Pseudo-contractions in belief revision

Abstract

My experience in the Master's was pleasing for the most part, and as such I think I owe some people their credit.In first place, I must say that I am very grateful to my advisor, professor Renata, who has guided me with all support needed to complete this Master's. She has a remarkable combination of wisdom and kindness which are unparalleled. Likewise, I need to thank my co-advisor, professor Márcio, whose ideas and insights have been always appropriate and who has helped me with important suggestions and corrections. I also want to express my gratitude to professors Marcelo Finger, Flávio Soares, Leliane de Barros and Eduardo Fermé, who have given me relevant advice. I also would like to thank Glauber de Bona, who participated in some discussions with us and found an application in his work for the theory developed here.Next, I want to thank my friends and colleagues from LIAMF, who have helped me to enjoy a good time these two years, and have also discussed with me some matters about my project, without which the quality of this work would not be the same. I am glad to have had the opportunity to meet such awesome friends. I must thank CNPq for the financial support and USP for the opportunity, and all the people who pay their taxes and believe in public education.Last, I want to thank my family, for, despite having not much to do directly with this project, without them I would not even be here to start it.iii iv Abstract Belief Revision addresses the problem of how to change epistemic states, usually represented in the literature by sets of logical sentences. Solid theoretical results were consolidated with the AGM paradigm, which deals with theories (logically closed sets of sentences). After that, the theory was extended to belief bases, that is, arbitrary sets of sentences. Besides all this theoretical framework, AI researchers face serious difficulties when trying to implement belief revision systems. One of the major complications is the closure required by AGM theory, which cannot be easily computed.Even belief bases, which do not require closure, seem to be improper for practical purposes, since their changes are usually very rigid (syntax dependent).Some operations, known as pseudo-contractions, are in the middle ground between belief set change and belief base change. In the present work we have proposed a new pseudo-contraction operation, studied its properties and characterized it. We have also found connections between this operator and some other pseudo-contractions.Keywords: belief revision, pseudo-contractions. v vi ResumoRevisão de Crenças aborda o problema de como alterar estados epistêmicos, normalmente representados na literatura como conjuntos de sentenças lógicas. Resultados teóricos sólidos foram consolidados com o paradigma AGM, que lida com teorias (conjuntos de sentenças logicamente fechados). Depois disso, a teoria foi estendida para bases de crenças, isto é, conjuntos arbitrários de sentenças. Apesar de todo esse arcabouço teórico, pesquisadores de IA enfrentam sérias dificuldades ...

Pseudo-contraction operations for description logics

“…The ideas and results of this section were first presented in Santos's thesis [San16] and published in [SMRW18] 6 . We first introduce the operation, and then show its formal properties.…”

“…We first introduce the operation, and then show its formal properties. Definition 3.3 (Cn* partial meet pseudo-contraction [San16]). Let B be a set of sentences, Cn* a consequence relation and g a selection function for Cn*(B).…”

“…It is not surprising that, in order to characterise this operation, which embeds two different logics, we would need to mention this second logic Cn* in the postulates. The representation theorem is the following: Theorem 3.5 (Cn* partial meet pseudo-contraction: representation theorem [San16]). Provided that Cn* satisfies inclusion, idempotence and subclassicality, an operation is a Cn* partial meet pseudo-contraction if and only if it satisfies success, inclusion*, relevance* and uniformity*.…”

“…Proposition 3.10. [San16] If Cn* satisfies subclassicality, an operation that satisfies inclusion* and relevance* 10 also satisfies vacuity*.…”

“…In[SMRW18], the operation was denoted by − * and was called Cn*-pseudo-contraction. In this text, we added 'partial meet' to its name because we shall also discuss a similar construction that is based on kernel contraction.7 This is analogous to the consequence operator that 'breaks conjunctions into conjuncts' presented in[San16].8 Although this proof is not new (it was presented in [San16]), we did not omit it here, because it is the most important result about Cn* partial meet pseudo-contractions; moreover, it might be a good demonstration of the notation that we are proposing.…”