Iterated Belief Contraction from First Principles ∗ Abhaya C. Nayak Department of Computing Macquarie University Sydney, NSW Australia 2109 abhaya@ics.mq.edu.au Randy Goebel Department of Comp. Sc. University of Alberta Edmonton, Alberta Canada T6G 2H1 goebel@cs.ualberta.ca Mehmet A. Orgun Department of Computing Macquarie University Sydney, NSW Australia 2109 mehmet@ics.mq.edu.au Abstract Importance of contraction for belief change notwithstanding, literature on iterated belief change has by and large centered around the issue of iter- ated belief revision, ignoring the problem of iter- ated belief contraction. In this paper we examine iterated belief contraction in a principled way, start- ing with Qualified Insertion, a proposal by Hans Rott. We show that a judicious combination of Qualified Insertion with a well-known Factoring principle leads to what is arguably a pivotal prin- ciple of iterated belief contraction. We show that this principle is satisfied by the account of iterated belief contraction modelled by Lexicographic State Contraction, and outline its connection with Lex- icographic Revision, Darwiche-Pearl’s account of revision as well as Spohn’s Ordinal ranking theory. Keywords: Belief Change, Information State Change, Iterated Belief Contraction. Literature on belief change deals with the problem of how new evidence impinges upon the current knowledge of a ra- tional agent. The pioneering works in the area such as [Al- chourr´ on et al., 1985] provide the formal framework, and the solution for “one-shot belief change”. Follow up work in the area, e.g. [Williams, 1994; Nayak, 1994; Boutilier, 1996] have explored the related issue of Iterated Belief Change that deals with sequential changes in belief. This latter research has by and large been confined to iterated belief revision: Given belief corpus K, two sequential pieces of evidence x, y and revision operator ∗, how do we construct the resul- tant corpus (K ∗ x ) ∗ y ? However, the accompanying problem of iterated belief contraction, namely, Given belief corpus K, two beliefs x, y that are to be sequentially removed from K, and contraction operator −, how do we construct the resul- tant corpus (K − x ) − y ? has, for no obvious reason, been rather neglected. Very few reseach works, e.g., [Bochman, 2001; Chopra et al., 2002; Rott, 2001; 2006] and [Nayak et al., 2006], have addressed this problem. The primary aim of this ∗ The authors acknowledge the research support from the Aus- tralian Research Council (ARC), and thank Maurice Pagnucco and the referees for very helpful suggestions. paper is to examine this problem from what we may call the “first principles” of belief change. In Section 1, we introduce the problem of iterated belief contraction, and as a starting point, take up a proposal by Hans Rott [Rott, 2001] called Qualified Insertion. In the next section, we examine this principle and its variations, and show that when combined with Factoring, a well-known re- sult in belief change, Qualified Insertion can lead to a good account of iterated belief contraction. In Section 3 we study some interesting properties of this contraction, followed by its semantic modelling via Lexicographic state contraction in Section 4. We conclude with a short summary. 1 Background The theory of belief change purports to model how a current theory or body of beliefs, K, can be rationally modified in order to accommodate a new observation x. An observation, such as x, is represented as a sentence in a propositional lan- guage L, and a theory, such as K, is assumed to be a set of sentences in L, closed under a supra-classical consequence operation, Cn. Since the new piece of information x may contravene some current beliefs in K, chances are, some be- liefs in K will be discarded before x is eased into it. Accord- ingly, three forms of belief change are recognised in the belief change framework: 1. CONTRACTION: K − x is the result of discarding some unwanted information x from the theory K 2. EXPANSION: K + x is the result of simple-mindedly in- corporating some information x into the theory K, and 3. REVISION: K ∗ x is the result of incorporating some in- formation x into the theory K in a manner so as to avoid internal contradiction in K ∗ x . The intuitive connection among these operators is captured by the following two identities named, respectively, after Isaac Levi and William Harper: LEVI I DENTITY: K ∗ x =(K − ¬x ) + x , and HARPER I DENTITY: K − x = K ∗ ¬x ∩ K. A belief change (revision, contraction and expansion) operation is AGM- RATIONAL if it satisfies the corre- sponding set of AGM rationality postulates. The three sets of postulates, along with motivation and interpreta- tion for them, may be found in [G¨ ardenfors, 1988]. It is IJCAI-07 2568