Conditionals in nonmonotonic reasoning and belief revision. Considering conditionals as agents.

*(English)*Zbl 0978.03014
Lecture Notes in Computer Science. 2087. Berlin: Springer. x, 190 p. (2001).

Relationships amongst propositions may be represented in a most general form by (if-then) conditionals. The present book proposes a new approach to conditionals which capture both the propositional as well as the dynamic, non-propositional nature of conditionals, considered as agents shifting possible worlds in order to establish relationships and beliefs. This view on conditionals yields a rich methodological theory and a unifying framework for knowledge representation, nonmonotonic reasoning, belief revision, and even knowledge discovery. Separating structural from numerical aspects, the basic techniques introduced in this book may be applied both in qualitative and in numerical settings, contributing to the elaboration of the fundamental lines of reasoning.

The ten chapters of the book are organized as follows: Chapter 1 (Introduction) includes a general presentation of the problems, basic definitions, and necessary notations. Chapter 2 (Belief revision and nonmonotonic reasoning – state of the art) outlines the currently existing approaches to belief revision and nonmonotonic reasoning. The standard AGM-theory on belief revision is recalled, and the difference between revision and updating in the sense of Katsuno and Mendelzon is explained. A picture of belief revision and nonmonotonic reasoning is presented from a probabilistic point of view, featuring revisions and inferences as based on the principles of optimum entropy. Chapter 3 (Conditionals) focuses on formal definitions and results related to conditionals: connections between conditionals and epistemic states, conditional evaluation functions, subconditionality and perpendicularity relations on conditionals, conditionals represented by generators of a (free-abelian) group and the conditional structure of a world, conditional indifference of a conditional evaluation function with respect to a set of conditionals.

In Chapter 4 (Revisiting epistemic states by conditional beliefs), conditional indifference is proved to be the essential ingredient to formalizing a quantitative principle of conditional preservation for revising conditional valuation functions. \(c\)-revisions (by sets of conditionals) and \(c\)-representation (of sets of conditionals) are introduced and their properties investigated. In order to preserve conditional beliefs in a purely quantitative setting, postulates for revising an epistemic state by a conditional are formulated, and representation theorems for these postulates are given. Chapter 5 (Characterizing the principle of minimum cross-entropy) examines in a probabilistic environment the idea of revisions based on the principle of conditional preservation. Such a revision should satisfy three more postulates: (1) a functional concept for the revised probability distribution, (2) the logical coherence postulates, and (3) the postulate for representation invariance. This axiomatic framework provides a new characterization of revisions based on the principle of minimum cross-entropy (ME-revisions). Chapter 6 (Reasoning at optimum entropy) analyzes how ME-reasoning works. The nonmonotonic inference operations and properties satisfied by ME-inference are examined, and ME-deduction schemes are used to illustrate ME-reasoning in some simple but typical situations such as transitive chaining, cautions monotonicity, and reasoning by cases. Chapter 7 (Belief revision and nonmonotonic reasoning – revisited) returns to the general belief revision and nonmonotonic reasoning, but in the extended framework of numerical ME-inference applied to epistemic states and conditionals. Chapter 8 (Knowledge discovery by following conditional structures) is devoted to probabilistic knowledge discovery by following conditional patterns within conditional valuation functions. Chapter 9 (Algorithms and implementations) presents briefly a selection of the various computational approaches to ME-reasoning with probabilistic conditionals, to probabilistic knowledge discovery, and to probabilistic belief revision. The final Chapter 10 (Conclusion) summarizes the main results contained in the book.

The ten chapters of the book are organized as follows: Chapter 1 (Introduction) includes a general presentation of the problems, basic definitions, and necessary notations. Chapter 2 (Belief revision and nonmonotonic reasoning – state of the art) outlines the currently existing approaches to belief revision and nonmonotonic reasoning. The standard AGM-theory on belief revision is recalled, and the difference between revision and updating in the sense of Katsuno and Mendelzon is explained. A picture of belief revision and nonmonotonic reasoning is presented from a probabilistic point of view, featuring revisions and inferences as based on the principles of optimum entropy. Chapter 3 (Conditionals) focuses on formal definitions and results related to conditionals: connections between conditionals and epistemic states, conditional evaluation functions, subconditionality and perpendicularity relations on conditionals, conditionals represented by generators of a (free-abelian) group and the conditional structure of a world, conditional indifference of a conditional evaluation function with respect to a set of conditionals.

In Chapter 4 (Revisiting epistemic states by conditional beliefs), conditional indifference is proved to be the essential ingredient to formalizing a quantitative principle of conditional preservation for revising conditional valuation functions. \(c\)-revisions (by sets of conditionals) and \(c\)-representation (of sets of conditionals) are introduced and their properties investigated. In order to preserve conditional beliefs in a purely quantitative setting, postulates for revising an epistemic state by a conditional are formulated, and representation theorems for these postulates are given. Chapter 5 (Characterizing the principle of minimum cross-entropy) examines in a probabilistic environment the idea of revisions based on the principle of conditional preservation. Such a revision should satisfy three more postulates: (1) a functional concept for the revised probability distribution, (2) the logical coherence postulates, and (3) the postulate for representation invariance. This axiomatic framework provides a new characterization of revisions based on the principle of minimum cross-entropy (ME-revisions). Chapter 6 (Reasoning at optimum entropy) analyzes how ME-reasoning works. The nonmonotonic inference operations and properties satisfied by ME-inference are examined, and ME-deduction schemes are used to illustrate ME-reasoning in some simple but typical situations such as transitive chaining, cautions monotonicity, and reasoning by cases. Chapter 7 (Belief revision and nonmonotonic reasoning – revisited) returns to the general belief revision and nonmonotonic reasoning, but in the extended framework of numerical ME-inference applied to epistemic states and conditionals. Chapter 8 (Knowledge discovery by following conditional structures) is devoted to probabilistic knowledge discovery by following conditional patterns within conditional valuation functions. Chapter 9 (Algorithms and implementations) presents briefly a selection of the various computational approaches to ME-reasoning with probabilistic conditionals, to probabilistic knowledge discovery, and to probabilistic belief revision. The final Chapter 10 (Conclusion) summarizes the main results contained in the book.

Reviewer: Neculai Curteanu (Iaşi)

##### MSC:

03B42 | Logics of knowledge and belief (including belief change) |

68T27 | Logic in artificial intelligence |

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

68T30 | Knowledge representation |