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A logical characterization of coherence for imprecise probabilities. (English) Zbl 1244.03082
This dense paper provides a logico-algebraic framework which allows for a logical characterization of coherence for imprecise improbability.
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.
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.
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.
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.
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.
Chapter 5 contains various complexity results concerning UMV-algebras. For instance, it is proved that satisfiability of certain UMV-equations is NP-complete.
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 $$[0,1]\subset\mathbb R.$$

##### MSC:
 03B48 Probability and inductive logic 03B50 Many-valued logic 06D35 MV-algebras 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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