An Axiomatic Utility Theory for Dempster-Shafer Belief Functions

International audienceThe main goal of this paper is to describe an axiomatic utility theory for Dempster-Shafer belief function lotteries. The axiomatic framework used is analogous to von Neumann-Morgenstern's utility theory for proba-bilistic lotteries as described by Luce and Raiffa. Unlike the probabilistic case, our axiomatic framework leads to interval-valued utilities, and therefore, to a partial (incomplete) preference order on the set of all belief function lotteries. If the belief function reference lotteries we use are Bayesian belief functions, then our representation theorem coincides with Jaffray's representation theorem for his linear utility theory for belief functions. We illustrate our framework using some examples discussed in the literature. Finally, we compare our decision theory with those proposed by Jaffray and Smets

Formalizing restriction categories

Restriction categories are an abstract axiomatic framework by Cockett and Lack for reasoning about (generalizations of the idea of) partiality of functions. In a restriction category, every map defines an endomap on its domain, the corresponding partial identity map. Restriction categories cover a number of examples of different flavors and are sound and complete with respect to the more synthetic and concrete partial map categories. A partial map category is based on a given category (of total maps) and a map in it is a map from a subobject of the domain. In this paper, we report on an Agda formalization of the first chapters of the theory of restriction categories, including the challenging completeness result. We explain the mathematics formalized, comment on the design decisions we made for the formalization, and illustrate them at work

Judging the Rationality of Decisions in the Presence of Vague Alternatives

The standard framework of the decision theory is subjected to partial revision in regard to the usage of the notion of alternative. An approach to judging the rationality of decision-maker's behavior is suggested for various cases of incomplete observability and/or controllability of alternatives. The approach stems from the conventional axiomatic treatment of rationality in the general choice theory and proceeds via modifying the description of alternative modes of behavior into a generalized model that requires no explicit consideration of alternatives. The criteria of rationality in the generalized decision model are proposed. For the conventional model in the choice theory, these criteria can be reduced to the well known criteria of the regularity (binariness) of choice functions. Game and economic examples are considered

Theorem proving support in programming language semantics

We describe several views of the semantics of a simple programming language as formal documents in the calculus of inductive constructions that can be verified by the Coq proof system. Covered aspects are natural semantics, denotational semantics, axiomatic semantics, and abstract interpretation. Descriptions as recursive functions are also provided whenever suitable, thus yielding a a verification condition generator and a static analyser that can be run inside the theorem prover for use in reflective proofs. Extraction of an interpreter from the denotational semantics is also described. All different aspects are formally proved sound with respect to the natural semantics specification.Comment: Propos\'e pour publication dans l'ouvrage \`a la m\'emoire de Gilles Kah

Complete Axioms for Categorical Fixed-Point Operators

We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !-continuous functions between !-complete pointed partial orders. This possesses a least-fixed-point oper..

Axiomatic Testing of Structure Metrics

Axiomatic testing of software metrics is described, based on axioms from representational measurement theory. In a case study, the axioms are given for the formal relational structure and the empirical relational structure. Two approaches to axiomatic testing are elaborated: deterministic testing and probabilistic testin