA 2007. LNCS, vol. 4583, pp. 351–365. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73228-0_25 CrossRefGoogle Scholar
  63. 63.
    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). doi: 10.1007/978-3-319-08867-9_50 Google Scholar
  64. 64.
    Sohn, K., Gelder, A.V.: Termination detection in logic programs using argument sizes. In: Rosenkrantz, D.J. (ed.) PODS, pp. 216–226. ACM Press (1991)Google Scholar
  65. 65.
    Srikanth, A., Sahin, B., Harris, W.R.: Complexity verification using guided theorem enumeration. In: Castagna and Gordon [11], pp. 639–652Google Scholar
  66. 66.
    Urban, C.: The abstract domain of segmented ranking functions. In: Logozzo, F., Fähndrich, M. (eds.) SAS 2013. LNCS, vol. 7935, pp. 43–62. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38856-9_5 CrossRefGoogle Scholar
  67. 67.
    Wilhelm, R., et al.: The worst-case execution-time problem - overview of methods and survey of tools. ACM Trans. Embed. Comput. Syst. 7(3), 1–53 (2008)CrossRefGoogle Scholar
  68. 68.
    Yang, L., Zhou, C., Zhan, N., Xia, B.: Recent advances in program verification through computer algebra. Front. Comput. Sci. China 4(1), 1–16 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Hongfei Fu
    • 2
    Email author
  • Amir Kafshdar Goharshady
    • 1
  1. 1.IST AustriaKlosterneuburgAustria
  2. 2.State Key Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingPeople’s Republic of China

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