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Automating Agential Reasoning: Proof-Calculi and Syntactic Decidability for STIT Logics

  • Tim LyonEmail author
  • Kees van Berkel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11873)

Abstract

This work provides proof-search algorithms and automated counter-model extraction for a class of \(\mathsf {STIT}\) logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for \(\mathsf {STIT}\) logics. A new class of cut-free complete labelled sequent calculi \(\mathsf {G3Ldm}_{n}^{m}\), for multi-agent \(\mathsf {STIT}\) with at most n-many choices, is introduced. We refine the calculi \(\mathsf {G3Ldm}_{n}^{m}\) through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in the auxiliary calculi \(\mathsf {Ldm}_{n}^{m}\mathsf {L}\). In the single-agent case, we show that the refined calculi \(\mathsf {Ldm}_{n}^{m}\mathsf {L}\) derive theorems within a restricted class of (forestlike) sequents, allowing us to provide proof-search algorithms that decide single-agent \(\mathsf {STIT}\) logics. We prove that the proof-search algorithms are correct and terminate.

Keywords

Decidability Labelled calculus Logics of agency Proof search Proof theory Propagation rules Sequent \(\mathsf {STIT}\) logic 

References

  1. 1.
    Arkoudas, K., Bringsjord S., Bello, P.: Toward ethical robots via mechanized deontic logic. In: AAAI Fall Symposium on Machine Ethics, pp. 17–23 (2005)Google Scholar
  2. 2.
    Balbiani, P., Herzig, A., Troquard, N.: Alternative axiomatics and complexity of deliberative STIT theories. J. Philos. Logic 37(4), 387–406 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Belnap, N., Perloff, M., Xu, M.: Facing the Future: Agents and Choices in Our Indeterminist World. Oxford University Press on Demand, Oxford (2001)Google Scholar
  4. 4.
    van Berkel, K., Lyon, T.: Cut-free calculi and relational semantics for temporal STIT logics. In: Calimeri, F., Leone, N., Manna, M. (eds.) JELIA 2019. LNCS (LNAI), vol. 11468, pp. 803–819. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-19570-0_52CrossRefGoogle Scholar
  5. 5.
    Broersen, J.: Deontic epistemic stit logic distinguishing modes of mens rea. J. Appl. Logic 9(2), 137–152 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ciabattoni, A., Lyon, T., Ramanayake, R.: From display to labelled proofs for tense logics. In: Artemov, S., Nerode, A. (eds.) LFCS 2018. LNCS, vol. 10703, pp. 120–139. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-72056-2_8CrossRefGoogle Scholar
  7. 7.
    Gentzen, G.: Untersuchungen über das logische Schließen. Math. Z. 39(3), 405–431 (1935)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L., Vitalis, Q.: VINTE: an implementation of internal calculi for lewis’ logics of counterfactual reasoning. In: Schmidt, R.A., Nalon, C. (eds.) TABLEAUX 2017. LNCS (LNAI), vol. 10501, pp. 149–159. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66902-1_9CrossRefzbMATHGoogle Scholar
  9. 9.
    Grossi, D., Lorini, E., Schwarzentruber, F.: The ceteris paribus structure of logics of game forms. J. Artif. Intell. Res. 53, 91–126 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Herzig, A., Schwarzentruber, F.: Properties of logics of individual and group agency. Adv. Modal Logic 7, 133–149 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Horty, J.: Agency and Deontic Logic. Oxford University Press, Oxford (2001)CrossRefGoogle Scholar
  12. 12.
    Lorini, E., Sartor, G.: Influence and responsibility: a logical analysis. In: Legal Knowledge and Information Systems, pp. 51–60. IOS Press (2015)Google Scholar
  13. 13.
    Murakami, Y.: Utilitarian deontic logic. Adv. Modal Logic 5, 211–230 (2005)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34(5–6), 507–544 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Schwarzentruber, F.: Complexity results of stit fragments. Stud. Logica 100(5), 1001–1045 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sipser, M.: Introduction to the Theory of Computation. Course Technology (2006)Google Scholar
  17. 17.
    Tiu, A., Ianovski, E., Goré, R.: Grammar Logics in Nested Sequent Calculus: Proof Theory and Decision Procedures. CoRR (2012)Google Scholar
  18. 18.
    Viganò, L.: Labelled Non-classical Logics. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  19. 19.
    Wansing, H.: Tableaux for multi-agent deliberative-stit logic. Adv. Modal Logic 6, 503–520 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xu, M.: Decidability of deliberative stit theories with multiple agents. In: Gabbay, D.M., Ohlbach, H.J. (eds.) ICTL 1994. LNCS, vol. 827, pp. 332–348. Springer, Heidelberg (1994).  https://doi.org/10.1007/BFb0013997CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Logic and ComputationTechnische Universität WienWienAustria

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