A logical characterization of coherence for imprecise probabilities
This dense paper provides a logico-algebraic framework which allows for a logical characterization of coherence for imprecise improbability.\newline The first chapter begins with a discussion of first-order and second-order uncertainty. It is argued that in many interesting situations single-valued probabilities and/or binary events cannot satisfy practical needs. To combine statistical and logical approaches to second-order uncertainty the authors aim to develop a framework allowing to represent key concepts of both traditions.\newline After a discussion of de Finetti's Dutch Book argument, the authors give brief introductions to the logical tradition using probability over many-valued events and MV-algebras and the statistical tradition using previsions.\newline The first part of the second chapter contains a highly condensed presentation of algebraic tools such as MV-algebra topology and filters on MV-algebras. In the second part a logical characterization of coherence of assessments of upper/lower probability and previsions are given.\newline In Chapter 3 external operators of upper and lower probability are internalized in the algebra yielding a UMV-algebra. These UMV-algebras are then used to characterize coherence.\newline Chapter 4 begins with the definition of UG-algebras, which are obtained from UMV-algebras internalizing real-valued upper previsions. It is then shown that there exists a categorical equivalence between UG-algebras and UMV-algebras. This equivalence is then exploited to prove the existence of PTIME-computable maps \(\circ\) and \(*\) between UG-equations and UMV-equations which preserve semantic consequence and satisfiability.\newline Chapter 5 contains various complexity results concerning UMV-algebras. For instance, it is proved that satisfiability of certain UMV-equations is NP-complete.\newline In Chapter 6 the authors conclude and point out further promising lines of research, such as a logico-algebraic framework with conditioning events taking values in \([0,1]\subset\mathbb R.\)
scientific article; zbMATH DE number 5976304
scientific article