A process for generating quantitative belief functions

We present a structured methodology for transforming qualitative preference relationships among propositions into appropriate numeric representations. This approach will be useful in the difficult process of knowledge acquisition from experts on the degree of belief in various propositions or the probability of the truthfulness of those propositions. The approach implicitly (through the qualitative assignments) and explicitly (through the vague interval pairwise comparisons) provides for different levels of preference relationships. Among its advantages, it permits the expert to: explore the given problem situation, using linguistic quantifiers; avoid the premature use of numeric measures; and identify input data that are inconsistent with the theory of belief functions.     

A decision theory for partially consonant belief functions

Partially consonant belief functions (pcb), studied by Walley, are the only class of Dempster-Shafer belief functions that are consistent with the likelihood principle of statistics. Structurally, the set of foci of a pcb is partitioned into non-overlapping groups and within each group, foci are nested. The pcb class includes both probability function and Zadeh’s possibility function as special cases. This paper studies decision making under uncertainty described by pcb. We prove a representation theorem for preference relation over pcb lotteries to satisfy an axiomatic system that is similar in spirit to von Neumann and Morgenstern’s axioms of the linear utility theory. The closed-form expression of utility of a pcb lottery is a combination of linear utility for probabilistic lottery and two-component (binary) utility for possibilistic lottery. In our model, the uncertainty information, risk attitude and ambiguity attitude are separately represented. A tractable technique to extract ambiguity attitude from a decision maker behavior is also discussed.     

Weak MV-algebras

In a recent paper [CHAJDA, I.—KüHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property. The financial support by the grant of Czech Government MSM 6198959214 is gratefully acknowledged.     

Propagating belief functions in qualitative Markov trees

This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempster's rule.The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables.We show that efficient implementation of Dempster's rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempster's rule can be implemented by propagating belief functions through the tree.     

Ensemble clustering in the belief functions framework

In this paper, belief functions, defined on the lattice of intervals partitions of a set of objects, are investigated as a suitable framework for combining multiple clusterings. We first show how to represent clustering results as masses of evidence allocated to sets of partitions. Then a consensus belief function is obtained using a suitable combination rule. Tools for synthesizing the results are also proposed. The approach is illustrated using synthetic and real data sets.     

Dominance of capacities by k-additive belief functions

In this paper we deal with the set of k-additive belief functions dominating a given capacity. We follow the line introduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general k-additive measures. First, we show that the conditions for the general k-additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid. On the other hand, we show that these conditions cannot be improved. We solve this situation by imposing additional constraints on the dominating functions. Then, we consider the more restrictive case of k-additive belief functions. In this case, a similar result with stronger conditions is proved. Although better, this result is not completely satisfactory and, as before, the conditions cannot be strengthened. However, when the initial capacity is a belief function, we find a subfamily of the set of dominating k-additive belief functions from which it is possible to derive any other dominant k-additive belief function, and such that the conditions are even more restrictive, obtaining the natural extension of the result for probabilities. Finally, we apply these results in the fields of Social Welfare Theory and Decision Under Risk.     

Generalized MV-algebras

We generalize the notion of an MV-algebra in the context of residuated lattices to include non-commutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from lattice-ordered groups via a truncation construction that generalizes the Chang–Mundici Γ functor. This correspondence extends to a categorical equivalence that generalizes the ones established by D. Mundici and A. Dvurečenskij. The decidability of the equational theory of the variety of generalized MV-algebras follows from our analysis.     

State-morphism MV-algebras

We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such (state-morphism) state MV-algebras with the category of unital Abelian ?-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.     

On a homogeneity condition for MV-algebras

In this paper we deal with a homogeneity condition for an MV-algebra concerning a generalized cardinal property. As an application, we consider the homogeneity with respect to α-completeness, where α runs over the class of all infinite cardinals. This work was supported by VEGA grant 1/9056/02.     

Independence of the axiomatic system for MV-algebras

We prove that the well-known axiomatic system of MV-algebras is not independent. The axiom of commutativity can be deleted and the remaining axioms are shown to be independent.     

Implication in MV-algebras

We set up an axiomatic system for the logical connective implication within the framework of MV-algebras which generalizes implication in classical logic described similarly by J. C. Abbott. The induced structure (weak implication algebra) turns to be a join-semilattice whose principal filters are MV-algebras.Received October 2, 2003; accepted in final form March 17, 2004.     

State morphism MV-algebras

We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.     

Cauchy completions of MV-algebras

Abstract. An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras.?We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra. Received August 8, 2001; accepted in final form October 18, 2001.     

Weakly linear quantum MV-algebras

Quantum MV-algebras (QMV-algebras) are a non lattice-theoretic generalization of MV-algebras (multi-valued algebras) and a non-idempotent generalization of orthomodular lattices. In this paper we construct a finite basis for the variety generated by the class of all weakly linear quantum MV-algebras.Dedicated to the memory of Wim BlokReceived October 12, 2000; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Relative subalgebras of MV-algebras

Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\). These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra.     

On implication in MV-algebras

In [1], the authors introduced the notion of a weak implication algebra, which reflects properties of implication in MV-algebras, and demonstrated that the class of weak implication algebras is definitionally equivalent to the class of upper semilattices whose principal filters are compatible MV-algebras. It is easily seen that weak implication algebras are just duals of commutative BCK-algebras. We show here that most results of [1] are, in fact, immediate consequences of two well-known facts: (i) a bounded commutative BCK-algebra possesses a natural upper semilattice structure, (ii) the class of MV-algebras and that of bounded commutative BCK-algebras are definitionally equivalent. Presented by I. Hodkinson. Received November 11, 2005; accepted in final form November 26, 2005.