Several forms of iterable belief change exist, differing in the kind of change and its strength: some operators introduce formulae, others remove them; some add formulae unconditionally, others only as additions to the previous beliefs; some only relative to the current situation, others in all possible cases. A sequence of changes may involve several of them: for example, the first step is a revision, the second a contraction and the third a refinement of the previous beliefs. The ten operators considered in this article are shown to be all reducible to three: lexicographic revision, refinement and severe withdrawal. In turn, these three can be expressed in terms of lexicographic revision at the cost of restructuring the sequence. This restructuring needs not to be done explicitly: an algorithm that works on the original sequence is shown. The complexity of mixed sequences of belief change operators is also analyzed. Most of them require only a polynomial number of calls to a satisfiability checker, some are even easier.

Merging beliefs requires the plausibility of the sources of the information to be merged. They are typically assumed equally reliable in lack of hints indicating otherwise; yet, a recent line of research spun from the idea of deriving this information from the revision process itself. In particular, the history of previous revisions and previous merging examples provide information for performing subsequent mergings. Yet, no examples or previous revisions may be available. In spite of the apparent lack of information, something can still be inferred by a try-and-check approach: a relative reliability ordering is assumed, the merging process is performed based on it, and the result is compared with the original information. The outcome of this check may be incoherent with the initial assumption, like when a completely reliable source is rejected some of the information it provided. In such cases, the reliability ordering assumed in the first place can be excluded from consideration. The first theorem of this article proves that such a scenario is indeed possible. Other results are obtained under various definition of reliability and merging.

This article proposes a solution to the problem of obtaining plausibility information, which is necessary to perform belief revision: given a sequence of revisions, together with their results, derive a possible initial order that has generated them; this is different from the usual assumption of starting from an all-equal initial order and modifying it by a sequence of revisions. Four semantics for iterated revision are considered: natural, restrained, lexicographic and reinforcement. For each, a necessary and sufficient condition to the existence of an order generating a given history of revisions and results is proved. Complexity is proved coNP complete in all cases but one (reinforcement revision with unbounded sequence length).

When integrating information coming from different sources, a distinction is made between revision [13, 5, 14, 28, 6] (new information more reliable than old) and merging [22, 4, 18] (same reliability). More generally, priorities or weights are assigned to the sources to indicate their reliability [26, 27, 30, 7]. Measures and aggregation functions allow for fine-grained policies of integration [16, 11, 18]. Families of operators are then defined, all depending in a way or another from the relative reliability of the sources. The two basic cases of non-iterated revision and merging result from giving priority to the new information or the same to all pieces of information to be incorporated, respectively. The strenght of information sources has been studied in the field of cognitive psychology, where it was determined to depend on the order in which the information is given [32], on the size of the group generating it [25] and other social factors [31]. The first time merging is done, the relative reliability of the pieces of information to be integrated cannot come other than from sources external to the merging process. However, subsequent mergings may then take advantage from the previous results.