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.

Equivalences between subcategories of MTL-algebras via Boolean algebras and prelinear semihoops

Stefano Aguzzoli, Tommaso Flaminio, Sara Ugolini

This article studies the class of strongly perfect MTL-algebras, i.e. MTL-algebras having an involutive co-radical, and the variety they generate, namely SBP0. Once these structures will be introduced, we will first establish categorical equivalences for several of their relevant proper subvarieties by employing a generalized notion of triplets whose main components are a Boolean algebra and a prelinear semihoop. When triplets are further expanded by a suitable operation between their semihoop reducts, we define a category of quadruples that are equivalent to the whole category of SBP0-algebras. Finally, we will provide an explicit representation of SBP0-algebras in terms of (weak) Boolean products.

Keywords: Strongly perfect MTL-algebras, Boolean Algebras, Prelinear Semihoops, Categorical Equivalence

Journal of Logic and Computation, exx014, https://doi.org/10.1093/logcom/exx014

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.

ManyVal 2017 – Deadline Extension

ManyVal is a series of international workshops on the logical and algebraic aspects of many-valued reasoning.
The aim of the workshops is to gather both established and young researchers sharing an interest for this topic. Accordingly, each edition has a sharp focus.
If you wish to give a talk should submit an extended abstract (max. 3 pages written in plain LaTeX) by 25 of June 2017 via the Easychair website