Abstract
In this paper we study AGM contraction and revision of rules using input/output logical theories. We replace propositional formulas in the AGM framework of theory change by pairs of propositional formulas, representing the rule based character of theories, and we replace the classical consequence operator Cn by an input/output logic. The results in this paper suggest that, in general, results from belief base dynamics can be transferred to rule base dynamics, but that a similar transfer of AGM theory change to rule change is much more problematic. First, we generalise belief base contraction to rule base contraction, and show that two representation results of Hansson still hold for rule base contraction. Second, we show that the six so-called basic postulates of AGM contraction are consistent only for some input/output logics, but not for others. In particular, we show that the notorious recovery postulate can be satisfied only by basic output, but not by simple-minded output. Third, we show how AGM rule revision can be defined in terms of AGM rule contraction using the Levi identity. We highlight various topics for further research.
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Notes
K-7 and K-8 can be formulated on contraction with a set of formulas, so, the contraction of \(a\wedge b\) can be understood as the contraction of \(\{a,b\}\) see, e.g., Billington and Antoniou (1999).
This follows directly from Proposition 1, see also Makinson and Torre (2000).
However, there may be special cases where it holds, for example for \({ out}_2\).
References
Alchourrón, C. E., & Makinson, D. (1981). Hierarchies of regulations and their logic in Hilpinen, pp. 125–148.
Alchourrón, C. E., & Makinson, D. (1982). On the logic of theory change: Contraction functions and their associated revision functions. Theoria, 48, 14–37.
Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50(2), 510–530.
Billington, D., Antoniou, G., Governatori, G., & Maher, M. (1999). Revising non-monotonic belief sets: The case of defeasible logic. In: KI-99: Advances in artificial intelligence, Berlin: Springer, pp. 101–112.
Boella, G., Pigozzi, G., & van der Torre, L. (2009). Normative framework for normative system change. In: 8th International joint conference on autonomous agents and multiagent systems (AAMAS 2009), Budapest, Hungary, May 10–15, vol. 1, pp. 169–176.
Corapi, D., De Vos, M., Padget, J., Russo, A., & Satoh, K. (2011). Norm refinement and design through inductive learning. In M. De Vos, N. Fornara, J. Pitt, & G. Vouros (Eds.), Coordination, organizations, institutions, and norms in agent systems VI (vol. 6541, pp. 77–94). Lecture Notes in Computer Science Berlin Heidelberg: Springer.
Delgrande, J. (2010). A program-level approach to revising logic programs under the answer set semantics. Theroy and Practice of Logic Programming, 10, 565–580.
Delgrande, J., Schaub, T., Tompits, H., & Woltran, S. (2008). Belief revision of logic programs under answer set semantics. In G. Brewka, J. Lang (Eds.), (pp. 411–421). KR: AAAI Press.
Gärdenfors, P. (1978). Conditionals and changes of belief. Acta Philosophica Fennica, 30(1), 381–404.
Gärdenfors, P., & Rott, H. (1995). Belief revision. In: D. M. Gabbay, C. J. Hogger, & J. Robinson (Eds.) Handbook of logic in artificial intelligence and logic programming. vol. IV: Epistemic and temporal reasoning. Oxford: Oxford University Press, pp. 35–132.
Governatori, G., & DiGiusto, P. (1999). Modifying is better than deleting: A new approach to base revision. In: E. Lamma, & P. Mello (Eds.) AI*IA 99, Pitagora, pp. 145–154.
Governatori, G., & Rotolo, A. (2010). Changing legal systems: Legal abrogations and annulments in defeasible logic. Logic Journal of IGPL 18(1):157–194, http://jigpal.oxfordjournals.org/cgi/content/abstract/jzp075v1
Governatori, G., Rotolo, A., Olivieri, F., & Scannapieco, S. (2013). Legal contractions: a logical analysis. In E. Francesconi & B. Verheij (Eds.) (pp. 63–72). ACM: ICAIL.
Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170.
Hansson, S. (1993). Reversing the Levi identity. Journal of Philosophical Logic, 22, 637–669.
Makinson, D., & van der Torre, L. (2000). Input–output logics. Journal of Philosophical Logic, 29, 383–408.
Makinson, D., & van der Torre, L. (2001). Constraints for input–output logics. Journal of Philosophical Logic, 30(2), 155–185.
Makinson, D., & van der Torre, L. (2003). Permissions from an input–output perspective. Journal of Philosophical Logic, 32(4), 391–416.
Nute, D. (1984). Conditional logic. In: Handbook of philosophical logic, synthese library, vol. 165, Berlin: Springer, pp. 387–439.
Parent, X., & van der Torre, L. (2013). Input/output logics. In D. Gabbay, J. Horty, X. Parent, R. van der Meyden, & L. van der Torre (Eds.), Handbook of deontic logic and normative systems (pp. 499–544). London: College Publications.
Stolpe, A. (2010). Norm-system revision: Theory and application. Artificial Intelligence and Law, 18, 247–283.
Acknowledgments
Thanks to David Makinson and Jörg Hansen for discussions on the issues raised in this paper. We also thank the two anonymous referees for their valuable comments and suggestions that helped us improving the content and readability of the paper. Gabriella Pigozzi benefited from the support of the project AMANDE ANR-13-BS02-0004 of the French National Research Agency (ANR).
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Boella, G., Pigozzi, G. & van der Torre, L. AGM Contraction and Revision of Rules. J of Log Lang and Inf 25, 273–297 (2016). https://doi.org/10.1007/s10849-016-9244-9
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DOI: https://doi.org/10.1007/s10849-016-9244-9