Semiring-Weighted Automata and Recursive Path Counting for Multi-State Reliability in Discrete Infrastructures

Year : 2026 | Volume : 13 | Issue : 02 | Page : 30 36
By

S. Vairachilai,

Markala Karthik,

Mohammed Almakki,

Mohammed El Khider,

  1. Associate Professor, School of Computer Science and Artificial Intelligence, SR University, Warangal, Telangana, India
  2. Assistant Professor, Department of Electrical and Electronics Engineering, SR University, Warangal, Telangana, India
  3. Assistant Professor, School of Engineering, Architecture and Interior Design, Amity University Dubai, Dubai, United Arab Emirates
  4. Assistant Professor, Department of General Undergraduate Curriculum Requirements, University of Dubai, Dubai, United Arab Emirates

Abstract

Multi-state infrastructures such as communication backbones, microgrids, warehouse routing systems, and sensor-actuator pipelines evolve through discrete event sequences rather than through a single binary “working/failed” transition. This paper develops a semiring-weighted automata framework for reliability analysis in which state changes, repair actions, and degraded operating modes are represented by weighted transitions on a finite automaton. A path valuation is defined over an additively idempotent reliability semiring and extended to a hesitation-aware fuzzy semiring so that uncertainty in maintenance quality and observational ambiguity can be included without abandoning discrete rigor. We derive recursive path-counting identities, a transfer-matrix representation, and a closed-form generating function for horizon-limited mission reliability. A min-plus reliability distance is introduced to characterize the least costly restoration sequence, while a max-times acceptance functional captures the best feasible service quality over all admissible event words. We further prove monotonicity, subadditivity, and convergence of a truncated Kleene-star approximation under a contractive spectral condition on the transition operator. Numerical experiments on a five-zone distribution network show that the proposed method separates availability, recoverability, and path diversity more effectively than crisp Markov aggregation.

 

Keywords: weighted automata, semiring, discrete reliability, recursive path counting, multi-state infrastructure, fuzzy semiring

[This article belongs to Research & Reviews: Discrete Mathematical Structures ]

How to cite this article: S. Vairachilai, Markala Karthik, Mohammed Almakki, Mohammed El Khider. Semiring-Weighted Automata and Recursive Path Counting for Multi-State Reliability in Discrete Infrastructures. Research & Reviews: Discrete Mathematical Structures. 2026; 13(02):30-36.
How to cite this URL: S. Vairachilai, Markala Karthik, Mohammed Almakki, Mohammed El Khider. Semiring-Weighted Automata and Recursive Path Counting for Multi-State Reliability in Discrete Infrastructures. Research & Reviews: Discrete Mathematical Structures. 2026; 13(02):30-36. Available from: https://journals.stmjournals.com/rrdms/article=2026/view=250999

