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Verified analysis of random binary tree structures. (English) Zbl 07268890
Summary: This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.
MSC:
68V15 Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.)
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[1] Aslam, J.A.: A simple bound on the expected height of a randomly built binary search tree. Technical Report TR2001-387, Dartmouth College, Hanover, NH (2001). Abstract and paper lost
[2] Audebaud, P.; Paulin-Mohring, C., Proofs of randomized algorithms in Coq, Sci. Comput. Program., 74, 8, 568-589 (2009) · Zbl 1178.68667
[3] Barthe, G., Grégoire, B., Béguelin, S.Z.: Formal certification of code-based cryptographic proofs. In: Proceedings of the 36th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2009, pp. 90-101 (2009). 10.1145/1480881.1480894 · Zbl 1315.68081
[4] Basin, D.A., Lochbihler, A., Sefidgar, S.R.: CryptHOL: Game-based proofs in higher-order logic. Cryptology ePrint Archive, Report 2017/753 (2017). 10.1007/978-3-662-49498-1_20. https://eprint.iacr.org/2017/753 · Zbl 1455.94121
[5] Chatterjee, K., Fu, H., Murhekar, A.: Automated recurrence analysis for almost-linear expected-runtime bounds. In: Computer Aided Verification: 29th International Conference, CAV 2017, pp. 118-139 (2017). 10.1007/978-3-319-63387-9_6
[6] Cichoń, J.: Quick Sort: average complexity. http://cs.pwr.edu.pl/cichon/Math/QSortAvg.pdf Accessed 13 Mar 2017
[7] Cormen, TH; Stein, C.; Rivest, RL; Leiserson, CE, Introduction to Algorithms (2001), New York: McGraw-Hill Higher Education, New York
[8] Eberl, M.: Expected shape of random binary search trees. Archive of Formal Proofs (2017). http://isa-afp.org/entries/Random_BSTs.html, Formal proof development
[9] Eberl, M.: The number of comparisons in QuickSort. Archive of Formal Proofs (2017). http://isa-afp.org/entries/Quick_Sort_Cost.html, Formal proof development
[10] Eberl, M.: Randomised binary search trees. Archive of Formal Proofs (2018). http://isa-afp.org/entries/Randomised_BSTs.html, Formal proof development
[11] Eberl, M., Haslbeck, M.W., Nipkow, T.: Verified analysis of random trees. In: Proceedings of the 9th International Conference on Interactive Theorem Proving (2018). 10.1007/978-3-319-94821-8 · Zbl 06946981
[12] Eberl, M., Hölzl, J., Nipkow, T.: A verified compiler for probability density functions. In: J. Vitek (ed.) Proceedings of the 24th European Symposium on Programming, pp. 80-104. Springer, Berlin Heidelberg (2015). 10.1007/978-3-662-46669-8_4 · Zbl 1335.68037
[13] Flajolet, P., Salvy, B., Zimmermann, P.: Lambda - Upsilon - Omega: An assistant algorithms analyzer. In: 6th International Conference Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-6, Rome, Italy, July 4-8, 1988, Proceedings, pp. 201-212 (1988). 10.1007/3-540-51083-4_60 · Zbl 0681.68064
[14] Giry, M.: A categorical approach to probability theory. In: Categorical Aspects of Topology and Analysis, Lecture Notes in Mathematics, vol. 915, pp. 68-85. Springer Berlin (1982). 10.1007/BFb0092872 · Zbl 0486.60034
[15] Gouëzel, S.: Ergodic theory. Archive of Formal Proofs (2015). http://isa-afp.org/entries/Ergodic_Theory.html, Formal proof development
[16] Haslbeck, M., Eberl, M., Nipkow, T.: Treaps. Archive of Formal Proofs (2018). http://isa-afp.org/entries/Treaps.html, Formal proof development · Zbl 06946981
[17] Hoare, CAR, Quicksort, Comput. J., 5, 1, 10 (1962) · Zbl 0108.13601
[18] Hölzl, J.: Formalising semantics for expected running time of probabilistic programs. In: J.C. Blanchette, S. Merz (eds.) Interactive Theorem Proving (ITP 2016), pp. 475-482. Springer, Berlin (2016). 10.1007/978-3-319-43144-4_30 · Zbl 06644762
[19] Hölzl, J., Markov chains and Markov decision processes in Isabelle/HOL, J. Autom. Reason. (2017) · Zbl 1425.68375
[20] Hölzl, J., Heller, A.: Three chapters of measure theory in Isabelle/HOL. In: Interactive Theorem Proving—Second International Conference, ITP 2011, Berg en Dal, The Netherlands, August 22-25, 2011. Proceedings, pp. 135-151 (2011). 10.1007/978-3-642-22863-6_12 · Zbl 1342.68287
