On the logic of theory change: Partial meet contraction and revision functions, Journal of Symbolic Logic, vol.50, pp.510-530, 1985. ,
On the link between partial meet, kernel, and infra contraction and its application to Horn logic, Journal of Artificial Intelligence Research, vol.42, pp.31-53, 2011. ,
A unified model of qualitative belief change: A dynamical systems perspective, Artificial Intelligence, vol.98, issue.1-2, pp.281-316, 1998. ,
Semantical and computational aspects of Horn approximations, Artificial Intelligence, vol.119, issue.1-2, pp.1-17, 2000. ,
Belief contraction within fragments of propositional logic, Proc. ECAI, pp.390-398, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01490118
Belief revision within fragments of propositional logic, Journal of Computer and System Sciences, vol.80, issue.2, pp.427-449, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-01194410
Belief merging within fragments of propositional logic, A preliminary version appeared in Proc. of ECAI, vol.17, p.20, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-01194343
Do hard sat-related reasoning tasks become easier in the Krom fragment? CoRR, 2017. ,
Boolean constraint satisfaction problems: When does post's lattice help?, Complexity of Constraints -An Overview of Current Research Themes, vol.5250, pp.3-37, 2008. ,
Investigations into theory of knowledge base revision, Proc. AAAI, pp.449-479, 1988. ,
A unified view of belief revision and update, Journal of Logic and Computation, vol.4, issue.5, pp.797-810, 1994. ,
Compositional belief update, Journal of Artificial Intelligence Research, vol.32, pp.757-791, 2014. ,
Belief revision in Horn theories, Artificial Intelligence, vol.218, pp.1-22, 2015. ,
Horn clause contraction functions, Journal of Artificial Intelligence Research, vol.48, pp.457-511, 2013. ,
The PMA and relativizing minimal change for action update, Fundamenta Informaticae, vol.44, issue.1-2, pp.95-131, 2000. ,
Belief revision and updates in numerical formalisms: An overview, with new results for the possibilistic framework, Proc. IJCAI, pp.620-625, 1993. ,
On the complexity of propositional knowledge base revision, updates, and counterfactuals, Artificial Intelligence, vol.57, issue.2-3, pp.227-270, 1992. ,
On The Semantic of Updates in Databases, Proc. ACM SIGACT SIGMOD, pp.352-365, 1983. ,
Introducing actions into qualitative simulation, Proc. IJCAI, pp.1273-1278, 1989. ,
Modeling belief in dynamic systems, part II: Revision and update, Journal of Artificial Intelligence Research, vol.10, pp.117-167, 1999. ,
Merging in the Horn fragment, ACM Transactions on Computational Logic, vol.18, issue.1, pp.1-6, 2017. ,
Specification and implementation of programs for updating incomplete information databases, Proc. ACM SIGACT-SIGMOD-SIGART, pp.146-158, 1987. ,
Propositional belief base update and minimal change, Artificial Intelligence, vol.115, issue.1, pp.107-138, 1999. ,
On sentences which are true of direct unions of algebras, Journal of Symbolic Logic, vol.16, pp.14-21, 1951. ,
Propositional knowledge base revision and minimal change, Artificial Intelligence, vol.52, issue.3, pp.263-294, 1991. ,
On the difference between updating a knowledge base and revising it, Belief revision, pp.183-203, 1992. ,
On the use of an extended relational model to handle changing incomplete information, IEEE Transactions of Software Engineering, vol.11, issue.7, pp.620-633, 1985. ,
Changement de croyances dans des fragments de la logique propositionnelle, vol.5, 2016. ,
Belief update revisited, Proc. IJCAI, pp.2517-2522, 2007. ,
Belief revision and update: Complexity of model checking, Journal of Computer and System Sciences, vol.62, issue.1, pp.43-72, 2001. ,
The two-valued iterative systems of mathematical logic, Annals of Mathematical Studies, vol.5, pp.1-122, 1941. ,
Prime implicates and relevant belief revision, Journal of Logic and Computation, vol.23, issue.1, pp.109-119, 2013. ,
Nonmonotonic reasoning by minimal belief revision, Proc. FGCS, pp.455-462, 1988. ,
The complexity of satisfiability problems, Proc. STOC, pp.216-226, 1978. ,
Reasoning about action using a possible models approach, Proc. AAAI, pp.89-93, 1988. ,
Updates with disjunctive information: From syntactical and semantical perspectives, Computational Intelligence, vol.16, issue.1, pp.29-52, 2000. ,
Entrenchment-based Horn contraction, Journal of Artificial Intelligence Research, vol.51, pp.227-254, 2014. ,
Definability of Horn revision from Horn contraction, Proc. IJCAI, pp.1205-1212, 2013. ,
Also note that Modp?q ? Modpµq. We get Modp?pµ^ ?qq " Modp??q " f pModp? ? ?qq " f pttau, ta, b, euuq " N (N is closed under ^, f pN q " N holds by definition of refined operators), but Modpp?µq ^ ?q " f pMq X Modp?q " ttauu. Otherwise tau R f pMq. Since f pMq ? H and f pMq is closed under ^, by symmetry of the role played by the variables b and c, it is sufficient to examine three possibilities for f pMq: either f pMq " tta, Let ? and µ in LHorn such Modp?q ,
thus proving that p?µq ^ ? is satisfiable, whereas ?pµ ^ ?q |ù p?µq ^ ? in LHorn ,
Let us consider the possibilities for Modp?µq " f pMq. By definition of refined operators, we know that ta, bu R f pMq since ta, bu R Clmaj 3 pMq. We consider two cases: First assume tb, cu P f pMq: Let ? be such that Modp?q " ttb, cu, ta, buu " N . Clearly such a ? exists in LKrom . Besides note that Modp?q ? Modpµq. We get Modp?pµ ^ ?qq " Modp??q " f pModp? ? ?qq " ttb, cu, ta, buu " N , whereas Modpp?µq ^ ?q " ttb, cuu. Otherwise, we have tb, cu R f pMq. Since f pMq ? H and f pMq is already closed under maj 3 , by symmetry of the role played by the variables a, For L 1 " LKrom , the formulas ?, µ P LKrom with Modp?q ,
, It is then clear that in any case Modpp?µq ^ ?q ? H and Modp?pµ ^ ?qq ? Modpp?µq^?q, thus showing eventually that p?µq^? is satisfiable, whereas ?pµ^ ?q |ù p?µq ^ ? in LKrom
Observe that the set of models of µ is the set of solutions of the following equations system ,
,
, L affine . Also note that Modp?q ? Modpµq. We obtain on the one hand Modp?pµ ^ ?qq " Modp??q " f pModp? ? ?qq " tta, buu and on the other hand Modpp?µq ^ ?q contains tta
If f pMq " tta, buu or f pMq " tta, b, c, euu, we consider the formula ? such that Modp?q " tta, bu, ta, b, c, euu. Clearly, such a ? exists in L affine . We obnot closed by intersection. Observe that Cl ? pModp? ? µ1qq is at distance 1 from Modp? ? µ1q, and hence Cl^pModp? ? µ1qq P FppModp? ? µ1qq. Thus, Modp? ? Prox^ µ1q, Since f pMq ? H and f pMq is closed under '3, by symmetry of the role played by the variables d and e, it is sufficient to distinguish four cases for f pMq: either f pMq " tta, buu or f pMq " tta ,
, It induces the following lexicographical order on the sets of models of FppModp? ? µ2qq: tta, bu