498
E. Bartocci et al.
References
1. Barthe, G., Espitau, T., Fioriti, L.M.F., Hsu, J.: Synthesizing Probabilistic Invariants via
Doob’s Decomposition. In: CAV. LNCS, vol. 9779, pp. 43–61. Springer (2016)
2. Bartocci, E., Kov´ acs, L., Stankovic, M.: Automatic generation of moment-based invariants
for prob-solvable loops. In: Proc. of ATVA 2019: the 17th International Symposium on Automated Technology for Verification and Analysis. LNCS, vol. 11781, pp. 255–276 (2019)
3. Chakarov, A., Sankaranarayanan, S.: Expectation Invariants for Probabilistic Program Loops
as Fixed Points. In: SAS. LNCS, vol. 8723, pp. 85–100 (2014)
4. Gehr, T., Misailovic, S., Vechev, M.T.: PSI: Exact Symbolic Inference for Probabilistic Programs. In: CAV. LNCS, vol. 9779, pp. 62–83 (2016)
5. Ghahramani, Z.: Probabilistic Machine Learning and Artificial Intelligence. Nature
521(7553), 452–459 (2015)
6. Humenberger, A., Jaroschek, M., Kov´ acs, L.: Aligator.jl - A Julia Package for Loop Invariant
Generation. In: CICM. LNCS, vol. 11006, pp. 111–117 (2018)
7. Katoen, J.P., McIver, A.K., Meinicke, L.A., Morgan, C.C.: Linear-Invariant Generation for
Probabilistic Programs: Automated Support for Proof-Based Methods. In: SAS. LNCS,
vol. 6337, pp. 390–406 (2010)
8. Kauers, M., Paule, P.: The Concrete Tetrahedron - Symbolic Sums, Recurrence Equations,
Generating Functions, Asymptotic Estimates. Texts & Monographs in Symbolic Computation, Springer (2011)
9. Kura, S., Urabe, N., Hasuo, I.: Tail Probabilities for Randomized Program Runtimes via
Martingales for Higher Moments. In: TACAS. LNCS, vol. 11428, pp. 135–153 (2019)
10. McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science, Springer (2005)
Open Access This chapter is licensed under the terms of the Creative Commons
Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons license and your intended
use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain
permission directly from the copyright holder.
E. Bartocci et al.
References
1. Barthe, G., Espitau, T., Fioriti, L.M.F., Hsu, J.: Synthesizing Probabilistic Invariants via
Doob’s Decomposition. In: CAV. LNCS, vol. 9779, pp. 43–61. Springer (2016)
2. Bartocci, E., Kov´ acs, L., Stankovic, M.: Automatic generation of moment-based invariants
for prob-solvable loops. In: Proc. of ATVA 2019: the 17th International Symposium on Automated Technology for Verification and Analysis. LNCS, vol. 11781, pp. 255–276 (2019)
3. Chakarov, A., Sankaranarayanan, S.: Expectation Invariants for Probabilistic Program Loops
as Fixed Points. In: SAS. LNCS, vol. 8723, pp. 85–100 (2014)
4. Gehr, T., Misailovic, S., Vechev, M.T.: PSI: Exact Symbolic Inference for Probabilistic Programs. In: CAV. LNCS, vol. 9779, pp. 62–83 (2016)
5. Ghahramani, Z.: Probabilistic Machine Learning and Artificial Intelligence. Nature
521(7553), 452–459 (2015)
6. Humenberger, A., Jaroschek, M., Kov´ acs, L.: Aligator.jl - A Julia Package for Loop Invariant
Generation. In: CICM. LNCS, vol. 11006, pp. 111–117 (2018)
7. Katoen, J.P., McIver, A.K., Meinicke, L.A., Morgan, C.C.: Linear-Invariant Generation for
Probabilistic Programs: Automated Support for Proof-Based Methods. In: SAS. LNCS,
vol. 6337, pp. 390–406 (2010)
8. Kauers, M., Paule, P.: The Concrete Tetrahedron - Symbolic Sums, Recurrence Equations,
Generating Functions, Asymptotic Estimates. Texts & Monographs in Symbolic Computation, Springer (2011)
9. Kura, S., Urabe, N., Hasuo, I.: Tail Probabilities for Randomized Program Runtimes via
Martingales for Higher Moments. In: TACAS. LNCS, vol. 11428, pp. 135–153 (2019)
10. McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science, Springer (2005)
Open Access This chapter is licensed under the terms of the Creative Commons
Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons license and your intended
use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain
permission directly from the copyright holder.