[21] Hurd, J.: Formal verification of probabilistic algorithms. Ph.D. thesis, University of Cambridge (2002) · Zbl 1013.68193
[22] Kaminski, B.L., Katoen, J.P., Matheja, C., Olmedo, F.: Weakest precondition reasoning for expected run—times of probabilistic programs. In: Proceedings of the 25th European Symposium on Programming Languages and Systems: volume 9632, pp. 364-389. Springer-Verlag New York, Inc., New York, NY, USA (2016). 10.1007/978-3-662-49498-1_15 · Zbl 1335.68058
[23] Karp, RM, Probabilistic recurrence relations, J. ACM, 41, 6, 1136-1150 (1994) · Zbl 0830.68046
[24] Knuth, DE, The Art of Computer Programming, Volume 3: Sorting and Searching (1998), Redwood City: Addison Wesley Longman Publishing Co., Inc., Redwood City
[25] Kwiatkowska, MZ; Norman, G.; Parker, D., Quantitative analysis with the probabilistic model checker PRISM, Electr. Notes Theor. Comput. Sci., 153, 2, 5-31 (2006)
[26] Lochbihler, A.: Probabilistic functions and cryptographic oracles in higher order logic. In: P. Thiemann (ed.) Programming Languages and Systems (ESOP 2016), LNCS, vol. 9632, pp. 503-531. Springer (2016). 10.1007/978-3-662-49498-1_20 · Zbl 1335.68033
[27] Martínez, C.; Roura, S., Randomized binary search trees, J. ACM, 45, 288 (1997) · Zbl 0904.68074
[28] Nipkow, T.; Urban, C.; Zhang, X., Amortized complexity verified, Interactive Theorem Proving (ITP 2015). LNCS, 310-324 (2015), Berlin: Springer, Berlin · Zbl 06481872
[29] Nipkow, T.; Blanchette, J.; Merz, S., Automatic functional correctness proofs for functional search trees, Interactive Theorem Proving (ITP 2016), LNCS, 307-322 (2016), Berlin: Springer, Berlin · Zbl 06644751
[30] Nipkow, T.: Verified root-balanced trees. In: Chang, B.Y.E. (ed.) Asian Symposium on Programming Languages and Systems, APLAS 2017, LNCS, vol. 10695, pp. 255-272. Springer, Berlin (2017)
[31] Nipkow, T.; Klein, G., Concrete Semantics with Isabelle/HOL (2014), Berlin: Springer, Berlin · Zbl 1410.68004
[32] Nipkow, T.; Paulson, L.; Wenzel, M., Isabelle/HOL: A Proof Assistant for Higher-Order Logic, LNCS (2002), Berlin: Springer, Berlin · Zbl 0994.68131
[33] Ottmann, T., Widmayer, P.: Algorithmen und Datenstrukturen, 5. Auflage. Spektrum Akademischer Verlag (2012). 10.1007/978-3-8274-2804-2
[34] Petcher, A., Morrisett, G.: The foundational cryptography framework. In: R. Focardi, A.C. Myers (eds.) Principles of Security and Trust: 4th International Conference, POST 2015, Lecture Notes in Computer Science, vol. 9036, pp. 53-72. Springer (2015). 10.1007/978-3-662-46666-7_4
[35] Reed, B., The height of a random binary search tree, J. ACM, 50, 3, 306-332 (2003) · Zbl 1325.68076
[36] Schneider, J., Eberl, M., Lochbihler, A.: Monad normalisation. Archive of Formal Proofs (2017). http://isa-afp.org/entries/Monad_Normalisation.html, Formal proof development
[37] Sedgewick, R., The analysis of Quicksort programs, Acta Inf., 7, 4, 327-355 (1977) · Zbl 0325.68016
[38] Seidel, R.; Aragon, CR, Randomized search trees, Algorithmica, 16, 4, 464-497 (1996) · Zbl 0857.68030
[39] Stüwe, D., Eberl, M.: Probabilistic primality testing. Archive of Formal Proofs (2019). http://isa-afp.org/entries/Probabilistic_Prime_Tests.html, Formal proof development
[40] Tassarotti, J.; Harper, R.; Avigad, J.; Mahboubi, A., Verified tail bounds for randomized programs, Interactive Theorem Proving (2018), Cham: Springer, Cham · Zbl 06947002
[41] Vuillemin, J., A unifying look at data structures, Commun. ACM, 23, 4, 229-239 (1980) · Zbl 0434.68047
[42] van der Weegen, E.; McKinna, J., A Machine-Checked Proof of the Average-Case Complexity of Quicksort in Coq, 256-271 (2009), Berlin: Springer, Berlin · Zbl 1246.68201
[43] Wenzel, M.: Isabelle/Isar: a versatile environment for human-readable formal proof documents. Ph.D. thesis, Institut für Informatik, Technische Universität München (2002). https://mediatum.ub.tum.de/node?id=601724
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