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Architecture-Guided Test Resource Allocation via Logic

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Part of the Lecture Notes in Computer Science book series (LNPSE,volume 12740)


We introduce a new logic named Quantitative Confidence Logic (QCL) that quantifies the level of confidence one has in the conclusion of a proof. By translating a fault tree representing a system’s architecture to a proof, we show how to use QCL to give a solution to the test resource allocation problem that takes the given architecture into account. We implemented a tool called Astrahl and compared our results to other testing resource allocation strategies.


  • Reliability
  • Test resources allocation
  • Logic

The authors are supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603).

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  • DOI: 10.1007/978-3-030-79379-1_2
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  1. 1.

    Here, we mean fuzzy logics interpreted in [0, 1].

  2. 2.

    The code and experimental data are publicly available on

  3. 3.

    Functional optimisation may not be as efficient in larger dimensions, but even a naive estimate should give a better result than completely ignoring system structure.

  4. 4.

    It would equally be possible to assume a test costs one resource and scale the budget.

  5. 5.

    We used smaller confidence so that components will usually contain faults.


  1. Bergmann, M.: An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems. Cambridge University Press, Cambridge (2008).

    CrossRef  MATH  Google Scholar 

  2. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Cambridge (2014)

    MATH  Google Scholar 

  3. Carrozza, G., Pietrantuono, R., Russo, S.: Dynamic test planning: a study in an industrial context. Int. J. Softw. Tools Technol. Transfer 16(5), 593–607 (2014).

    CrossRef  Google Scholar 

  4. Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 36, 325–339 (1967).

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Ericson, C.A.: Fault tree analysis. In: System Safety Conference, Orlando, Florida, vol. 1, pp. 1–9 (1999)

    Google Scholar 

  6. Esteva, F., Godo, L.: Monoidal T-norm based logic: towards a logic for left-continuous T-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001).

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Goel, A.L., Okumoto, K.: Time-dependent error-detection rate model for software reliability and other performance measures. IEEE Trans. Reliab. 28(3), 206–211 (1979).

    CrossRef  MATH  Google Scholar 

  8. Hájek, P.: Metamathematics of Fuzzy Logic, vol. 4. Springer Science & Business Media, Dordrecht (2013).

    CrossRef  MATH  Google Scholar 

  9. Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001).

    CrossRef  Google Scholar 

  10. Huang, C.Y., Lo, J.H.: Optimal resource allocation for cost and reliability of modular software systems in the testing phase. J. Syst. Softw. 79(5), 653–664 (2006).

    MathSciNet  CrossRef  Google Scholar 

  11. Kamei, Y., Shihab, E.: Defect Prediction: Accomplishments and Future Challenges. In: 2016 IEEE 23rd International Conference on Software Analysis, Evolution, and Reengineering (SANER), vol. 5, pp. 33–45. IEEE (2016).

  12. Pietrantuono, R.: On the testing resource allocation problem: research trends and perspectives. J. Syst. Softw. 161, 110462 (2020).

    CrossRef  Google Scholar 

  13. Pietrantuono, R., Russo, S., Trivedi, K.S.: Software reliability and testing time allocation: an architecture-based approach. IEEE Trans. Softw. Eng. 36(3), 323–337 (2010).

    CrossRef  Google Scholar 

  14. Sallak, M., Schön, W., Aguirre, F.: Reliability assessment for multi-state systems under uncertainties based on the Dempster-Shafer theory. IIE Trans. 45(9), 995–1007 (2013).

    CrossRef  Google Scholar 

  15. Shafer, G.: A Mathematical Theory of Evidence, vol. 42. Princeton University Press, Princeton (1976).

    CrossRef  MATH  Google Scholar 

  16. Van Laarhoven, P.J., Aarts, E.H.: Simulated annealing. In: Simulated Annealing: Theory and Applications. MAIA, vol. 37, pp. 7–15. Springer, Dordrecht (1987).

    CrossRef  MATH  Google Scholar 

  17. Vesely, W.E., Goldberg, F.F., Roberts, N.H., Haasl, D.F.: Fault Tree Handbook. Technical Report, Nuclear Regulatory Commission Washington DC (1981)

    Google Scholar 

  18. Yamada, S., Osaki, S.: Software reliability growth modeling: models and applications. IEEE Trans. Softw. Eng. SE-11(12), 1431–1437 (1985).

  19. Zhang, G., Su, Z., Li, M., Yue, F., Jiang, J., Yao, X.: Constraint handling in NSGA-II for solving optimal testing resource allocation problems. IEEE Trans. Reliab. 66(4), 1193–1212 (2017).

    CrossRef  Google Scholar 

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Eberhart, C. et al. (2021). Architecture-Guided Test Resource Allocation via Logic. In: Loulergue, F., Wotawa, F. (eds) Tests and Proofs. TAP 2021. Lecture Notes in Computer Science(), vol 12740. Springer, Cham.

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  • Print ISBN: 978-3-030-79378-4

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