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Paraconsistent transition structures: compositional principles and a modal logic

Published online by Cambridge University Press:  04 July 2025

Juliana Cunha ,
Alexandre Madeira Open the ORCID record for Alexandre Madeira [Opens in a new window]  and
Luis S. Barbosa
Juliana Cunha
Affiliation:
CIDMA, University of Aveiro, Aveiro, Portugal HASLab INESC-TEC, University of Minho, Braga, Portugal
Alexandre Madeira*
Affiliation:
CIDMA, University of Aveiro, Aveiro, Portugal
Luis S. Barbosa
Affiliation:
HASLab INESC-TEC, University of Minho, Braga, Portugal
*
Corresponding author: Alexandre Madeira; Email: madeira@ua.pt
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Abstract

Often in Software Engineering, a modeling formalism has to support scenarios of inconsistency in which several requirements either reinforce or contradict each other. Paraconsistent transition systems are proposed in this paper as one such formalism: states evolve through two accessibility relations capturing weighted evidence of a transition or its absence, respectively. Their weights come, parametrically, from a residuated lattice. This paper explores both i) a category of these systems, and the corresponding compositional operators and ii) a modal logic to reason upon them. Furthermore, two notions of crisp and graded simulation and bisimulation are introduced in order to relate two paraconsistent transition systems. Finally, results of modal invariance, for specific subsets of formulas, are discussed.

Keywords

Paraconsistent transition systemsparaconsistent logicmodal logicsimulationbisimulation

