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

 

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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.

Layers of zero probability and stable coherence over Łukasiewicz events

Tommaso Flaminio, Lluis Godo

The notion of stable coherence has been recently introduced to characterize coherent assignments to conditional many-valued events by means of hyperreal-valued states. In a nutshell, an assignment, or book, β on a finite set of conditional events is stably coherent if there exists a coherent variant β of β such that β maps all antecedents of conditional events to a strictly positive hyperreal number, and such that β and β differ by an infinitesimal. In this paper, we provide a characterization of stable coherence in terms of layers of zero probability for books on Łukasiewicz logic events.

Keywords: Layers of zero probability, Conditional probability, Stable coherence MV-algebras. 

Volume 21, Issue 1, pp. 113–123.

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.