References

  1. Mohri M. Weighted automata algorithms. In: Droste M, Kuich W, Vogler H, eds. Handbook of Weighted Automata. Springer; 2009:213–250. doi:10.1007/978-3-642-01492-5_6.
  2. Droste M, Kuich W. Semirings and formal power series. In: Droste M, Kuich W, Vogler H, eds. Handbook of Weighted Automata. Springer; 2009:3–28. doi:10.1007/978-3-642-01492-5_1.
  3. Mohri M, Pereira F, Riley M. Weighted finite-state transducers in speech recognition. Computer Speech & Language. 2002;16(1):69–88. doi:10.1006/csla.2001.0184.
  4. Kostolányi P. Weighted automata and logics meet computational complexity. Information and Computation. 2024;301:105213. doi:10.1016/j.ic.2024.105213.
  5. Daviaud L, Johnson M. Feasibility of learning weighted automata on a semiring. Logical Methods in Computer Science. 2025;21(3:15). doi:10.46298/Imcs-21(3:15)2025.
  6. Otero Sánchez Á, Camazón Portela D, López-Ramos JA. On the solutions of linear systems over additively idempotent semirings. Mathematics. 2024;12(18):2904. doi:10.3390/math12182904.
  7. Stanimirović PS, Ćirić M, Mourtas SD, Brzaković P, Karabašević D. Simulations and bisimulations between weighted finite automata based on time-varying models over real numbers. Mathematics. 2024;12(13):2110. doi:10.3390/math12132110.
  8. Buchholz P, Kemper P. Model checking for a class of weighted automata. Discrete Event Dynamic Systems. 2009;19:429–461. doi:10.1007/s10626-008-0057-0.
  9. Zadeh LA. Fuzzy sets. Information and Control. 1965;8(3):338-353. doi:10.1016/S0019-9958(65)90241-X.
  10. Mendel JM. Fuzzy logic systems for engineering: a tutorial. Proceedings of the IEEE. 1995;83(3):345377. doi:10.1109/5.364485.
  11. Yogeesh N, et al. Improving speech privacy with fuzzy logic-based encryption. In: 2023 IEEE 2nd International Conference on Integrated Intelligence and Communication Systems for Digital Transformation (ICIDeA). 2023:217–222. doi:10.1109/ICIDeA59866.2023.10295183.
  12. Yogeesh N, et al. Maximizing efficiency using fuzzy matrix optimization for wireless resource allocation. Applied Mathematics & Information Sciences. 2024;18(6):1495–1506. doi:10.18576/amis/180625.
  13. Yogeesh N, et al. Leveraging fuzzy logic for habitat suitability analysis: a comprehensive case study in digital ecosystems. Applied Mathematics & Information Sciences. 2025;19(2):335–347. doi:10.18576/amis/190210.
  14. Yogeesh N, et al. A mathematical fuzzy model for syntax-pragmatics interface. Forum for Linguistic Studies. 2025;7(6):26–41. doi:10.30564/fls.v7i6.9618.
  15. Yogeesh N, et al. Quantum implementation of fuzzy logic conjunction and disjunction using multiqubit gates. European Chemical Bulletin. 2023;12(5):2098–2108. doi:10.48047/ecb/2023.12.5.145.
  16. Yogeesh N, et al. Optimizing MIMO antenna performance using fuzzy logic algorithms. Applied Mathematics & Information Sciences. 2025;19(2):349–364. doi:10.18576/amis/190211.
  17. Yogeesh N, Sayed A. Theoretical framework of quantum perspectives on fuzzy mathematics: unveiling neural mechanisms of consciousness and cognition. NeuroQuantology. 2017;15(4):180–187. doi:10.48047/nq.2017.15.4.1148.
  18. Peng Y, Zhu X, Nie F, Kong W, Ge Y. Fuzzy graph clustering. Information Sciences. 2021;571:38–49. doi:10.1016/j.ins.2021.04.058.
  19. Cangemi D, D’Urso PS, De Giovanni L, Federico L, Vitale V. Fuzzy clustering and community detection: an integrated approach. Journal of Classification. 2025. doi:10.1007/s00357-025-09520-7.
  20. Zhang Y, Yan S, Zhang L, Du B. Fast projected fuzzy clustering with anchor guidance for multimodal remote sensing imagery. IEEE Transactions on Image Processing. 2024;33:4640–4653. doi:10.1109/TIP.2024.3444323.
  21. Dunn JC. A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. Journal of Cybernetics. 1973;3(3):32–57. doi:10.1080/01969727308546046.
  22. Bezdek JC, Ehrlich R, Full W. FCM: the fuzzy c-means clustering algorithm. Computers & Geosciences. 1984;10(2-3):191–203. doi:10.1016/0098-3004(84)90020-7.
  23. Krishnapuram R, Keller JM. The possibilistic c-means algorithm: insights and recommendations. IEEE Transactions on Fuzzy Systems. 1996;4(3):385–393. doi:10.1109/91.531779.
  24. Kaushal M, Lohani QMD, Castillo O. Weighted intuitionistic fuzzy c-means clustering algorithms. International Journal of Fuzzy Systems. 2024;26(3):943–977. doi:10.1007/s40815-023-01644-5.

Yan Z, Wang W, Liu J, et al. Robust unsupervised feature selection algorithm based on fuzzy neighborhood manifold learning. Entropy. 2025;27(8):827. doi:10.3390/e27080827.

Regular Issue Subscription Original Research
Volume 13
Issue 02
Received 13/04/2026
Accepted 19/05/2026
Published 30/05/2026
Publication Time 47 Days

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