Information

Type
Special issue: LSFA 2021 and LSFA 2022
Information
Mathematical Structures in Computer Science , Volume 35 , 2025 , e14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Abe, J. M., Torres, C. R., Lambert-Torres, G., da Silva Filho, J. I. and Martins, H. G. (2007). Paraconsistent autonomous mobile robot emmy III. In: Lambert-Torres, G., Abe, J. M., da Silva Filho, J. I. and Martins, H. G. (eds.) Advances in Technological Applications ofLogical and Intelligent Systems, Selected Papers from the Sixth Congress on Logic Applied to Technology,LAPTEC 2007, Unisanta, Santa Cecilia University, Santos, Brazil, November 21–23, 2007, volume 186 ofFrontiers in Artificial Intelligence and Applications, IOS Press, 236–258.Google Scholar
Agudelo, J. C. and Carnielli, W. A. (2010). Paraconsistent machines and their relation to quantum computing. Journal of Logic and Computation. 20 (2) 573–595.10.1093/logcom/exp072CrossRefGoogle Scholar
Akama, S. (ed.) (2016) Towards Paraconsistent Engineering, Volume 110 of Intelligent Systems Reference Library, Springer.Google Scholar
Badia, G., Caicedo, X. and Noguera, C. (2023). Frame definability in finitely valued modal logics. Annals of Pure and Applied Logic. 174 (7) 103273.10.1016/j.apal.2023.103273CrossRefGoogle Scholar
Badia, G. and Olkhovikov, G. K. (2020). A lindström theorem in many-valued modal logic over a finite mtl-chain. Fuzzy Sets and Systems. 388) 26–37.10.1016/j.fss.2019.03.002CrossRefGoogle Scholar
Baeten, J. C. M., Basten, T. and Reniers, M. A. (2010). Process Aalgebra: Eequational theories of communicating processes. Cambridge Tracts in Theoretical Computer Science, 50, Cambridge University Press.Google Scholar
Barbosa, L. S. and Madeira, A. (2023). Capturing qubit decoherence through paraconsistent transition systems. In: Chiba, S., Cong, Y. and Boix, E. G., Companion Proceedings of the 7th International Conference on the Art, Science, and Engineering of Programming, Programming 2023, Tokyo,Japan, March 13–17, 2023, ACM, 109–110.10.1145/3594671.3594689CrossRefGoogle Scholar
Beohar, H., Gurke, S., König, B. and Messing, K. (2023). Hennessy–Milner theorems via Galois connections. In: Klin, B. and Pimentel, E., 31st EACSL Annual Conference on Computer ScienceLogic (CSL 2023), volume 252 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2023, Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik, 12:1–12:18.Google Scholar
Bou, F., Esteva, F., Godo, L. and Rodríguez, R. O. (10 2009). On the minimum many-valued modal logic over a finite residuated lattice. Journal of Logic and Computation. 21 (5) 739–790.10.1093/logcom/exp062CrossRefGoogle Scholar
Caicedo, X., Metcalfe, G., Rodríguez, R. O. and Rogger, J. (2017). Decidability of order-based modal logics. Journal of Computer and System Sciences. 88 53–74.10.1016/j.jcss.2017.03.012CrossRefGoogle Scholar
Caicedo, X. and Rodríguez, R. O. (2010). Standard gödel modal logics. Studia Logica. 94 (2) 189–214.10.1007/s11225-010-9230-1CrossRefGoogle Scholar
Caicedo, X. and Rodríguez, R. O. (2015). Bi-modal gödel logic over [0, 1]-valued kripke frames. Journal of Logic and Computation. 25 (1) 37–55.10.1093/logcom/exs036CrossRefGoogle Scholar
Chiara, M. L. D. and Giuntini, R. (2000). Paraconsistent ideas in quantum logic. Synthese. 125 (1-2) 55–68.10.1023/A:1005296018904CrossRefGoogle Scholar
Costa, D. and Martins, M. A. (2016). Intelligent-based robot to deal with contradictions. 2016 International Conference on Autonomous Robot Systems and Competitions (ICARSC), 199–204.10.1109/ICARSC.2016.48CrossRefGoogle Scholar
Cruz, A., Madeira, A. and Barbosa, L. S. (2022). A logic for paraconsistent transition systems. In: Indrzejczak, A. and Zawidzki, M. (eds.) Proceedings of the 10th International Conference on Non-ClassicalLogics. Theory and Applications, NCL 2022, Łódź, Poland, 14–18 March 2022, Volume 358 of EPTCS, 270–284.10.4204/EPTCS.358.20CrossRefGoogle Scholar
Cruz, A., Madeira, A. and Barbosa, L. S. (2022). Paraconsistent transition systems. In: Nantes-Sobrinho, D. and Fontaine, P. (eds.) Proceedings 17th International Workshop on Logical and SemanticFrameworks with Applications, LSFA 2022, Belo Horizonte, Brazil (hybrid), 23–24 September 2022, Volume 376 of EPTCS, 3–15.Google Scholar
Cunha, J., Madeira, A. and Barbosa, L. S. (2023). Stepwise development of paraconsistent processes. In: David, C. and Sun, M. (eds.) Theoretical Aspects of Software Engineering - 17th InternationalSymposium, TASE 2023, Bristol, UK, July 4–6, 2023, Proceedings, Volume 13931 of Lecture Notes in Computer Science, Springer, 327–343.10.1007/978-3-031-35257-7_20CrossRefGoogle Scholar
Cunha, J., Madeira, A. and Barbosa, L. S. (2023) Structured specification of paraconsistent transition systems. In: Hojjat, H. and brahám, E. Á. (eds.) Fundamentals of Software Engineering - 10thInternational Conference, FSEN 2023, Tehran, Iran, May 4–5, 2023, Revised Selected Papers, Volume 14155 of Lecture Notes in Computer Science, Springer, pp. 1–17.Google Scholar
da Costa, N. C. A. and Krause, D. (2014). Physics, inconsistency, and quasi-truth. Synthese. 191 (13) 3041–3055.10.1007/s11229-014-0472-8CrossRefGoogle Scholar
da Costa, N. C. A., Krause, D. and Bueno, O. (2007) Paraconsistent logics and paraconsistency. In: Jacquette, D. (ed.), Handbook of the Philosophy of Science (Philosophy of Logic), Elsevier, 791–911.Google Scholar
Esteva, F. and Godo, L. (2001). Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems. 124 (3) 271–288.10.1016/S0165-0114(01)00098-7CrossRefGoogle Scholar
Jain, M., Madeira, A. and Martins, M. A. (2019). A fuzzy modal logic for fuzzy transition systems. Felty, A. P. and Marcos, J. (eds.) Proceedings of the 14th Workshop on Logical and Semantic Frameworkswith Applications, LSFA 2019, Natal, Brazil, August, 2019, volume 348 of Electronic Notes in Theoretical Computer Science, Elsevier, 85–103.Google Scholar
Jaśkowski, S. (1969). Propositional calculus for contradictory deductive systems. Studia Logica. 24 (1) 143–157.10.1007/BF02134311CrossRefGoogle Scholar
Kracht, M. (1998). On extensions of intermediate logics by strong negation. Journal of Philosophical Logic. 27 (1) 49–73.10.1023/A:1004222213212CrossRefGoogle Scholar
Madeira, A., Neves, R. and Martins, M. A. (2016). An exercise on the generation of many-valued dynamic logics. Journal of Logical and Algebraic Methods in Programming. 85 (5) 1011–1037.10.1016/j.jlamp.2016.03.004CrossRefGoogle Scholar
Marti, M. and Metcalfe, G. (2014) A hennessy-milner property for many-valued modal logics. In: Goré, R., Kooi, B. P. and Kurucz, A. (eds.), Advances in Modal Logic 10, Invited and Contributed Papers From the Tenth Conference On “Advances in Modal Logic,” Held in Groningen, The Netherlands, August 5–8, 2014, College Publications, 407–420.Google Scholar
Nguyen, L. A. (2022). Logical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices. Fuzzy Sets and Systems. 431 70–93.10.1016/j.fss.2021.08.009CrossRefGoogle Scholar
Preskill, J. (2018). Quantum computing in the nisq era and beyond. Quantum. 2 (79) 87–95.10.22331/q-2018-08-06-79CrossRefGoogle Scholar
Sannella, A. T. D. (2012). Foundations of Algebraic Specification and Formal Software Development, Springer-Verlag.10.1007/978-3-642-17336-3CrossRefGoogle Scholar
Urabe, N. and Hasuo, I. (2018). Coalgebraic infinite traces and kleisli simulations. Logical Methods in Computer Science. 14 (3).Google Scholar
Winskel, G. and Nielsen, M. (1995) Models for concurrency. In: Handbook of Logic in Computer Science (Vol. 4): Semantic Modelling, Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Oxford Science Publications, 1–148.Google Scholar