Strict coherence for infinite-valued events

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My paper “Three characterization of strict coherence for infinite-valued events” has been published online on the Review of Symbolic Logic.  If you are interest, please follow this link

IPMU 2018



My contribute paper titled “Logics for strict coherence and Carnap-regular probability functions” has been recently accepted to be presented at the 17th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems – IPMU 2018.  that will take place in Cádiz, Spain.

On the complexity of Strictly Coherent Books

IMG_20170828_111440_480.jpg Given an book on formulas (i.e., a partial map on SL to rational numbers of [0,1]), deciding if it is coherent is an NP-complete problem. The proof essentially uses Carathéodory theorem which characterizes the points of a convex set: given a convex set C=cl-co(X) whose affine dimension is n, then x belongs to C if and only if there is a finite subset Y of X of cardinality at most n+1, such that x is a convex combination of the elements of Y. Indeed a book is coherent if and only if it belongs to cl-co(H) being H the set of logical valuations.

Moving from coherence to strict-coherence essentially boils down, in geometrical terms, to providing a characterization, á la Carathéodory, for the relative interior of cl-co(H). Steinitz theorem gives a (unfortunately useless) direction: if a point x belongs to relint cl-co(H), then there exists a finite subset K of H of cardinality at most 2n such that x belongs to relint cl-co(K). The converse direction (which is key for the decidability of a strictly coherent book) is trivially false: consider a cube C in R3 and pick a point x lying in the relint of a face of C. Then x belongs to the relative interior of a square, but not to the relint of the cube.

In the last weeks I’m just trying to extend (in certain sense) Steinitz theorem to obtain a characterization of the relative interior of a convex set. I don’t know if I will get it, but I like very much the drawings.